We welcome our readers back to our continuing technical series on the Concepts of Light and Optics. So far we’ve examined the nature of light and how it propagates, the principles of refraction and dispersion, material characteristics, interferometry, as well as, the fundamental specifications of plano optics. Building on all of these discussions, we now move on to a topic that is at the heart of optical fabrication – lenses.
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Lenses represent the workhorse of most optical systems. Properly designed and manufactured, these essential elements manipulate light, allowing scientists to examine everything from the minute details of cellular structure using optical microscopy, to the vast overall structure of the universe using powerful ground-based and orbiting telescopes such as the Hubble.
As the combined topic of lens characteristics would produce an article of great length, we’ve decided to separate it into three sections. The first, which we will discuss today, will cover the different types of lenses and their intended purpose. In subsequent articles we will then discuss the terminology and basic equations used to characterize lenses, as well as, the types of aberrations they are designed to minimize. Finally, we’ll examine the specifications used by optical designers and fabricators to precisely define the surface figure and performance requirements.
If you recall from our discussion of refraction and dispersion, all optical materials influence light in different ways. The refractive index defines the manner in which light will bend as it enters a new medium while the dispersive index more precisely defines how each individual wavelength (or frequency) will behave. Optical engineers choose materials based on these principles, however, this is only the first step in the design process. More importantly, they then must decide on the appropriate physical shape of an optic in order to achieve its desired effect. In essence, it the final curvature of a lens in tandem with its optical properties that ultimately determines how it influences light.
All lenses adhere to one of two basic principles as a function of their shape – they either focus or disperse light. As such, lenses which focus light are called positive lenses while those that disperse light are called negative lenses. As the purpose of this article is to first introduce the basic types of lenses, let us now examine the most common lens configurations.
As their name implies, both plano-convex and biconvex lenses have either one or two surfaces with positive spherical contours. Regardless of design, the thickness at the edge of the lens is always less than the thickness at the center. These lenses are used to focus light to a pre-defined point based on the amount of curvature of their surfaces. In the case of a plano-convex design, one surface remains flat while the second has a positive curve and for biconvex lenses, both surfaces are positively curved. In practical use, plano-convex lenses are most commonly employed where the object being imaged is far away from the lens. In optical terms this is called infinite conjugate and an example of this circumstance is focusing light from a distant object such as a star. Biconvex lenses are also used to focus light but are best employed where the object being imagined is much closer to the lens. This is called finite conjugate and microscopy is a practical example.
Plano-concave and biconcave lenses can be thought of as the opposite of plano-convex and biconvex lenses. Having negative curvature means that rather than focusing light, these optical elements disperse the incident energy. By design, the thickness at the edge of the lens is always greater than the thickness at the center. These optics are often called beam expanders and the below examples demonstrate how light traveling through them is dispersed from its original path.
Meniscus lenses have one surface that is concave and one that is convex. Depending on their contour, these optics can be either positive or negative. If the curvature of the convex side is steeper than the concave side, the resulting light path is refracted towards a predetermined focal point. Conversely, if the concave curvature is greater than the convex side, the light is then dispersed. Negative meniscus lenses can therefore also be thought of as beam expanders.
Aspheric lenses differ from all other optics in regards to their shape. As shown above, typical lenses are designed to have spherical contours. Regardless of their intended use, their convex or concave shape follows a consistent regular path across the surface of the lens. In the case of aspheres, optical designers intentionally create surface contours that no longer adhere to this standard type of curvature. The resulting shape allows a single lens to refract light in a manner that otherwise would require the use several lenses placed in series. Using aspheres in this manner typically reduces both the size and overall weight of an integrated lens system, however, these benefits must be weighed against the cost to manufacture such highly complex surface contours. Aspheres are more difficult to fabricate and measure and are therefore more expensive than traditional lenses. As they can be designed with either convex or concave surfaces, aspheres can therefore be either positive or negative in their refraction of light.
Cylinder lenses are unique in their design and function. The easiest way to conceptualize their shape is to first think of a circular rod. Slicing off a section of the rod along its length produces a lens with curvature in a single direction. This one-directional shape allows optical designers to focus light along a line instead of a single point. In practical use, they are often employed in scanning systems where a linear cross section allows for faster retrieval of data, such as barcode readers and inspection equipment.
Acylinders combine the line-focus of a cylinder lens with the benefits of an aspherical design. They assist in reducing what opticians call spherical aberration (which will be covered more in-depth in our next article), while allowing the resulting focused line to be thinner and more precise.
As shown from the different examples, lenses come in a wide variety of configurations. By combining a specific optical material with various types of surface contours, light can be harnessed, steered and manipulated for a given purpose. Esco Optics manufactures all of the above types of lenses using both traditional pitch polishing techniques, as well as, CNC-guided spherical and free-form generation equipment. In addition, all metrology to certify the surface figure and shape is performed in-house.
