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Optical Spectrum Analyzer:- About, Types, Operation theory & Selection:
An Optical Spectrum Analyzer (OSA) is an optical test and measurement instrument used to measure and displays the power level of the received optical signal in the vertical axis against the signal wavelength in the horizontal axis over a specified wavelength span.
Optical spectrum analyzers are used for the optical spectral analyses of an optical signals source or networks for the application such as light source characterization, WDM network analysis, optical amplifier assessment, OSNR measurements etc.
Optical spectrum analyzers are basically classified into two categories.
In the first type of OSA, the received signal is decomposed into different frequencies and wavelengths for measuring optical power.
The second type of optical spectrum analyzer is high-resolution OSA which is used to achieve spectral resolution below 1 pm.
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In this RF and Microwave Optical spectrum Analyzer article, we will cover,
What is an Optical SpectrumAnalyzer?
Types and application of Optical SpectrumAnalyzer
Theory of operation of Optical Spectrum Analyzer:
Specifications need to consider in the selection of Optical Spectrum Analyzer:
RF and microwave Optical SpectrumAnalyzer Manufactures.
Theory of operation of Optical Spectrum Analyzer:
An Optical Spectrum Analyser (or OSA) is designed to trace and display the distributed power level in the vertical scale of an optical signal over a specified wavelength span in the horizontal scale.
An OSA is used to study the spectral composition of an optical signal and provide the characteristics of optical signals within the specified wavelength of the DUT.
The principle of operation of the normal optical analyzer is based on the assumption that the properties remain approximately constant during the sweep time for the input continuous-wave laser lights into it.
However, for the pulsed laser signals, the measuring instrument may be completely unusable. To measure high pulse repetition rates of pulsed laser sources, the optical spectrum analyzers with dual-monochromator or double-pass setup are required to rotate the polarization state by 90° between the two stages.
Types of Optical Spectrum Analyzer:
Optical spectrum analyzers are classified on the basis of technology used and form factors. Optical spectrum analyzers are basically three categories on the basis of analysis techniques used. They are diffraction grating, Fabry-Perot interferometer and Michelson interferometer technology-based Optical spectrum analyzers.
Diffraction Grating based Optical Spectrum Analyzers:
In a Diffraction grating-based optical spectrum analyzer the polychromatic light signal is spatially dispersed with a diffraction grating. Then these dispersed signals are sent to the multi-channel photodetector, in the form of a photodiode array. As this technology will be suitable for limited wavelength regions, different gratings are used for each different wavelength range and resolution.
Diffraction grating-based optical spectrum analyzer will have high acquisition speed however often it will not span in the infrared region.
These high-performance optical spectrum analyzers are used as Scanning Instruments by sending the, input light through a tunable band-pass filter, and then to a high-dynamic-range photodetector and the results will be displayed on a digital screen.
Fabry&#;Perot Interferometers based Optical Spectrum Analyzers:
Fabry&#;Perot interferometer based optical spectrum analyzers offer fine wavelength resolution by using a wavelength-tunable band-pass filter. In Fabry&#;Perot interferometer based optical spectrum analyzers, true Fabry&#;Perot with flat mirrors, or a resonator with curved mirrors where mode matching is required to detect and measure the input light signal.
The Fabry&#;Perot based OSA is mainly used for specific high-precision applications, as it has limited free spectral range, it is not suitable as a universal optical spectrum analyzer.
Michelson Interferometers based Optical Spectrum Analyzers:
Michelson interferometer based optical spectrum analyzer worked on the operation principle of Fourier transform spectroscopy. It does not involve any type of tunable band-pass filter, the spatial period of the input signal is used to calculate the optical wavelength and the result can be interpreted by utilizing the interferometric autocorrelation and as the function of Fourier transform, based on FFT algorithm using a microprocessor.
OSA is classified as Benchtop and PXI based rack-mountable units on the base of form factor.
Bench-top Optical spectrum analyzer:
Bench-top Optical spectrum analyzers are generally AC power operated suitable for the field and Lab environment for R&D and production. Bench-top optical spectrum analyzers will have an internal display for monitoring and a control panel for control and configuration settings.
Rack-Mountable PXI based Optical spectrum Analyzer:
Rack-Mountable PXI based Optical spectrum Analyzer are future capability expandable type optical spectrum analyzers worked on the base of PXI based measurement cards. The range of measurements can be extended by adding the corresponding PXI card to the PXI chaise.
Application of Optical Spectrum Analyzers:
Optical spectrum analyzers (OSA) are one of the prime required instruments for measuring optical spectra in the optical communication domain, few of the industrial applications are listed below.
Test and Characterization of light sources such as light-emitting diodes(LEDs), lasers source etc.
Testing of optical systems, components such as wavelength division multiplexing systems, optical splitter etc in communication networks.
Measurements of the wavelength-dependent reflectivity or transmissivity optical devices or systems.
Test and characterization of optical fibre amplifiers in terms of wavelength-dependent gain.
Main specifications need to consider in the selection of an Optical Spectrum Analyzer:
The Important selection figure of an Optical Spectrum Analyzers is its operating frequency, wavelength range and resolution. Even though there are many performance figures that need to consider in the selection of OSA, a few of important are briefly described below.
Operating Frequency (GHz):
The operating frequency of the optical spectrum analyzer is its fundamental frequency range over which it will functions. Need to select as per the test requirement.
Wavelength Range:
Optical spectrum analyzers are always designed to a certain optical wavelengths range due to the wavelength limitation of a diffraction grating or the Photodetector. For different wavelengths, a photo-detector needs to be replaced, to match the requirement. OSA need to select as per the required signal wavelength.
Wavelength Resolution and Filter Shape:
The wavelength resolution of most of the optical spectrum analyzers will be a fixed value in the range of about 0.1 nm and 5 nm. For very High-performance OSA can reach in the order of 0.01 nm (= 10 pm) resolution. Higher the resolution better will be measurements. Inbuilt optical filters help to tune the optical signal for better measurements in their spectral range.
Dynamic Range and Sensitivity:
The dynamic range of the optical spectrum analyzer is based on the internal photodetector sensitivity.
The Sensitivity of the analyzer indicates the minimum amount of power level required for the optical input signal to detect by the photo-detector. In simple terms, it indicates how much weaker signal OSA can measure above the noise level. As the photodetector also has a maximum optical input level it can tolerate without failure, there is an upper limit to the OSA.
The dynamic range of the optical spectrum analyzer indicates the lower and the upper measurement limit of the instrument.
Wavelength Accuracy of Optical Spectrum Analyzer:
Wavelength accuracy is the measurement accuracy of the optical spectrum analyzer. It is also based on the calibration of the instrument. After a long use in a rough environment, the wavelength calibration may be lost and it may lead to consistently wrong measurements. Higher the accuracy better for the measurements and increase the reliability.
Power Accuracy of Optical Spectrum Analyzer:
Power accuracy of an optical spectrum analyzers indicate the accuracy in the measurement of the input signal power level, it is also based on the accuracy of the instrument calibration. For an optical fibre communications network, it is ideal to use high power accuracy optical spectrum analyzer or a dedicated optical power meter for the measurements to determine channel powers.
Acquisition Speed:
The acquisition speed of an optical spectrum analyzer is substantially based on other features like wavelength range, resolution, dynamic range and sensitivity. Higher the acquisition and measurement speed better for the applications.
Conclusion:
An Optical Spectrum Analyser is a precision and high accurate instrument for measuring and displaying the distributed power levels of an optical signal from a source in a specified wavelength span for R&D and manufacturing applications. The application of the optical spectrum analyzer includes consumer electronics, telecommunications, healthcare, R&D in science and medical industry, security, environmental monitoring etc.
In the manufactures section, you can see the top optical spectrum analyzers manufacturers, who offer a wide range like wide-band wavelength, high-resolution OSA for your LAB and field test of the optical source.
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Scientists are increasingly relying on astronomical and remote sensing technologies to gain deeper insights into the Earth and the universe. In these fields, the optical spectrum analyzer (OSA) or spectrometer plays a pivotal role. This Review offers a comprehensive overview of the fundamental principles, key parameters, and applications of various branches of traditional OSAs, including prisms, gratings, interferometers, tunable filters, and reconstructive spectrometers. We specifically focus on their latest major applications in astronomy and remote sensing. Additionally, we present a mathematical model of the generalized reconstructive spectrometer and provide a summary of its principles pertaining to spectral mapping, reconstruction, and imaging. Despite its limited aperture and étendue, the reconstructive spectrometer holds great potential for future use in astronomy and remote sensing due to its compact size and exceptional ultrahigh spectral resolution.
A spectrometer is an instrument used for spectral analysis of optical signals, enabling the determination of the spectral composition of emitted, absorbed, or scattered optical radiation by substances. It allows for the measurement of parameters such as wavelength, intensity, and profile, among others. Within the realm of astronomical and remote sensing instruments, the optical spectrum analyzer (OSA), 1 or spectrometer, holds significant importance. Spectrometers can be classified into several types, including prism, grating, Fourier transform (such as Michelson, Mach&#;Zehnder, and Sagnac interferometers), tunable filter (such as Fabry&#;Pérot, acousto-optic and liquid crystal tunable filters) spectrometers, as well as reconstructive spectrometers 2 (comprising spectral response and spectral-to-spatial mapping types). In Sec. II , we present a systematic introduction to the fundamental principles, classifications, and typical applications of spectrometers in the fields of astronomy and remote sensing. In Sec. III , we construct a generalized reconstructive spectrometer model and introduce the principles behind spectral mapping, reconstruction, and imaging. Section IV delves into the important parameters of spectrometers, including spectral range, bandwidth, spectral resolution or spectral resolving power (SRP), étendue, and calibration from a theoretical standpoint. In Sec. V , we explore the diverse applications of spectrometers. Finally, in Sec. VI , we provide concluding remarks and outline future prospects.
The BASS system incorporates an aplanatic optical design with transmissive Féry prisms made of NaCl and CaF 2 . The CaF 2 prism has radii of curvature of 84.923 and 59.621 mm, a thickness of 30 mm, and an included angle of 27.29°. The spectral range covered by the BASS is 2.9&#;6.5 and 6.5&#;13.5 µm, respectively. This system is employed to study the infrared spectroscopy of celestial bodies such as α Orionis, planetary nebulae and comets. In contrast, the NIHTS utilizes a ZnS transmissive Féry prism with a maximum thickness of 44 mm and a 96 in. single slit. It covers a spectral range of 0.86&#;2.45 µm, and its optical throughput is &#;40%. The NIHTS is designed for observing time-varying celestial bodies, such as the Kuiper Belt, asteroids, and brown dwarfs. The FPIS employs a BaF 2 transmissive Féry prism with radii of curvature of 58.71 and 84.16 mm, respectively, and a thickness of 27.66 mm. It covers a spectral range of &#; cm &#;1 with a spectral resolution of 0.5&#;5.5 cm &#;1 . The FPIS utilizes a high-brightness synchrotron radiation infrared light source, which significantly increases the light energy. This allows for the use of a micrometer-level entrance aperture to achieve high spectral resolution and luminous flux. The integration time is 1&#;4 µs, and the time resolution is 30 µs. The FPIS is particularly suitable for studying the dynamic processes in biological macromolecules, such as the conformational changes of retinal chromophore proteins.
The catadioptric Féry prism is known to suffer from significant image astigmatism, and the curved spectral plane can pose challenges for planar array detectors. Consequently, the transmissive Féry prism is often preferred in various applications, including the BASS 16,17 (Broadband Array Spectrometer System) shown in Fig. 2(b) , the NIHTS 18&#;20 (Near-Infrared High-Throughput Spectrometer) installed on the 4.3 m Discovery Channel Telescope LDT depicted in Fig. 2(c) , and the FPIS 21,22 (Féry prism Infrared Spectrometer) illustrated in Fig. 2(d) .
The TSIS-1-SIM is designed to continuously monitor solar spectral irradiance (SSI) over the spectral range of 200&#; nm. It has a spectral resolution of 2 nm (<280 nm), 5 nm (280&#;400 nm), and 45 nm (>400 nm). In recent years, the TSIS-1-HSRS has been developed using a modified spectral ratio method to normalize very high spectral resolution solar line data to the absolute irradiance scale of the TSIS-1-SIM and the CSIM. The TSIS-1-HSRS spans 202&#; nm, covering an integrated energy that exceeds 97% of the total solar irradiance. Its accuracy ranges from 0.3% to 1.3%, and its spectral resolution ranges from 0.001&#;0.01 nm.
The SORCE-SIM instrument is designed to measure x rays, ultraviolet rays, visible light, and near-infrared radiation emitted by the sun that reaches the Earth. It enables the study of long-term climate change, natural fluctuations, atmospheric ozone, and UV-B radiation. The instrument covers a spectral range of 200&#; nm with a spectral resolution ranging from 0.25 to 33 nm. To achieve this, the SORCE-SIM employs a fused quartz prism, with radii of curvature of 421.48 and 441.27 mm, respectively. The prism has a central thickness of 12.30 mm and an included angle of 34.35°.
The SPS utilizes a fused quartz prism, with radii of curvature of 423 and 440 mm, a thickness of 12.08 mm, and an included angle of 33.44°. It is equipped with a scanning four-exit-slit structure, consisting of one UV, two VNIR, and an IR channel. These channels cover spectral ranges of 250&#;400 nm, 400&#; nm, and &#; nm, respectively. The full-width half-maximums (FWHMs) for these channels range from 0.7 to 3.5 nm, 3.5&#;35.0 nm, and 28.5&#;41.2 nm, respectively. It is worth noting that the FWHM varies with wavelength according to Eqs. (1) and (2) . The SPS is used in the study of solar physics, weather forecasting, and climate change.
Féry prism dispersion spectrometers. (a) The SPS, 11 Reproduced with permission from Zhang et al., Acta Opt. Sin 33 , (). Copyright Chinese Laser Press. (b) The BASS, 17 Reproduced with permission from Warren, Hackwell, and Gutierrez, Opt. Eng. 36 (4), &#; (). Copyright SPIE. (c) The NIHTS, 19 Reproduced with permission from Gustafsson et al., Publ. Astron. Soc. Pac. 133 , (). Copyright The Astronomical Society of the Pacific. (d) The FPIS, 21 Reproduced with permission from Ritter et al., J. Phys. Chem. Lett. 10 , &#; (). Copyright American Chemical Society.
Féry prism dispersion spectrometers. (a) The SPS, 11 Reproduced with permission from Zhang et al., Acta Opt. Sin 33 , (). Copyright Chinese Laser Press. (b) The BASS, 17 Reproduced with permission from Warren, Hackwell, and Gutierrez, Opt. Eng. 36 (4), &#; (). Copyright SPIE. (c) The NIHTS, 19 Reproduced with permission from Gustafsson et al., Publ. Astron. Soc. Pac. 133 , (). Copyright The Astronomical Society of the Pacific. (d) The FPIS, 21 Reproduced with permission from Ritter et al., J. Phys. Chem. Lett. 10 , &#; (). Copyright American Chemical Society.
In a catadioptric Féry prism, the rear surface typically features a highly reflective coating, while the transmissive Féry prism uses a mirror positioned behind it to redirect the light path, resulting in two reflections of the light passing through the prism. If the curvature of the two curved surfaces of the prism is relatively large, the prism can be approximated as a triangular prism. Consequently, the angular dispersion of a Féry prism at its minimum declination can be estimated using the following approximation:When the apex angle ɛ of the Féry prism is small, its angular dispersion can be larger than that of a triangular prism with the same refractive index n and material dispersion dn/dλ. However, the spectral resolution of a Féry prism can be similar to or larger than that of a triangular prism, depending on the prism&#;s size and the curvature of its two surfaces.
The Féry prism 10 is characterized by its curved surfaces and is defined by parameters such as the curvature radii of its front and rear light transmission surfaces, its thickness (which represents the light transmission distance inside the prism), and the apex angle (which is the angle between the two tangents at the intersection of the light and the prism&#;s curved surfaces). Due to its design, the Féry prism exhibits several advantageous properties, including collimation, aberration correction, and imaging. There are two main types of Féry prisms: catadioptric and transmissive.
The APEX and PRISMA spectrometers are employed in remote sensing applications for soil, water, and atmosphere. The APEX instrument consists of VNIR (Visible-Near-Infrared) and SWIR (Short-Wave-Infrared) channels, realized by CaF 2 /ZnS and CaF 2 /sapphire prisms, respectively. The spectral range for VNIR is 380.5&#;971.7 nm (334 bands), while for SWIR, it is 941.2&#;.5 nm (198 bands). On the other hand, the PRISMA instrument uses three prisms to achieve VNIR and SWIR channels with a spectral range of 400&#; nm (66 bands) and 920&#; nm (174 bands), respectively. The CSIM spectrometer is utilized for long-term measurements of solar spectral irradiance (SSI) to understand how solar variability affects Earth&#;s climate. The CSIM system incorporates a fused silica triangular prism that disperses the incoming light. By rotating the prism, the dispersed light is detected by Si (200&#;950 nm), InGaAs (900&#; nm), and extended InGaAs (&#; nm) detectors, covering a spectral range of 200&#; nm. This range accounts for over 97% of the total solar irradiance.
The dispersive spectrometer operates on the principle of obtaining the spatial distribution function of wavelength by imaging the Fraunhofer diffraction pattern of the entrance slit onto the focal plane. Among the earliest optical devices used in dispersive spectrometers is the prism. Prisms offer several advantages, including high light transmittance, ease of processing and adjustment, excellent long-term stability, absence of spectral order overlap, and a large free spectral range (FSR). Two commonly used types of prisms are the triangular prism and the Féry prism. The angular dispersion of a triangular prism, denoted as dθ/dλ, can be expressed by the following equation:Here, &#; represents the apex angle of the prism, and n denotes the refractive index. The material dispersion of the prism can be approximated by dn/dλ &#; &#;2b/λ&#; 4c/λ, where b and c are constants related to the material used. It is evident that dn/dλ is inversely proportional to the wavelength λ of the incident light. The theoretical maximum spectral resolving power (SRP) at the minimum deviation can be calculated using the following equation:Here, g represents the size of the bottom edge of the prism, and δλ represents the spectral resolution. The triangular prism continues to be an essential component for spectral analysis in astronomy and remote sensing applications. It has been utilized in various projects, such as the Airborne Prism Experiment (APEX) 3,4 conducted by the European Space Agency shown in Fig. 1 , the PRecursore IperSpettrale della Missione Applicativa (PRISMA) hyperspectral solar synchronous orbit satellite developed by the Italian Space Agency, 5&#;7 and the Compact Spectral Irradiance Monitor (CSIM) 6U CubeSat as part of the SpaceX SSO-A: SmallSat Express Mission. 8,9
The AES spectrometer employs a convex grating with a spectrally partitioned structure, as depicted in Fig. 7(e) . The radius of curvature of the grating is given by R 2 = Δ l / m g Δ λ &#;, where Δl, m, g, and Δλ represent the width of the spectral plane, the diffraction order, the groove density of the grating, and the spectral range, respectively. The central and outer regions of the grating correspond to the VNIR and SWIR channels, respectively, with groove densities of 118 and 40 lines/mm. The blazed angles for these regions are 1.25°/3.5° and 1.9°, respectively. The spectral range of the AES spans from 0.4 to 1 µm in the VNIR channel and from 1 to 2.7 µm in the SWIR channel. The spectral resolution is 3.63 and 9.64 nm for the VNIR and SWIR channels, respectively. The AES is primarily used for studying the evolution of the solar system by detecting asteroid bodies.
The AHSI spectrometer employs an improved Offner structure that incorporates a meniscus lens positioned at the location of the convex grating. This lens helps to correct for spectral bending, additional astigmatism, and field curvature caused by the ultra-long slit (60 mm). The dispersion width of the AHSI reaches 30 mm, while its swath width is 60 km and its spatial resolution is 30 m. The instrument&#;s spectral range spans from 400 to nm, with 330 spectral channels. The spectral resolution is higher than 5 nm in the VNIR channel and 10 nm in the SWIR channel. The AHSI is renowned for its ability to detect and identify ground objects, making it well-suited for precision applications, such as ecological environment monitoring, land and resource surveying, and oil/gas exploration.
The VIMS-V spectrometer utilizes a convex grating that is holographically recorded in a Rowland circle configuration, as shown in Fig. 7(b) . The grating features a rectangular laminar groove profile, with 67.5% and 32.5% of the surface area covered by grooves with depths of 300 and 440 nm, respectively. The spectral range of the VIMS-V spans from 300 to nm. This instrument is employed to create spatial maps of the distribution of minerals and chemicals on the surfaces of Saturn, its rings, and its moons.
Convex grating dispersion spectrometers. (a) The VIMS-V 43 and its convex grating (b), Reproduced with permission from Reininger et al., Proc. SPIE , 239&#;250 (). Copyright SPIE. (c) The AHSI, 46 Reproduced with permission from Liu et al., J. Remote Sens. 24 (4), 333&#;334 (). Copyright Chinese Academy of Sciences. (d) The AES 47 and its convex grating (e), Reproduced with permission from Yang et al., Acta Opt. Sin. 40 , (). Copyright Chinese Laser Press.
Convex grating dispersion spectrometers. (a) The VIMS-V 43 and its convex grating (b), Reproduced with permission from Reininger et al., Proc. SPIE , 239&#;250 (). Copyright SPIE. (c) The AHSI, 46 Reproduced with permission from Liu et al., J. Remote Sens. 24 (4), 333&#;334 (). Copyright Chinese Academy of Sciences. (d) The AES 47 and its convex grating (e), Reproduced with permission from Yang et al., Acta Opt. Sin. 40 , (). Copyright Chinese Laser Press.
The convex grating 40,41 is well-known for its strong spectral bending and its ability to correct optical aberrations, such as astigmatism and coma. These characteristics make it a popular choice for spectrometer applications. The Offner structure 42 is a commonly used design in convex grating spectrometers, which find extensive applications in astronomy and remote sensing. Examples of these applications include the Visual and Infrared Mapping Spectrometer (VIMS-V) 43&#;45 on the Cassini Saturn probe depicted in Fig. 7(a) , the Advanced Hyperspectral Imager (AHSI) 46 on China's GF-5 satellite shown in Fig. 7(c) , and the Asteroid Exploration Spectrometer (AES) 47 illustrated in Fig. 7(d) .
The flat-field holographic concave grating (FHCG) possesses the capability to both diffract and image incident light onto a plane, making it a valuable tool for detecting array CCDs. One notable application of the FHCG is in the extreme ultraviolet spectrometer (EUVS), 38,39 as depicted in Fig. 6 . The EUVS utilizes a gold-plated concave variable line spacing (VLS) grating with a groove density of lines/mm, an area of 46 × 26 mm 2 , a curvature radius of 15 920 mm, a blaze angle of 1.9°, and a blaze wavelength of 30 nm. With the FHCG, the EUVS is capable of covering a spectral range of 10&#;130 Å, enabling the study of high-temperature fusion plasma radiation, including elements such as H, B, C, O, Fe, and W.
The FOCAS incorporates a grism structure consisting of a VHG measuring 110 × 106 mm 2 , which is sandwiched between two right-angle prisms. The spectral range covered by FOCAS is 200&#; nm. It is used to investigate the formation, evolution, and large-scale structure of high-redshift quasars and galaxies. On the other hand, the DESI employs three VHGs to realize three spectral channels. Each VHG is sandwiched between two fused quartz plates, as depicted in Fig. 5(b) . The three gratings have groove densities of .5 ± 21, .4 ± 21, and 992.5 ± 21 lines/mm, respectively. The spectral ranges covered by the three channels are 360&#;593, 566&#;772, and 747&#;980 nm, respectively. DESI is used to observe dark energy in the large-scale structure of the universe.
The volume holographic grating (VHG) is produced using high-precision holographic photography technology, 33 resulting in a grating with high density, a large production area, high dispersion, high SRP, and low stray light. However, according to the standard grating equation (sin θ = λm/d), a holographic grating with a higher diffraction order (m) and a smaller grating pitch (d) will have a larger m/d or diffraction angle (θ) than an echelle grating, which can lead to high-order dispersion interference. VHGs find applications in systems such as the Faint Object Camera and Spectrometer (FOCAS) 34,35 on the 8.2 m Subaru Optical Telescope, as shown in Fig. 5(a) , the Dark Energy Spectroscopic Instrument (DESI) 36 for the Stage IV ground-based dark energy experiment on the Mayall 4 m Telescope, and the Raman Spectrometer for MMX (RAX), which is part of the JAXA Martian Moons eXploration (MMX) mission scheduled for launch in . 37
The ELT-HIRES incorporates four R4 echelle gratings with groove densities of 67.00, 34.50, 16.00, and 11.95 lines/mm. These gratings operate at diffraction orders of 72&#;87, 68&#;140, 68&#;146, and 72&#;86, respectively, enabling the U, BVRI, ZYJH, and K modules. Each of the U, B, V, R, I, Z, Y, J, H, and K arms has a specific spectral range: 330&#;405, 400&#;467, 483&#;555, 549&#;674, 666&#;833, 827&#;955, 950&#;, &#;, &#;, and &#; nm, respectively. Additionally, each arm utilizes a transmission grating as a cross-disperser with groove densities of , , , , , , 930, 730, 465, and 360 lines/mm, respectively. The ELT-HIRES possesses SRPs for different modes: 100 000 for HR-modes, 150 000 for UHR-modes, and 20 000 for MR-modes. The primary scientific objectives of the ELT-HIRES are the detection of bio-signatures in Earth-like exoplanets and the direct observation of cosmic expansion re-acceleration.
The G-CLEF instrument incorporates three echelle gratings, each measuring 300 × mm 2 , with a groove density of 31.6 lines/mm. Cross-dispersion is a process that disperses light in a direction perpendicular to the main dispersion axis, spatially separating the light or separating it based on another property orthogonal to the wavelength axis. In the G-CLEF system, to achieve cross-dispersion and generate a two-dimensional continuous spectrum, two volume holographic gratings with groove densities of 910 and 370 lines/mm are employed. The G-CLEF covers a spectral range of 350&#;540 and 540&#; nm, enabling observations of low-mass exoplanets, stellar abundances, and high-redshift phenomena. The ACS-MIR utilizes an echelle grating with a groove density of 3.03 lines/mm and a blazed angle of 63.43°. The diffraction orders used range from 142 to 258. Cross-dispersion is accomplished using two ordinary blazed gratings of 361 and 180 lines/mm. The ACS-MIR covers a spectral range of 2.3&#;4.3 µm and is designed to detect trace gases, such as C 2 H 6 , HCl, H 2 S, N 2 O, CH 4 , and 13 CH 4 , with a sensitivity of less than 1 ppm. Its goal is to explore the planetary chemistry and search for life-related information on Mars. The CryoNIRSP incorporates an R2 echelle grating with a ruled width of 408 mm and a diffraction order range of 12&#;107. It covers a spectral range of 530&#; nm and is utilized to study the solar magnetic field and its thermodynamic properties.
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The echelle grating 25 is a type of Littrow grating with a larger groove spacing compared to the wavelength. It features a larger blaze angle, greater angular dispersion, and the ability to utilize higher diffraction orders, allowing for higher spectral resolution compared to ordinary blazed gratings. However, it should be noted that echelle gratings typically have a smaller FSR compared to ordinary blazed gratings. Echelle gratings have found various applications in astronomy and remote sensing. Notable examples include the GMT-Consortium Large Earth Finder (G-CLEF) 26 on the Giant Magellan Telescope (GMT), as depicted in Fig. 4(a) ; the mid-infrared channel of the ExoMars Mars probe Atmospheric Chemistry Suite (ACS); 27 the Cryogenic Near-Infrared-Spectro-Polarimeter (CryoNIRSP) 28&#;30 on the Daniel K. Inouye Solar Telescope (DKIST) currently been constructed at the Haleakala Observatory, as shown in Fig. 4(b) ; and the High-Resolution Spectrograph (HIRES) in the European Extremely Large Telescope (ELT). 31,32
An example of a significant application of blazed gratings is the Reflection Grating Spectrometer (RGS), 23,24 which is installed on the XMM-Newton satellite by the European Space Agency, as depicted in Fig. 3 . The RGS employs 182 reflection blazed gratings, with a groove density of 645.6 lines/mm. It has a first-order blaze wavelength of 1.5 nm and a blaze angle of 0.°. The RGS achieves a high SRP ranging from 100 to 500 across a wide spectral range of 0.6&#;3.8 nm in the extreme ultraviolet band. Its primary purpose is to observe galaxy clusters and detect K-shell spectral transitions of elements such as C, N, O, Ne, Me, and Si, as well as L-shell spectral transitions of Fe.
Gratings operate based on the principles of diffraction and multi-beam interference. They can be categorized into various types, including plane blazed, echelle, volume holographic, flat-field concave holographic, and convex gratings. The theoretical SRP (R) and the FSR (Δλ) of a grating can be expressed as follows:Here, λ represents the wavelength of the incident light, δλ represents the spectral resolution, m is the diffraction order of the grating, and N denotes the total number of grooves. The value of Δλ is inversely proportional to m. For ordinary blazed grating, the typical working range is m = 1, 2, which imposes significant limitations on the SRP of the grating.
Furthermore, the Sagnac Fourier Spectrometer (SAFOS) 83 incorporates a transmission grating with a groove density of lines/mm into the Sagnac optical path, as illustrated in Fig. 13(d) . This configuration forms a spatial heterodyne spectrometer (SHS). The diffraction beam of the grating at a specific diffraction order β generates a small angle divergent beam, with its optical axis positioned at the center of the diffraction wavelength λ 0 . The angular dispersion can be expressed as Δβ/Δλ = g/cos β, where g represents the groove density. By scanning a wide range of input angles α, it obtains a series of interferograms with an angle accuracy better than 0.01°. The SRP can reach 40 000 in the spectral range of 230&#;530 nm.
The Sagnac interferometric spectrometer (SIS) with a compact half-pentaprism structure is particularly well-suited for aerospace and harsh mechanical environments. An example of this is the China&#;s Chang&#;e&#;1 Lunar Exploration Sagnac Fourier Transform Spectrometer (CE1-SFTS), 79&#;81 presented in Fig. 13(b) . The CE1-SFTS has a maximum OPD of 15.37 µm, corresponding to a spectral resolution of 325.25 cm &#;1 in the spectral range of 0.48&#;0.96 µm, utilizing 32 channels. This spectrometer is employed for studying the chemical composition and mineral distribution of the lunar surface. Similarly, the Ultraviolet Sagnac Fourier Transform Spectrometer (UV-SFTS), 82 which shares a similar structure with the CE1-SFTS, achieves a maximum OPD of 64 µm. This corresponds to a spectral resolution of 78.125 cm &#;1 in the UV band ranging from 250 to 400 nm. The UV-SFTS is particularly significant due to the numerous challenges involved in developing an ultraviolet imaging spectrometer, including low incident power, large chromatic aberration, and relatively low quantum efficiency of imaging sensors in the UV band.
Sagnac interferometric spectrometers. (a) The SIFTS, 78 Reproduced with permission from Cho, Lee, and Jang, &#;Improvement of spectral resolution by signal padding method in the spatially modulated Fourier transform spectrometer based on a Sagnac interferometer,&#; Appl. Opt. 58 (25), (). Copyright Optical Society of America. (b) The CE1-SFTS, 81 Reproduced with permission from Zhao et al., Acta Photonica Sin. 38 , 474&#;478 (). Copyright Acta Photonica Sinic. (c) The UV-SFTS, 82 Reproduced with permission from Yang et al., &#;Large-aperture UV (250&#;400 nm) imaging spectrometer based on a solid Sagnac interferometer,&#; Opt. Express 26 (26), &#; (). Copyright Optical Society of America. (d) The SAFOS, 83 Reproduced with permission from M. Lenzner and J.-C. Diels, Opt. Express 25 , A447&#;A453 (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license.
Sagnac interferometric spectrometers. (a) The SIFTS, 78 Reproduced with permission from Cho, Lee, and Jang, &#;Improvement of spectral resolution by signal padding method in the spatially modulated Fourier transform spectrometer based on a Sagnac interferometer,&#; Appl. Opt. 58 (25), (). Copyright Optical Society of America. (b) The CE1-SFTS, 81 Reproduced with permission from Zhao et al., Acta Photonica Sin. 38 , 474&#;478 (). Copyright Acta Photonica Sinic. (c) The UV-SFTS, 82 Reproduced with permission from Yang et al., &#;Large-aperture UV (250&#;400 nm) imaging spectrometer based on a solid Sagnac interferometer,&#; Opt. Express 26 (26), &#; (). Copyright Optical Society of America. (d) The SAFOS, 83 Reproduced with permission from M. Lenzner and J.-C. Diels, Opt. Express 25 , A447&#;A453 (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license.
The Sagnac interferometer is a transverse shearing interferometer characterized by its triangular common optical path structure. An example of this is the Sagnac Interferometer Fourier Transform Spectrometer (SIFTS), 78 as depicted in Fig. 13(a) . It splits the incoming light into two beams that are spatially misaligned. The separation distance, denoted as l, is related to the displacement, a, of mirror 2, with l = 2 a &#;. Consequently, the two beams overlap and interfere on the detector, forming an interference pattern with an increasing OPD from the center to both sides. With a maximum OPD of 8 mm, the SIFTS achieves a spectral resolution of 238 cm &#;1 at a wavenumber of cm &#;1 . Signal padding can extend the spectral range up to 27 cm &#;1 .
The Multi-Aperture Fourier Transform Plane Waveguide Spectrometer (Ma-PWS), 77 as depicted in Fig. 12(d) , consists of 200 waveguide-based MZIs with OPDs of different integer multiples of 2π. The wavenumber spectral resolution can be expressed as follows:where nrepresents the effective refractive index of the waveguide and ΔLis the maximum interference delay in the array of MZIs, measuring 21 mm. This results in a spectral resolution of 0.025 nm in the spectral range of .5&#;.5 nm. Compared to a single MZI, the Ma-PWS increases the optical throughput by a factor of 200.
The spectral resolution of the MZI can be significantly enhanced by incorporating gratings, as demonstrated in the Mach&#;Zehnder spatial heterodyne spectrometer (MZ-SHS) 76 illustrated in Fig. 12(c) . The MZ-SHS replaces one of the mirrors in the MZI with a diffraction grating having a width (W) of 28 mm, a groove density of lines/mm, and a working diffraction order (m) of 1. The maximum OPD is determined by OPD= W sin 2α/2, and the SRP is calculated asHere, α represents the grating inclination angle. The MZ-SHS operates in the spectral range of 572.55&#;590 nm, achieving a maximum OPD of &#;10 mm, which corresponds to a spectral resolution of 0.017 nm or 0. cm
Mach&#;Zehnder interferometric spectrometers. (a) The MZFTS, 73 Reproduced with permission from Naylor et al., Proc. SPIE , 540&#;551 (). Copyright SPIE. (b) The OCFTS, 75 Reproduced with permission from Li et al., Opt. Commun. 460 , (). Copyright Elsevier B.V. (c) The MZ-SHS, 76 Reproduced with permission from Cai, Xiangli, and Du, Opt. Commun. 355 , 239&#;245 (). Copyright Elsevier B.V. (d) The Ma-PWS, 77 Reproduced with permission from Florjańczyk et al., Opt. Express 15 , &#; (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license.
Mach&#;Zehnder interferometric spectrometers. (a) The MZFTS, 73 Reproduced with permission from Naylor et al., Proc. SPIE , 540&#;551 (). Copyright SPIE. (b) The OCFTS, 75 Reproduced with permission from Li et al., Opt. Commun. 460 , (). Copyright Elsevier B.V. (c) The MZ-SHS, 76 Reproduced with permission from Cai, Xiangli, and Du, Opt. Commun. 355 , 239&#;245 (). Copyright Elsevier B.V. (d) The Ma-PWS, 77 Reproduced with permission from Florjańczyk et al., Opt. Express 15 , &#; (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license.
The Michelson interferometer suffers from a loss of half the light throughput and high incoherent intensity. In contrast, the Mach&#;Zehnder interferometer (MZI) 70 exhibits a lower incoherent intensity (A) and utilizes nearly all of the light throughput, resulting in a significantly enhanced signal-to-noise ratio (SNR). This characteristic makes it particularly advantageous for satellite remote sensing applications. An example is the Mach&#;Zehnder Fourier Transform Spectrometer (MZFTS) 73,74 installed on the James Clark Maxwell Telescope (JCMT), as depicted in Fig. 12(a) . The MZFTS utilizes two beam splitters to enable optical double input and double output. It operates in the submillimeter spectral range at wavelengths of 350, 450, 750, and 850 µm. With a maximum OPD of 1.2 m, it achieves a spectral resolution of 0.005 cm &#;1 or 150 MHz. The MZFTS is employed for studying planetary atmospheres, molecular clouds, and extragalactic sources. Another example is the On-Chip Fourier transform Spectrometer (OCFTS) 75 depicted in Fig. 12(b) , which utilizes TE-mode rectangular silicon-on-insulator (SOI) waveguides and employs digital microfluidic technology. By gradually introducing silicone oil into a 3 m hollow sensing arm to scan the OPD, it achieves a maximum OPD of 0.57 mm and a spectral resolution of 3.1 nm in the spectral range of 1.5&#;1.7 µm.
The Mach&#;Zehnder interferometer 72 can also serve as a time-modulated Fourier transform spectrometer. When considering a broadband light source, the differential output signal can be expressed as, whereand. The spectrum of the light source,, can be reconstructed using the Fourier transform as follows:
The BLC-FTS utilizes a liquid crystal to generate electronically controlled birefringence. The spectral resolution (δλ) and the SRP (R) can be expressed as follows:In these equations, Δnrepresents the maximum birefringence, and d denotes the thickness of the liquid crystal. For a liquid crystal with a thickness of 100 µm, the maximum OPD is 23.4 µm, which corresponds to a spectral resolution of 8.6 nm. When a liquid crystal with a thickness of 125 µm is placed in the F&#;P cavity, the maximum OPD increases to 210 µm, resulting in a spectral resolution of 1 nm. 71
In the WP-FTS, a Wollaston Prism is utilized to introduce birefringence. The two polarized light beams overlap on the detector array, and the OPD linearly changes with the deflection distance y on the array, similar to Eq. (10) . The OPD can be mathematically expressed asHere, nand nrepresent the refractive indices of the birefringent e-beam and o-beam, respectively, while M denotes the magnification. The maximum OPD is 19 µm, which corresponds to a spectral resolution of 526 cmat the center wavenumber of cm. With signal padding, a spectral resolution of 27 cmcan be achieved. The TD-ISP utilizes a Wollaston prism and a ferroelectric liquid crystal polarization state analyzer to simultaneously acquire both the spectrum and polarization information. 70 The maximum OPD in this instrument is 30 µm, corresponding to a spectral resolution of 341.9 cmin the spectral range of 400&#; nm.
In the SM-FTIR spectrometer, the scanning mirror of the Michelson interferometer is replaced with a Fourier lens. It generates a two-dimensional interferogram on a microbolometer array, and the OPD can be expressed asHere, f represents the focal length of the Fourier lens, while s denotes a constant associated with the refractive index and thickness of the beam splitter, as well as the incident angle (α) of the beam. The OPD varies linearly with the y coordinate at thepoint. The maximum value of OPD is 0.81 mm, which corresponds to a spectral resolution of 12 cmwithin the spectral range of 800 to cm
Equivalent Michelson interferometric spectrometers with spatial modulation. (a) The SM-FTIR, 64 Reproduced with permission from Köhler et al., OSA Continuum 2 , &#; (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license. (b) The WP-FTS, 66 Reproduced with permission from Cho, Lee, and Jang, Opt. Eng. 60 , (). Copyright SPIE. (c) The BLC-FTS, 69 Reproduced with permission from Xia et al., Opt. Eng. 53 , (). Copyright SPIE.
Equivalent Michelson interferometric spectrometers with spatial modulation. (a) The SM-FTIR, 64 Reproduced with permission from Köhler et al., OSA Continuum 2 , &#; (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license. (b) The WP-FTS, 66 Reproduced with permission from Cho, Lee, and Jang, Opt. Eng. 60 , (). Copyright SPIE. (c) The BLC-FTS, 69 Reproduced with permission from Xia et al., Opt. Eng. 53 , (). Copyright SPIE.
The spatially modulated Michelson interferometer can be implemented using various components, such as a single mirror, a Wollaston prism, or a liquid crystal. In this configuration, the incident light is divided into two spatially separated beams that overlap and interfere on the detector. The OPD in this setup varies linearly with the deflection distance of the beams. Notable examples of spatially modulated Michelson interferometers include the Single-Mirror Fourier Transform Infrared Spectrometer (SM-FTIR) 64,65 shown in Fig. 11(a) , the Wollaston Prism Fourier Transform Spectrometer (WP-FTS) 66,67 shown in Fig. 11(b) , the Time Division Fourier Transform Imaging Spectropolarimeter (TD-ISP), 68 and the Birefringent Liquid Crystal Fourier Transform Spectrometer (BLC-FTS) 69 shown in Fig. 11(c) .
The SM-FTS replaces the two mirrors of the Michelson interferometer with two stepped mirrors. One mirror has 19 tall steps, while the other has 24 small steps. This configuration allows the SM-FTS to generate 456 OPD samples in space and achieve a maximum OPD of 6.3 cm. The resulting SRP is &#;65 000 at cm &#;1 , which is comparable to a time-modulated spectrometer. The SM-FTS is primarily used for detecting greenhouse gases in low-Earth orbit, &#;700 km above the Earth&#;s surface. Similarly, the UHRIS utilizes a static Michelson interferometer with two stepped mirrors. One mirror has 25 tall steps, and the other has 25 small steps. This setup enables the UHRIS to generate 625 OPD samples in space, with a maximum OPD of 10 cm. By combining with a scanning Fabry&#;Pérot interferometer (FPI) with specific parameters, such as a cavity length of 2.5 cm, a scanning step of 25 nm, and 50 scanning steps, the SRP can reach 1 000 000 at the center wavenumber of cm &#;1 . The CaHyD instrument is installed on a moving aircraft, such as a helicopter flying at an altitude of m and a velocity of 25 m/s. It achieves OPD scanning by continuous acquiring ground scene images through three sine filters with different periods, resulting in three distinct OPD signals. The CaHyD instrument achieves an OPD range of &#;20 to 40 µm and a spectral resolution of 250 cm &#;1 . Its purpose is to detect the vertical atmospheric spectrum along a 2 km transmission path.
Standard Michelson interferometric spectrometers with spatial modulation. (a) the SM-FTS, 61 Reproduced with permission from Lacan et al., Opt. Express 18 , &#; (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license. (b) The UHRIS, 62 Reproduced with permission from Q. Yang, Opt. Express 27 , &#; (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license. (c) The CaHyD, 63 Reproduced with permission from Ferrec et al., &#;Experimental results from an airborne static Fourier transform imaging spectrometer,&#; Appl. Opt. 50 (30), &#; (). Copyright Optical Society of America.
Standard Michelson interferometric spectrometers with spatial modulation. (a) the SM-FTS, 61 Reproduced with permission from Lacan et al., Opt. Express 18 , &#; (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license. (b) The UHRIS, 62 Reproduced with permission from Q. Yang, Opt. Express 27 , &#; (). Copyright Optical Society of America; licensed under a Creative Commons Attribution (CC BY) license. (c) The CaHyD, 63 Reproduced with permission from Ferrec et al., &#;Experimental results from an airborne static Fourier transform imaging spectrometer,&#; Appl. Opt. 50 (30), &#; (). Copyright Optical Society of America.
The FTS based on a time-modulated Michelson interferometer can achieve a large OPD and high SRP by scanning the moving mirror. However, this scanning mechanism introduces complexity, instability, and maintenance requirements to the system. In contrast, a spatially modulated FTS eliminates the requirement of a scanning component, leading to improved reliability and ease of maintenance. Notable examples of spatially modulated FTS include the Stepped Mirror Fourier Transform Spectrometer (SM-FTS) 61 shown in Fig. 10(a) , the Ultra-High-Resolution Interferometric Spectrometer (UHRIS) 62 shown in Fig. 10(b) , and the Caméra Hyperspectrale de Démonstration (CaHyD) 63 shown in Fig. 10(c) .
The IFTSUV instrument replaces the beam splitter of the Michelson interferometer with two reflection diffraction gratings, each having a density of lines/mm. This modification enables the instrument to achieve a maximum OPD of 1.784 253 mm and over a spectral range of 80&#;145 nm. It achieves a high SRP of 24 000 and is primarily utilized for studying the solar dynamic evolution and characteristics by observing the Ly-α 121.567 nm line. On the other hand, the CUSRIS instrument replaces one of the mirrors in the Michelson interferometer with a Littrow structured grating consisting of 106 lines/mm. When combined with a Fabry&#;Pérot cavity, it achieves a maximum OPD of 12.5 mm and an ultra-high SRP of 1 000 000, equivalent to a standard Michelson interferometer with a maximum OPD of 100 cm.
The SRP of the FTS can be significantly enhanced through the incorporation of gratings, as demonstrated by the Imaging Fourier Transform Spectrometer working in the far UV (IFTSUV) 59 depicted in Fig. 9(a) and the Compact Ultrahigh-Spectral-Resolution Imaging Spectrometer (CUSRIS) 60 shown in Fig. 9(b) .
The IMG instrument utilizes a 30° off-axis Michelson interferometer with a maximum OPD of 10 cm and a spectral range of 660&#; cm &#;1 . It is specifically designed for monitoring greenhouse gases, such as CH 4 , H 2 O, N 2 O, CO 2 , and O 3 . On the other hand, the SITELLE instrument incorporates a 15.5° off-axis Michelson interferometer, covering a spectral range of 350&#;900 nm. Its primary application is the study of galaxy cluster radiation and the analysis of nebular jet models, including phenomena such as the double peaks of [OII] , 29 in Orion and [NII] λ in the M1-67 nebula star WR124, which exhibits an escape velocity of 190.0 ± 7.4 km/s.
The standard Michelson interferometer suffers from a drawback where half of its energy returns to the light source, leading to interference with the spectral signal. This limitation can be overcome by employing an off-axis configuration, which introduces an angle between the interferometer and detector axes. Examples of instruments utilizing this approach include the Interferometric Monitor for Greenhouse Gases (IMG) 55,56 on the ADEOS satellite platform, as depicted in Fig. 8(a) , and the imaging Fourier transform spectrometer SITELLE 57,58 installed on the Canada&#;France&#;Hawaii Telescope (CFHT), as shown in Fig. 8(b) .
The ACE instrument is an FTS with a spectral range of 750&#; cm &#;1 , offering a high spectral resolution of 0.02 cm &#;1 and a FOV of 1.25 mrad. Its primary objective is to investigate atmospheric chemistry and dynamics, particularly related to stratospheric ozone depletion and the interplay between chemistry and climate change. The IASI instrument operates within an OPD range of &#;2 to 2 cm, covering a spectral range of 645&#; cm &#;1 with channels. It is employed for weather forecasting and climate studies, with the capability to detect various gases including CO, CO 2 , N 2 O, CH 4 , SO 2 , HNO 3 , NH 3 , OCS and CF 4 . The ACS-TIRVIM instrument operates in both nadir and solar occultation modes, achieving maximum OPDs of ±1 and ±5 cm, respectively. The corresponding spectral ranges for these modes are 590&#; and 590&#; cm &#;1 . The instrument is dedicated to investigating atmospheric chemistry by measuring various molecular species. The HIRAS instrument features distinct channels for the Long-Wave-Infrared (LWIR), Middle-Wave-Infrared (MWIR), and SWIR spectral regions, with maximum OPDs of 0.8, 0.4, and 0.2 cm, respectively. The spectral ranges covered by these channels are 650&#;, &#;, and &#; cm &#;1 . HIRAS is primarily employed to monitor atmospheric temperature, water vapor, and greenhouse gases, contributing to the improvement of numerical weather forecasting.
The theoretical SRP of the collimated light source Rand the extended light sources Rcan be expressed asHere, N represents the maximum interference order on the detector, and Ω is the solid angle of the extended light source. The standard time-modulated Michelson interferometer-based FTS is widely utilized in various astronomy and remote sensing applications. It finds application in the Canadian satellite mission known as the Atmospheric Chemistry Experiment (ACE), 49,50 the Infrared Atmospheric Sounding Interferometer (IASI) 51,52 present on the METOP series of European meteorological polar-orbit satellites, the Thermal InfraRed in honor of Professor Vassily Ivanovich Moroz (TIRVIM) channel of the Atmospheric Chemistry Suite (ACS), 27 and the High Spectral Infrared Atmospheric Sounder (HIRAS) 53,54 aboard China&#;s FY-3D satellite.
In the case of a Michelson interferometer, the spectrum of the light source [] can be reconstructed by applying a cosine Fourier transform to the detected light intensity []. The reconstruction equation can be expressed asHere, δ represents the optical path difference (OPD), which is defined as the discrepancy in the optical path traveled by two beams of light reaching a specific point. In practical systems, to reconstruct, the integration formula is substituted with an accumulation formula, where multiple OPD signalswith different OPD values (δ) are added together using weights determined by the cosine function factor, thereby performing a cosine Fourier transform. To obtain the OPD signal, the moving mirror is uniformly scanned across a range from &#;L to +L. This scanning process introduces a rectangular function, whereequals 1 whenand 0 when. As a result, the reconstructed spectrum is modified to, whereis known as the instrument line function (ILS). However, the presence of the sinc function in, characterized by positive and negative sidelobes, can interfere with the spectral signal. To address this interference, apodization techniques are employed to eliminate these sidelobes and enhance the quality of the reconstructed spectral signal. The spectral resolution can be expressed as
The working principle of a Fourier transform spectrometer (FTS) typically involves the utilization of interferometers, such as the traditional Michelson, Mach&#;Zehnder, and Sagnac interferometers. 48 These interferometers rely on the phenomenon of two-beam interference and make use of the mathematical concepts of Fourier transform. FTS offers various advantages, including high spectral resolution, wide spectral range, and the capability to measure the full spectrum simultaneously.
In addition to the aforementioned conventional tunable filter-based spectrometers, a new on-chip thermally tunable spectrometer (TTS) 99 has been developed. The TTS relies on a thermally tunable ultra-high-Q resonator (precision filter) and a wideband resonator array (coarse filters) connected in cascade across N channels. This compact device, with a size of 0.35 mm 2 , offers an ultra-high spectral resolution of 5 pm over a spectral range of .8&#;.5 nm.
W λ represents the optical response bandwidth, μ m is an initial integer guess determined by the transmission function and sparse representation, and Δλ m is the spectral interval. The LCPS system covers a spectral range of 500&#;700 nm, with a FWHM of 0.2 nm at the wavelength of 650 nm. It has the capability to reconstruct any complex spectrum using an adaptive sparse dictionary, eliminating the need for a filter array.
The nematic liquid crystal exhibits optical properties similar to a uniaxial crystal, where the alignment of its molecular long axis determines the optical axis. The birefringence of this crystal can be electrically controlled, as demonstrated in the Liquid Crystal Pixilated Spectrometer (LCPS) shown in Fig. 16 97,98 The LCPS utilizes a nematic liquid crystal subjected to n applied voltages at m incident wavelengths, resulting in an n × m dimensional sparse matrix optical transfer function, effectively creating a pixilated spectrometer. The theoretical spectral resolution can be expressed as
The CE4-AOTF, designed for lunar exploration missions, is equipped with two AOTFs featuring ultrasonic driving frequencies of 71.2&#;178.7 and 42&#;118.8 MHz, respectively. This configuration enables the CE4-AOTF to cover a spectral range of 450&#;950 and 900&#; nm, with corresponding FWHM values of 2.8&#;6.4 and 4.2&#;9.6 nm, respectively. The SPICAV instrument utilizes two AOTFs with ultrasonic driving frequencies of 140&#;250 and 80&#;140 MHz, respectively. Its spectral range spans from 0.65 to 1.05 µm and 1.05&#;1.7 µm, with spectral resolutions of 7.8 and 5.2 cm &#;1 , respectively. The SPICAV instrument is primarily employed for observing Venus&#;s atmosphere, studying various components, such as H 2 O, HDO, CO 2 , and aerosols, through solar occultation, nadir, and limb observation modes. In the ACS-NIR instrument, an AOTF is employed to select the diffraction orders of the echelle grating, which has a blazed angle of 70° (R3), a groove density of 24.3 lines/mm, an effective area of 46 × 102 mm 2 , and a diffraction order range of 48&#;105. The ultrasonic driving frequency in this instrument ranges from 64 to 156 MHz, while the spectral range covers 0.73&#;1.6 µm. The SRP achieved by the ACS-NIR ranges from 20 000 to 27 000.
Acousto-optic tunable filter spectrometers. (a) The CE4-AOTF, 94 Reproduced with permission from Li et al., Materials 14 , (). Copyright by The Authors; licensed under a Creative Commons Attribution (CC BY) license. (b) The SPICAV, 95 Reproduced with permission from Bertaux et al., Planet. Space Sci. 55 , &#; (). Copyright Elsevier Ltd. (c) The ACS-NIR, 27 Reproduced with permission from Korablev et al., Space Sci. Rev. 214 , 7 (). Copyright The Authors; licensed under a Creative Commons Attribution (CC BY) license.
Acousto-optic tunable filter spectrometers. (a) The CE4-AOTF, 94 Reproduced with permission from Li et al., Materials 14 , (). Copyright by The Authors; licensed under a Creative Commons Attribution (CC BY) license. (b) The SPICAV, 95 Reproduced with permission from Bertaux et al., Planet. Space Sci. 55 , &#; (). Copyright Elsevier Ltd. (c) The ACS-NIR, 27 Reproduced with permission from Korablev et al., Space Sci. Rev. 214 , 7 (). Copyright The Authors; licensed under a Creative Commons Attribution (CC BY) license.
Compared to prism and grating-based spectrometers, the AOTF offers advantages such as a smaller volume, lighter weight, tunable wavelength, larger aperture, faster scanning speed, and higher diffraction efficiency. These characteristics make it particularly well-suited for astronomical and aerospace applications. For instance, the China&#;s Chang&#;e&#;3, 4, 5 AOTF (CE3, 4, 5-AOTF) 92&#;94 [ Fig. 15(a) ] spectrometers, the Spectroscopy for the Investigation of the Characteristics of the Atmosphere of Venus (SPICAV) 95,96 on the Venus Express (VEX) orbiter [ Fig. 15(b) ], and the ACS-NIR 27 [ Fig. 15(c) ] employ AOTFs in their setups.
The acousto-optic tunable filter (AOTF) operates based on the principle of anomalous Bragg acousto-optic diffraction. In this device, an ultrasonic wave is loaded in the [110] direction of a TeObirefringent crystal, 91 with the optical axis aligned in the [001] direction. The periodically changing refractive index of TeOdue to the ultrasonic wave forms an optical phase grating. By varying the frequency of the ultrasonic wave, denoted as f, with the driving voltage of the ultrasonic transducer, the central wavelength of the Bragg diffraction light, denoted as λ, can be adjusted. The relationship between these parameters is given by, where Vis the ultrasonic velocity and nand nare the refractive indices of the incident and diffracted light, respectively. It can be observed that the AOTF bandwidth is limited by the ultrasonic frequency as λis a constant. The SRP of the AOTF can be expressed asHere, b is the dispersion constant, L represents the length of the acousto-optic interaction, and θdenotes the incident angle.
The standard F&#;P interferometer (FPI) typically exhibits a low spectral contrast of less than 10 3 due to uneven field amplitudes on multiple interference paths. These amplitudes are associated with the transmission intensity distribution, which is described by the Airy function in the far-field interference pattern. The low spectral contrast reduces fringe visibility and significantly limits the detection sensitivity of weak signals. To overcome this issue, the Single-Pass Fabry&#;Pérot Spectrometer (SPFPS) 89,90 [ Fig. 14(c) ] employs a spatial light modulator (SLM) to equalize the output field of the virtual imaging phased array (VIPA) using a semi-Gaussian distribution intensity mask. The VIPA has a FSR of 30 GHz or 1 cm &#;1 , a fineness of 40, and an optical throughput larger than 50%. This configuration significantly improves the spectral contrast by a factor of times, allowing for a spectral resolution of 0.08 cm &#;1 or 800 MHz with an SRP of 800 000.
The GFPI utilizes two cascaded F&#;P étalons with cavity lengths of 1.1 and 1.4 mm. These configurations achieve FWHM values of 3.43 and 2.73 pm, respectively, at a wavelength of 617 nm. The corresponding FSR values are 0.158 and 0.126 nm, resulting in a combined FWHM of 1.95 pm. The fineness of the system is 46, and the two cavities can change synchronously. The GFPI has a spectral range of 530&#;860 nm, offering a theoretical SRP of &#;250 000. The GFPI is utilized to study the dynamic evolution process of the solar atmosphere and magnetic field. Similarly, the VTF also employs a dual F&#;P étalon structure. It achieves a FWHM of 6&#;8 pm, with a specific value of 6 pm at 600 nm. The spectral range of the VTF spans from 520 to 870 nm. This instrument is used for studying the structure and dynamics of the Sun and its atmosphere.
A single high-finesse F&#;P étalon with a small FSR can achieve high spectral resolution or a small FWHM. However, it requires a filter with a bandwidth of FSR/2 to separate specific interference orders, which can be challenging. Additionally, the transmissivity of a single F&#;P étalon is influenced by the reflectivity and cavity absorption, leading to the need for low finesse to achieve a large optical throughput. By combining two or more F&#;P étalons, the finesse requirement of each étalon can be reduced, resulting in improved effective FSR, spectral purity, out-of-band suppression performance, and optical throughput of the system. 85 Two notable examples utilizing this approach are the GREGOR Fabry&#;Pérot Interferometer (GFPI) 86,87 on the German 1.5 m Gregor solar telescope [ Fig. 14(a) ] and the Visible Light Tunable Filter (VTF) 88 on the DKIST [ Fig. 14(b) ].
The F&#;P cavity can function as a cavity length tunable filter or a scanning spectrometer. The spectral bandwidth or resolution is given by. Due to significant difference in orders of magnitude between λand h, the bandwidth of the F&#;P cavity is typically narrow. The SRP can be expressed asHere, m represents the interference order. The F&#;P tunable filter is widely used as a spectral analysis device in astronomy and remote sensing applications. One example is the Wide-Field Imaging Fabry&#;Pérot Spectrometer for Astronomy (TAURUS), 84 which is employed on telescopes such as the 3.89 m Anglo-Australian Telescope (AAT) and 4.2 m William Herschel Telescope (WHT). TAURUS utilizes a single F&#;P cavity with a cavity length ranging from 3 to 540 µm, a tuning range of 2 µm, an FSR of 41.7 to 0.25 nm, and a fineness of 35. Its spectral range covers 370&#;950 nm with a spectral resolution of 0.007&#;1.2 nm and a maximum SRP of . The TAURUS instrument is used to map the velocity field of astronomical emission line sources, including planetary HII regions, active irregular interacting galaxies, and redshift galaxies. The Doppler velocity&#;s bandwidth ranges from 150 to 25 000 km/s with a velocity resolution of 4&#;714 km/s.
The F&#;P cavity operates based on the principle of multi-beam interference, which generates a series of comb-like transmission peaks. Mathematically, the transmission spectrum can be expressed asHere,represents the finesse of the cavity, ϕ = 4πnh cos θ/λ is the phase difference between adjacent light rays, R is the reflectivity of the mirrors, and h is the cavity length. The fineness can be expressed as, and the FSR can be written as
The TDSM reconstructive spectrometer can also be implemented using a chirp modulated temporal Rayleigh speckle (CMTRS) fiber system. 116 In this system, a frequency sweep signal is generated using a single sideband electro-optic modulator (EOM). The Rayleigh speckle is produced through the fiber interferometer and detected by a balanced photodetector (BPD). The intensityis given byHere, Erepresents the amplitude of the incident light, Ris the reflection coefficient, τis the round-trip time of the ith Rayleigh scatter point, and fis the frequency to be measured. The pattern of the Rayleigh speckle remains unchanged and only shifts with variations in the light source frequency. Consequently, the CMTRS system requires only one reference frequency fto generate the reference Rayleigh speckle. A cross-correlation algorithm is employed to determine the frequency difference Δf = γΔt between fand f, where Δt represents the time shift of the Rayleigh speckle. The spectral range of the CMTRS system is &#; nm, offering an ultra-high resolution of 40 am.
By including the interference signals at all wavelengths and positions, a transmission matrix can be obtained, and the spectrum can be reconstructed by solving the transmission matrix Eq. (24) . This TDSM reconstructive spectrometer operates within the optical communication band of &#; nm, offering an ultra-high spectral resolution of 0.4 fm, corresponding to a 2 µs probe pulse. It can resolve dynamic multi-wavelength signals with a time resolution of 25 µs.
A time-domain spectral-to-spatial mapping (TDSM) reconstructive spectrometer can be achieved by utilizing wavelength-dependent Rayleigh speckle (RS). 115 The system operates based on the principle of optical time-domain reflectometry (OTDR), where the incident light is modulated by an acousto-optic modulator (AOM). The resulting pulsed light generates the Rayleigh backscattering signal (RBS) at different positions along the fiber. At position z, Wv/λ Rayleigh scattering signals interfere, where Wrepresents the pulse width and v is the speed of light in the optical fiber. Each wavelength λ results in a distinct interference spot at z. The detector sequentially receives the interference signals at different fiber positions. The intensity Ican be described as follows:
The time-domain spectral response reconstructive spectrometers. (a) The BPS, 113 Reproduced with permission from Yuan et al., &#;A wavelength-scale black phosphorus spectrometer,&#; Nat. Photonics 15 , 601&#;607 (). Copyright The Authors, under exclusive licence to Springer Nature Limited. (b) The VDWJ, 114 Reproduced with permission from Yoon et al., Science 378 , 296&#;299 (). Copyright The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science.
The time-domain spectral response reconstructive spectrometers. (a) The BPS, 113 Reproduced with permission from Yuan et al., &#;A wavelength-scale black phosphorus spectrometer,&#; Nat. Photonics 15 , 601&#;607 (). Copyright The Authors, under exclusive licence to Springer Nature Limited. (b) The VDWJ, 114 Reproduced with permission from Yoon et al., Science 378 , 296&#;299 (). Copyright The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science.
The time-domain spectral response (TDSR) reconstructive spectrometer provides a simpler, more affordable, and faster alternative to spatial-domain reconstructive spectrometers. It only requires a single detector and a tunable filter to obtain a complete set of spectral response functions. One example of a tunable filter used in TDSR spectrometers is the black phosphorus (BP), 113 as depicted in Fig. 19(a) . BP is a dark gray layered material, and its responsivity function R D , λ or matrix R D,λ D 1 &#; D n , λ 1 &#; λ n at each wavelength λ i is related to the electric displacement (D) of the BP. By using a response vector I = RS, the unknown incident spectrum (S) can be reconstructed by solving the matrix equation. The BP spectrometer (BPS) covers a spectral range of 2&#;9 µm with a FWHM of 90 nm and utilizes 81 photocurrent sampling points. Remarkably, the BPS is the world&#;s smallest spectrometer, with an active area footprint of only 9 × 16 µm 2 . Another example of a tunable filter used in TDSR spectrometers is a single MoS 2 /WSe 2 van der Waals junction (VDWJ) 114 shown in Fig. 19(b) . The VDWJ exhibits an electrically tunable transport-mediated spectral response, where different gate voltages correspond to different photo-responses of the van der Waals junction. Its spectral range spans from 405 to 845 nm with a spectral resolution of 3 nm and a peak wavelength accuracy of 0.36 nm.
The Micro-taper leaky-mode spectrometer (MTLMS) 112 is another variation, which employs a fiber taper tip to generate intricate leaky mode patterns within a 1 mm length. These patterns exhibit a distinct association with wavelengths and can effectively operate in the spectral range of 450&#; nm. The MTLMS achieves an ultra-high spectral resolution of &#;1 pm. To analyzing the complex frames captured by the CMOS image sensor (CIS) and associate light pattern images with wavelength information, a combination of convolutional neural networks (CNNs) and a lightweight vision transformer (ViT) network is employed in the MTLMS.
The SDSM reconstructive spectrometer utilizes different methods and random media to achieve spectral reconstruction. One example of such random media is the fiber-coupled integrating sphere (FCIS), 109,110 as illustrated in Fig. 18(a) . The FCIS acts as an interferometer with a long OPD of several meters, producing wavelength-dependent speckle patterns that are captured by a camera. To calibrate the speckle wavemeter, an F&#;P interferometer (REF) is employed, and the transmission matrix method (TMM) is used for spectrum reconstruction. This spectrometer offers an ultra-high spectral resolution of 0.3 fm at a wavelength of 780 nm and covers a spectral range of 488&#; nm. Another example of the SDSM reconstructive spectrometer is the Multiple Scattering Chip Spectrometer (MSCS) 111 depicted in Fig. 18(b) . It utilizes a photonic crystal lattice with random air holes as the random medium. The random medium induces light scattering, which is detected by an annular detector array to obtain the intensity distribution. A matrix equation, I = TS, is employed for spectrum reconstruction. Nonlinear optimization techniques are applied to minimize I &#; T S 2 &#;. The MSCS covers a spectral range of &#; nm with a spectral resolution of 0.75 nm at a wavelength of nm.
This can be expressed by the matrix equationHere, the transmission matrix T represents the unique patterns of the random medium for each wavelength. By solving Eq. (24) , it becomes possible to reconstruct the incident spectrum S.
The SDSR reconstructive spectrometer typically employs a micro-structured spatial array filter to achieve different spectral responses. For instance, a photonic crystal slab (PC-slab) is integrated on top of a CMOS sensor array, 102 as illustrated in Fig. 17(c) . It contains a random basis of a 6 × 6 different pattern structures to generate distinct spectral responsivities. The incident spectrum can be reconstructed using least-square and compressive sensing methods. This spectrometer covers a spectral range of 550&#;750 nm with a resolution of &#;1 nm. Another spatial spectral filter array used is the F&#;P micro-cavities (FPMCs) 103 integrated onto an InGaAs detector chip, as shown in Fig. 17(d) . Different cavity thicknesses correspond to different spectral responses. A compressed sensing-based accelerated proximal gradient (APG) algorithm is employed to reconstruct the spectrum. This spectrometer covers a spectral range of 900&#; nm with a spectral resolution of 2 nm for a 50 pixel set. The photon counting reconstructive spectrometer (PCRS) 104 utilizes metasurfaces and superconducting nanowire single-photon detectors (SNSPD) and is fabricated on a silicon-on-insulator substrate. The PCRS can reconstruct the spectrum of mono-color light with a resolution of 2 nm in the wavelength range of &#; nm, with a detection efficiency of 1.4%&#;3.2%. Other similar spatial spectral filters include a 2D thin-film filter array, 105 étalon array, 106 quantum dot filters, 107,108 etc.
The spatial-domain spectral response reconstructive spectrometers. (a) The SNS, 100 Reproduced with permission from Yang et al., Science 365 , &#; (). Copyright The Authors; licensed under a Creative Commons Attribution (CC BY) license. (b) The MQWH, 101 Reproduced with permission from Sarwa et al., Nano. Lett. 22 , &#; (). Copyright American Chemical Society. (c) The PC-slabs, 102 Reproduced with permission from Wang et al., Nat. Commun. 10 , (). Copyright The Authors; licensed under a Creative Commons Attribution (CC BY) license. (d) The FPMCs, 103 Reproduced with permission from Xuan et al., Adv. Opt. Mater. 10 , (). Copyright Wiley-VCH GmbH.
The spatial-domain spectral response reconstructive spectrometers. (a) The SNS, 100 Reproduced with permission from Yang et al., Science 365 , &#; (). Copyright The Authors; licensed under a Creative Commons Attribution (CC BY) license. (b) The MQWH, 101 Reproduced with permission from Sarwa et al., Nano. Lett. 22 , &#; (). Copyright American Chemical Society. (c) The PC-slabs, 102 Reproduced with permission from Wang et al., Nat. Commun. 10 , (). Copyright The Authors; licensed under a Creative Commons Attribution (CC BY) license. (d) The FPMCs, 103 Reproduced with permission from Xuan et al., Adv. Opt. Mater. 10 , (). Copyright Wiley-VCH GmbH.
An example of the SDSR reconstructive spectrometers is the Single Nanowire Spectrometer (SNS), 100 as depicted in Fig. 17(a) . It consists of a parallel In/Au electrode array on compositionally graded semiconductor CdS x Se 1&#;x nanowires. Each adjacent pair of electrodes forms a photodetector with a distinct spectral response, resulting in the generation of 38 equations. The target spectrum is fitted using the adaptive Tikhonov regularization algorithm, also known as the restricted least squares method. The SNS covers a spectral range of 500&#;630 nm with a spectral resolution of 15 nm at a wavelength of 570 nm. Another example of the SDSR reconstructive spectrometer is a compressively strained InGaN/GaN multiple quantum well hetero-structure (MQWH), 101 as shown in Fig. 17(b) . It utilizes the built-in GaN pn junction to detect photocurrents and reconstructs the spectrum using a non-negative least-squares (NNLS) algorithm with total-variation regularization and an appropriate kernel function. The MQWH spectrometer covers a spectral range of 400&#;645 nm with a wavelength accuracy of 4&#;6 nm.
The spatial-domain spectral response (SDSR) reconstructive spectrometer utilizes different spectral response curves for each detector unit. This leads to the generation of n spectral response equations, where each equation corresponds to a specific detector unit. Mathematically, these equations are expressed asHere, the parameters λand λdefine the operational wavelength range of the spectrometer. The incident light spectrum is represented by, while the optical response function of the idetector is denoted by. The photocurrent signal of the idetector is represented by I
The reconstructive spectrometer utilizes reconstructive algorithms to map the incident spectrum either in the spatial or temporal domains to the intensity matrix of the detector array, similar to tomography techniques. There are two types of reconstructive spectrometers: spatial-domain and time-domain. 2 The spatial-domain reconstructive spectrometer consists of the spatial-domain spectral response (SDSR) reconstructive spectrometer and the spatial-domain spectral-to-spatial mapping (SDSM) reconstructive spectrometer. The time-domain reconstructive spectrometer includes the time-domain spectral response (TDSR) reconstructive spectrometer and the time-domain spectral-to-spatial mapping (TDSM) reconstructive spectrometer.
All spectrometers can be considered as generalized reconstructive spectrometers (GRS), as depicted in Fig. 20 . They operate in a manner similar to a tomography system. 117 The GRS comprises two main procedures: the forward procedure for spectral mapping and the backward procedure for spectral reconstruction. In the forward procedure, the spectral mapping structure is constructed according to specific requirements. A calibrated or tunable narrowband light source is used to determine the transformation characteristics for each wavelength, resulting in the acquisition of a reference intensity matrix. Subsequently, the system measures the light source with an unknown spectrum, capturing the intensity matrix. In the backward procedure, the spectral reconstruction algorithm is employed to reconstruct the unknown spectrum based on the acquired intensity matrix.
The time-domain spectral-to-spatial mapping (TDSM) reconstructive spectrometer converts the incident spectrum into a wavelength-dependent temporal Rayleigh speckle intensity matrix by utilizing an AOM or an EOM for frequency scanning, as illustrated in Fig. 23(b) . On the other hand, the time-domain spectral response (TDSR) reconstructive spectrometer transforms the incident spectrum into a wavelength-dependent intensity matrix using an electrically tunable spectral micro-responder, such as black phosphorus and van der Waals junction, as shown in Fig. 23(c) . Therefore, both TDSR and TDSM reconstructive spectrometers fall under the time-domain holo-morphism mapping (TDHM).
The holo-morphism mapping of the reconstructive spectrometers. (a) Spatial domain for the SDSR and SDSM types, where the small illustrations represent SDSR reconstructive spectrometers (from Fig. 17 ) and SDSM reconstructive spectrometers (from Fig. 18 ). (b) Time domain for the TDSM types. (c) Time domain for the TDSR type, where the small illustrations represent TDSR reconstructive spectrometers (from Fig. 19 ).
The holo-morphism mapping of the reconstructive spectrometers. (a) Spatial domain for the SDSR and SDSM types, where the small illustrations represent SDSR reconstructive spectrometers (from Fig. 17 ) and SDSM reconstructive spectrometers (from Fig. 18 ). (b) Time domain for the TDSM types. (c) Time domain for the TDSR type, where the small illustrations represent TDSR reconstructive spectrometers (from Fig. 19 ).
The Michelson or Mach&#;Zehnder interferometer Fourier transform spectrometer (FTS) generates varying optical path differences (OPDs) by scanning the moving mirror. Each OPD signal encompasses the entire wavelength information of the spectrum and is projected onto the detector array, as shown in Fig. 22(a) . Consequently, it exhibits a time-domain holo-morphism mapping (TDHM). Additionally, spatial Fourier transform spectrometers can be constructed using Michelson, Mach&#;Zehnder, or Sagnac interferometers, along with Wollaston prisms and liquid crystals. These spectrometers project the two-beam interference signals onto the detector array, with the OPDs linearly changing along the interferogram. Each OPD contains the complete spectrum of wavelengths. Hence, within the spectral resolution, they fall under the spatial-domain holo-morphism mapping (SDHP), as shown in Fig. 22(b) .
The one-to-one mapping: (a) spatial-domain for the prism and grating spectrometers, where the small illustrations represent triangular prism, Féry prism, ordinary blazed grating, echelle grating, concave grating, volume holographic grating [from Figs. 5(a) and 5(c) ], and convex grating [from Figs. 7(b) and 7(e) ] and (b) time-domain for the tunable filter spectrometers, where the small illustrations represent Fabry&#;Pérot, acousto-optic, and liquid crystal tunable filters.
The one-to-one mapping: (a) spatial-domain for the prism and grating spectrometers, where the small illustrations represent triangular prism, Féry prism, ordinary blazed grating, echelle grating, concave grating, volume holographic grating [from Figs. 5(a) and 5(c) ], and convex grating [from Figs. 7(b) and 7(e) ] and (b) time-domain for the tunable filter spectrometers, where the small illustrations represent Fabry&#;Pérot, acousto-optic, and liquid crystal tunable filters.
The prism or grating spectrometer projects the incident slit onto a detector array, where each position corresponds to a different wavelength, as illustrated in Fig. 21(a) . Within the spectral resolution, this mapping follows a spatial-domain one-to-one mapping (SDOM), also known as bijection. On the other hand, the Fabry&#;Pérot, acousto-optic, or liquid crystal tunable filter spectrometers sequentially image each wavelength signal onto the detector array, exhibiting a time-domain one-to-one mapping (TDOM), as depicted in Fig. 21(b) .
The optical spectrum analyzer or spectrometer establishes a mapping relationship between the spectrum of the light source and the intensity measured by the detector. This mapping enables the analysis and characterization of the spectral content of the light source.
The prism, grating, or tunable filter spectrometer operates based on a one-to-one mapping structure. It is represented by a white transparent box, allowing for direct acquisition of the spectrum without intricate reconstruction processes. The Fourier transform spectrometer (FTS), on the other hand, follows a holo-morphism mapping structure. It is depicted as a semi-transparent box with a straightforward mathematical relation involving Fourier transforms, enabling spectrum reconstruction through a simple inverse Fourier transform operation. The reconstructive spectrometer also falls within the holo-morphism mapping structure, but it is more complex than the FTS and cannot be described by a simple theoretical formula. It is represented by a gray box and requires more intricate algorithms, such as the least square method and the transfer matrix method, for spectrum reconstruction.
The WIS offers the advantage of a large field of view (FOV) and relatively simple calibration. However, it includes moving parts, which can introduce system instability and complexity. In comparison, the PIS does not require any moving parts, resulting in a simpler and more compact structure. It provides higher SNR, superior spatial resolution, and simpler image processing compared to the WIS. The PIS is widely employed in astronomy and remote sensing applications and can be implemented using prism, grating, and Fourier transform spectrometers, such as APEX, PRISMA, AHSI, CUSRIS, UV-SFTS, and CE1-SFTS. The FIS enables simultaneous acquisition of two-dimensional spatial images and one-dimensional spectral information. It can be realized using tunable filter spectrometers, such as GFPI and CE4-AOTF. The SIS enables non-scanning, single capture imaging without the need for post-processing. This can be achieved through reconstructive spectrometers, such as SNS, PC-slabs, FCIS, and VDWJ. The SIS can also be implemented using the principle of diffractive rotation 120 and encoding reconstruction. 121 To achieve 4D snapshot hyperspectral imaging (3D space + spectrum), structural lighting and 3D point cloud reconstruction can be employed. 97 For 4D video snapshot spectral imaging (2D space + spectrum + time), a prism spectrometer with multi-field slits can be utilized. 122 Furthermore, the advanced deep learning broadband encoding stochastic hyperspectral camera 123 leverages artificial intelligence techniques for filter design and spectrum reconstruction. This innovative enables significantly faster signal processing (&#;11 000 times faster) and &#;10 times improved noise tolerance.
The whiskbroom imaging spectrometer (WIS) employs a linear-array detector to collect spectral data from a single ground pixel. By flying along the aircraft orbit and scanning across the orbit in a whiskbroom fashion, it is possible to obtain the two-dimensional spatial information, as illustrated in Fig. 24(b) . The pushbroom imaging spectrometer (PIS) uses an area-array detector to capture ground image information. One dimension is dedicated to spatial imaging, while the other dimension is utilized for spectrum acquisition. Flying along the aircraft orbit allows for the acquisition of two-dimensional spatial information, as depicted in Fig. 24(c) . The filter imaging spectrometer (FIS) relies on an area-array detector to directly obtain ground image information. By tuning the filter, spectrum information can be obtained, as shown in Fig. 24(a) . The snapshot imaging spectrometer (SIS) is capable of directly capturing the three-dimensional data cube, as depicted in Fig. 24(a) .
The imaging spectrometer 118,119 is capable of capturing a three-dimensional data cube comprising two spatial dimensions (x, y) and one spectral dimension (λ), as depicted in Fig. 24(a) . As a result, it enables comprehensive observation of spatial, spectral, and radiometric information. Depending on the method used to acquire the data cube, imaging spectrometers can be categorized as whiskbroom type, pushbroom type, filter type, and snapshot type.
While area-array micro-detectors, such as CCD and CMOS sensors, are commonly employed in spectrometers, it is important to note that not all spectrometers utilizing these detectors are considered imaging spectrometers. Prism and grating spectrometers capture the image of the incident slit on the detector, while Fourier transform spectrometers capture the interferogram. Reconstructive spectrometers, on the other hand, capture the complex spectral mapping signals on the detector. In all these cases, the area-array detectors are primarily used for receiving spectral mapping signals and facilitating spectral reconstruction, rather than traditional imaging purposes.
The spectral range of the reconstructive spectrometer depends on the specific material used. For example, the SNS has a spectral range of 500&#;630 nm, which is determined by the spectral response of the In/Au electrodes on the compositionally graded semiconducting CdS x Se 1&#;x . The MQWH covers a spectral range of 400&#;645 nm, determined by the spectral response of the compressively strained InGaN/GaN multiple quantum well heterostructures. The PC-slabs operates within the spectral range of 550&#;750 nm, determined by the spectral response of the silicon-on-sapphire (SOS) substrate. The FPMCs has a spectral range of 900&#; nm, determined by the spectral response of the monolithic integrated F&#;P cavity on an InGaAs detector chip. The FCIS covers a spectral range of 488&#; nm, determined by the spectral response of the fiber-coupled integrating sphere. The MSCS operates within the spectral range of &#; nm, determined by the spectral response of the silicon-on-insulator (SOI) wafer material. The MTLMS covers a spectral range of 450&#; nm, determined by the spectral responses of the multimode optical fiber and the CMOS detector utilized. The BPS has a spectral range of 2&#;9 µm, determined by the spectral response of black phosphorus. The VDWJ covers a spectral range of 405&#;845 nm, influenced by the wavelength-dependent absorption of MoS2 and WSe2 as well as the controllable charge carrier transport through the MoS2/WSe2 van der Waals junction interface. The CMTRS operates within the spectral range of &#; nm, determined by the spectral response of the optical fiber system.
The spectral range of the liquid crystal tunable spectrometer (LCTS) is determined by the bandwidth of the liquid crystal utilized. Generally, the LCTS is designed to operate within the visible band. For instance, the LCPS covers a spectral range of 500&#;700 nm with a relatively narrow bandwidth of only 200 nm, as the tuning range of liquid crystal birefringence is limited in this particular case.
The spectral range of an acousto-optic tunable spectrometer is limited by the acousto-optic crystal TeO 2 , which has a theoretical bandwidth of 0.35&#;5 µm. Examples include CE4-AOTF, SPICAVS, and ACS-NIR, which have spectral ranges of 450&#; nm, 0.65&#;1.7 µm, and 0.73&#;1.6 µm, respectively. To achieve a wider spectral range, segmental ultrasonic tuning is often necessary, as implemented in the CE4-AOTF. It has an ultrasonic tuning range of 71.2&#;178.7 and 42&#;118.8 MHz, corresponding to spectral ranges of 450&#;950 and 900&#; nm, with bandwidths of 500 and nm, respectively. Higher ultrasonic frequencies are required for longer operating wavelengths.
The F&#;P tunable filter spectrometer is commonly used in the visible spectral band, with examples such as TAURUS covering 430&#;570 nm, GFPI covering 530&#;860 nm, and DKIST-VTF covering 390&#;550 nm and 600&#;860 nm. The F&#;P étalon is typically tuned using a piezoelectric ceramic, and the tuning range determines the spectrometer&#;s bandwidth. For instance, TAURUS has an étalon tuning range of 2 µm, resulting in a bandwidth of only 140 nm. On the other hand, SPFPS utilizes a non-tunable VIPA, but it requires a single-longitudinal mode laser at 532 nm.
The Fourier transform spectrometer (FTS) offers a wide spectral range, known as the Fellgett advantage, limited only by the detector&#;s spectral response and the optical material&#;s performance. It is particularly suitable for mid-infrared wavebands that cannot be reached by prism and grating spectrometers. Examples of FTS applications in the mid-infrared range include ACE, METOP-IASI, ACS-TIRVIM, ADEOS-IMG, FY3D-HIRAS, and SM-FTIR. Furthermore, FTS can operate in the terahertz or submillimeter wavebands, as demonstrated by the JCMT-MZFTS, which employs a 0.3 K neutron transmutation-doped (NTD) bolometer to cover spectral bands at 350, 450, 750, and 850 µm. FTS can also be utilized in the remote ultraviolet waveband, such as the IFTSUV with a spectral range of 80&#;145 nm. Additionally, a larger maximum OPD corresponds to a wider bandwidth. For instance, the JCMT-MZFTS achieves a maximum OPD of 1.2 m, corresponding to a bandwidth of one hundred micrometers. In contrast, the CE-SFTS has an OPD of only 15.37 µm, resulting in a bandwidth of only a few hundred nanometers.
The spectral range of a grating is determined by the grating pitch and the blaze angle. As an example, the RGS utilizes a small blaze angle of 0.° and a blaze wavelength of 1.5 nm, enabling it to cover the x-ray band of 0.6&#;3.8 nm under grazing incidence. Similar to the prism, the grating is not commonly used for mid-infrared spectrometers. Although grating spectrometers have a small FSR, the spectral range can be expanded by scanning the incident slit. For instance, DKIST-CryoNIRSP operates in the diffraction order of 12&#;107, with an FSR of only tens of nm. However, it still achieves a wide spectral range of 530&#; nm and a broad bandwidth of nm.
The prism spectrometer projects the entrance slit onto the detector array, capturing the spectral information. However, its spectral range is limited by the dispersion characteristics of the prism materials. For instance, the SPS utilize a fused silica prism, offering a wide bandwidth of 0.25&#;2.5 µm, as shown in Table I . On the other hand, the BASS employs a NaCl prism to extend the spectrum up to 13.5 µm. It is uncommon to use a prism for mid-infrared spectrometers due to the decrease in angular dispersion and spectral resolving power (SRP) with increasing wavelength.
The spectral resolution δλ or spectral resolving power (SRP), denoted by R = λ/δλ, determines the ability of a spectrometer to analyze spectra. The theoretical SRP of a prism spectrometer is limited by the size and nonlinear material dispersion of the prism. For instance, the APEX spectrometer utilizes CaF2/ZnS and CaF2/sapphire prisms, with maximum SRPs of 634 and 227, respectively. The material dispersion of the prism is typically negative, resulting in a larger SRP at lower wavelengths.
The Féry prism features curved surfaces that enable focusing and imaging. It exhibits a larger angular dispersion compared to a triangular prism. However, its radius of curvature (ROC) is typically small, resulting in smaller collimating lens f1 and converging lens f2 compared to an ordinary prism spectrometer. The small f1 introduces limitations on slit diffraction, leading to reduced spectral resolution. Similarly, the small f2 reduces the linear dispersion f2dθ/dλ and, consequently, the spectral resolution. Moreover, the Féry prism has a smaller thickness and apex angle. Overall, considering these parameters, the Féry prism spectrometer achieves a larger or comparable SRP compared to a triangular prism spectrometer. For example, spectrometers such as SPS, BASS, TSIS-1-SIM, and SORCE-SIM employ Féry prisms, resulting in SRPs of &#;250, 100, 100, and 800, respectively. In contrast, th TSIS-1-HSRS utilizes a modified spectral ratio method to normalize solar line data with extremely high spectral resolution to the absolute irradiance scale of the TSIS-1-SIM and the CSIM, yielding an impressive SRP ranging from &#;202&#;000 to 273&#;000.
The SRP of a grating spectrometer is determined by the diffraction order (m) and the groove number (N) of the grating. However, the actual SRP is often lower than the theoretical value due to limitations imposed by the size of the clear aperture. Ordinary plane blazed gratings typically operate at m = 1 or 2 to avoid overlapping diffraction orders, which significantly restricts their SRP and applications. An important application that overcome these limitations is the RGS, which utilizes two grating arrays with a grating density of 645.6 lines/mm and a blazed angle of 0.° to achieve an SRP of 100&#;500.
In contrast, echelle gratings have a large blaze angle and divergence angle and can be fabricated on a larger surface area. This enables significant improvement in SRP through the use of high diffraction orders. For example, the G-GLEF employs an echelle grating with a groove density of 31.6 lines/mm and an area of 300 × mm2, resulting in an SRP of 100&#;000. The ACS-MIR utilizes an echelle grating with a density of 3.03 lines/mm and dimensions of 107 × 240 mm2, achieving an SRP of 30&#;000&#;50&#;000. The DKIST-CryoNIRSP employs an echelle grating with a width of 408 mm and operates at diffraction orders of 12&#;107, resulting in an impressive SRP of 116&#;275&#;132&#;500.
The volume holographic grating (VHG) is produced using holographic technology, but its diffraction efficiency tends to be low at high diffraction orders. Furthermore, the VHG has a smaller grating pitch compared to the echelle grating, leading to high-order dispersion interference. As a result, the VHG typically operates at the m = 1 diffraction order. However, the VHG offers advantages such as high groove density and a large surface area. Taking all these factors into account, the SRP of the VHG falls between that of an ordinary blazed grating and an echelle grating. For instance, the DESI utilizes a VHG with &#; lines/mm to achieve an SRP of &#;. Similarly, the Subaru-FOCAS employs a VHG measuring 110 × 106 mm2, resulting in an SRP of 250&#;.
The flat-field concave grating functions similar to the Féry prism. For instance, in EUV applications, a concave grating with a groove density of lines/mm, a curvature radius of 235 mm, and an effective width of 46 mm is utilized, resulting in an SRP of 421.
The convex grating spectrometer utilizes the concentric three-reflection imaging system, such as the VIMS-V. The traditional Offner structure, which includes a large concave mirror, is employed, resulting in a decrease in the SRP. The VIMS-V incorporates a convex grating with a groove density of 27.661 lines/mm, achieving an SRP of 150&#;525. On the other hand, the GF-F-AHSI and EAS employ improved Offner structures that replace the large concave mirror with two small concave mirrors and introduce the Rowland circle configuration. As a result, their SRPs can reach 500 and 280, respectively.
The spectral resolution of the time-modulated Fourier transform spectrometer (FTS) is determined by the maximum OPD, which is dictated by the scanning range of the moving mirror. For instance, the METOP-IASI, ACS-TIRVIM, ADEOS-IMG, and FY3D-HIRAS are Michelson interferometer FTS systems with maximum OPDs of 2, 5, 10, and 0.2/0.4/0.8 cm, respectively. Their corresponding SRPs are &#;, /45&#;385, 30&#;000, and //. On the other hand, the JCMT-MZFTS is a Mach&#;Zander interferometer FTS operating in the submillimeter waveband. It utilizes an OPD of 1.2 m, resulting in an SRP of .
The spatially modulated FTS (SFTS) is a system that does not require a moving mirror. Instead, it projects the interference signals onto a detector array. The resulting interferogram contains an OPD distribution. Various implementations of SFTS, such as WP-FTS-SP, WP-FTS-NSP, TD-ISP, CaHyD, SM-FTIR, BLC-FTS, OCFTS, SIFTS, CE-SFTS, and UV-SFTS, achieve maximum OPDs of 19, 123, 30, 20, 810, 23.4 µm/210, 570, , 15.37, and 23.44 µm, respectively. The corresponding maximum SRPs are 238, 257, 73, 100, 216, 60/633, 548, 208, 64, and 512, respectively. The relatively small maximum OPDs result in small SRPs for these systems. To enhance the SRP, the signal padding method can be used, where incoherent regions on the detector are filled with zeros. The technique is particularly effective for systems such as WP-FTS-SP, which achieves an SRP of 238 despite having an OPD of only 19 µm. This SRP is equivalent to the WP-FTS-NSP, which has no signal padding but a larger OPD of 123 µm. Another approach is the SM-FTS, which utilizes stepped mirrors to generate nearly 500 OPDs. The maximum OPD for this system is 6.3 cm, corresponding to an impressive SRP of 65&#;000. The Ma-PWS, on the other hand, employs 200 Mach&#;Zehnder waveguide arrays to form an interferometer. It achieves a maximum OPD of 7.3 cm and a corresponding SRP of 54&#;620.
The F&#;P tunable filter is known for its extremely narrow bandwidth, enabling spectrometers such as TAURUS, GFPI, DKIST-VTF, and SPFPS to achieve high SRPs. Specifically, these spectrometers have SRPs of 714&#;285, 250&#;000, 100&#;000, and 800&#;000, respectively. The bandwidth of the AOTF is limited by the ultrasonic frequency. The SRP of an AOTF is inversely proportional to the dispersion constant, the acousto-optic interaction length, and the central wavelength of the light. Consequently, the SRP of the VOTF is significantly lower than that of a tunable F&#;P filter spectrometer due to several limitations. For example, the CE4-AOTF and SPICAVS have SRPs of 250 and 326, respectively. Liquid crystal tunable spectrometers utilize the electronically induced birefringence of liquid crystals. The SRP of such a spectrometer is directly proportional to the maximum birefringence and the thickness of the liquid crystal. For instance, the LCPS utilizes a nematic liquid crystal with a thickness of 22 µm and a bandwidth filter with an FWHM of 9 nm. This configuration enables the LCPS to achieve an SRP of .
The reconstructive spectrometer offers several advantages over traditional spectrometers as it is not limited by the size and clear aperture of optical elements. Additionally, it can achieve spectral resolutions equivalent to those of traditional spectrometers. The spectral resolution of a reconstructive spectrometer is primarily determined by the spectral interval of the spectral mapping intensity signals. For instance, the SNS has a spectral resolution of 15 nm at a wavelength of 570 nm, which is approximately equal to the spectral interval of the adjacent photodetector unit. The MQWH spectrometer achieves an SRP of 4&#;6 nm due to 16 different spectral encoders within a bandwidth of 245 nm, with each encoder covering about 15 nm. According to the Rayleigh criterion, the spectral interval is close to the spectral resolution. In the case of the PC-slabs, the spectral resolution is 1 nm due to 100 pixels with different spectral responses within a bandwidth of 200 nm. The FPMCs achieves a spectral resolution of 2 nm with 50 F&#;P microcavity pixels, corresponding to 400 InGaAs detectors with different spectral responses within an 800 nm bandwidth. The PCRS achieves a spectral resolution of 2 nm by utilizing 101 sample points within a bandwidth of 100 nm. The FCIS achieves a spectral resolution of 0.3 fm as the sparkle wavemeter can identify wavelength modulation with an amplitude of 0.3 fm. The MSCS has a spectral resolution of 0.75 nm, determined by the half-width at half-maximum (HWHM) of 0.6 nm, which depends on the required change in wavelength to generate an uncorrelated intensity distribution on the detectors. The BPS achieves a spectral resolution of 90 nm as the reconstructed laser spectrum has 81 photocurrent sampling points within a 3 µm bandwidth. The OTDR-RS achieves a spectral resolution of 0.4 fm, which is inversely proportional to the probe pulse width. With a probe pulse width of 2 µs, the spectral resolution increases 15 times to 0.4 fm using the correlation coefficient method. Finally, the CMTRS achieves a spectral resolution of 40 am, determined by the length of the single-mode fiber (SMF).
Furthermore, the SRP of a spectrometer can be significantly improved by incorporating gratings or F&#;P étalons. For instance, the MZ-SHS utilizes a diffraction grating instead of a mirror in the M&#;Z interferometer. With a maximum OPD of 1 cm, it achieves an impressive SRP of 34&#;705, surpassing that of a regular Mach&#;Zehnder spatially modulated spectrometer. The SAFOS system inserts a transmission grating into the Sagnac interferometer, reaching an SRP of . The CUSRIS combines a tunable F&#;P interferometer and a spatially modulated static grating interferometer (SGI), resulting in an ultra-high SRP of 1&#;000&#;000. Similarly, the UHRIS combines a tunable F&#;P interferometer and a spatially modulated stepper mirror interferometer (SMI), achieving the same ultra-high SRP of 1&#;000&#;000. These systems are equivalent to a standard time-modulated Michelson interferometer with a maximum OPD of 100 cm. The ACS-NIR combines an AOTF with an echelle grating, resulting in an SRP of 27&#;000. In the case of the BLC-FTS, the insertion of an F&#;P cavity into the optical path increases the SRP from 73 to 633.
Generally, traditional spectrometers can be roughly ranked according to their SRPs from high to low as follows: F&#;P tunable filter, echelle grating, time-modulated Michelson interferometer, liquid crystal tunable filter, acousto-optic tunable filter, convex grating, concave grating, ordinary blazed grating, spatially modulated Fourier transform, and prism spectrometer. The SRPs of spectral response reconstructive spectrometers typically range from tens to hundreds, which is comparable to prism spectrometers, spatially modulated interferometric spectrometers, and convex grating spectrometers. However, reconstructive spectrometers are usually much smaller in size compared to traditional spectrometers. Spectral-to-spatial reconstructive spectrometers often achieve ultrahigh SRPs. For instance, the FCIS has an SRP of 2.6 × 109, the OTDR-RS has an SRP of 3.875 × 109, and the CMTRS has an SRP of 3.875 × . These spectrometers employ wavelength-dependent speckle and wavelength modulation technologies with ultra-small amplitude. The MTLMS, based on microtaper leaky mode, achieves an SRP of over 450&#;000, comparable to that of an F&#;P tunable-based spectrometer or a time-spatially modulated Michelson interferometer, such as the CUSRIS and UHRIS. The MSCS, which is based on optical random scattering, achieves an SRP of , comparable to an AOTF or liquid crystal tunable filter spectrometer.
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