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In our next article we will build upon these basic lens shapes by discussing the terminology and equations opticians use to define how a lens will function, i.e. focal length, radius of curvature, etc. We will also present various optical phenomena such as astigmatism, spherical aberration and coma, and relate how opticians employ various lens designs to minimize and eliminate these anomalies.
Concepts in Light and Optics – Lenses – Part 2
Concepts in Light and Optics – Lenses – Part 3
Biconvex lenses are a type of simple lens. It has a wide area of applications like controlling and focusing of laser beams, image quality and other use in optical instruments. It is also called a plano-convex lens. Here, a parallel beam of light passes through the lens and converges into a focus or spot behind the lens. Thus, the biconvex lens is also called the positive and converging lens. The distance from the lens to the spot behind the lens is called the focal length of the lens. There are two curvatures on both sides of the lens which will be around 2 focal points and 2 centres. There is a line called the principal axis which is drawn on the middle of the biconvex lens. These lenses are symmetrical lenses which have two convex lenses arranged in a spherical form. Each of these lenses has the same radius of curvature.
Images that are formed by the lenses are due to refraction of light. The convex lens is also known as the converging lens as it converges the rays coming towards its direction at a certain point. The image formed is thus, real. Thus, biconvex lenses have a wide range of usage in the optical industries and it has allowed us to capture images making it easier and more accessible.
We know that images formed by lenses are because of the reflection of light. A convex lens is a converging lens that converges the rays coming from the point object at a certain point; therefore, the image formed is real.
We know that the centre of the spherical convex lens is the optical centre. When the two spheres intersect each other in such a way that their optical centres coincide; this type of arrangement is called the biconvex lens.
Focal Point of Biconvex Lenses
From Fig.1, we can see that the red point is the optical centres of two spherical convex lenses coinciding with each other.
One thing to be noticed is that the distances PO1 and PO2 are equal. Here, O1 and O2 are the centres of curvatures of two convex lenses or biconvex lenses.
Since O1 and O2 are the centres of curvatures, i.e., PO1 and PO2 are the radii of curvatures, so PO1 = PO2. This proves that biconvex lenses have the same radius of curvatures.
The line joining the centres of curvatures of these two lenses viz: O1and O2, passing via optical centre ‘P’ is the principal axis of the biconvex lenses.
We know that a convex lens contains two focal points one is on the left-hand side and the other on the right-hand side.
The focal point lies in the centre (midpoint) of the line segment joining the optical centre and the centre of curvature. Since we are talking about biconvex, there will be two focal points viz: f1 and f2. We call these focal points the principal focus because this point lies on the principal axis.
So,
Pf1 = f1O1, so OC = 2f1. Similarly, Pf2 = f2O2, and therefore, PO2 = 2f2.
Biconvex Lens Formula
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We know that
\[ \frac{1}{f}=\frac{1}{u}−\frac{1}{v} \]
We know that the general equation for the refraction on the spherical surface is given by:
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\[ \frac{\mu_2}{v} - \frac{\mu_2}{u} = \frac{\mu_2 - \mu_1}{R} \]
Now, considering Fig. (2) to frame the biconvex lens formula:
After the first refraction, the image is formed at O1. So the equation becomes:
\[ \frac{\mu_{2}}{v} - \frac{\mu_{2}}{u} = \frac{\mu_2- \mu_1}{R_1} \]
After the second refraction, the image is formed at I. So the equation for the light ray coming from the medium \[μ_2\] to the medium \[μ_1\] is:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{v_2} \] = \[ \frac{\mu_2 - \mu_1}{R_2} \] \[ \frac{\mu_2 - \mu_1}{R_1} \]
Adding eq (a) and (b), we get:
\[ \frac{1}{v} - \frac{1}{u} = \left ( \frac{\mu_2}{\mu_1} - 1 \right ) \left ( \frac{1}{R_1} - \frac{1}{R_2} \right ) \]
As the light rays come from the object placed at infinity, so u = ∞ and v = f.
So, from eq (1) and (c), we get the biconvex lens formula as:
\[ \frac{1}{f} = \left ( \frac{\mu_2}{\mu_1} - 1 \right ) \left ( \frac{1}{R_1} - \frac{1}{R_2} \right ) \]
Here,
\[R_1\] = radius of curvature of lens 1
\[R_2\] = radius of curvature of lens 2
How to Make Biconvex Lenses?
Take cardboard and cut out a circle of diameter 2.5 cm.
Now, place this circle on the plastic water bottle. Draw its shape, and cut out two circles from the bottle with the help of scissors.
Join these circles:
For joining these circles take a glue gun. Join their edges by leaving a space in :
between these two, as shown in the image below
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Take a bowl of water and let the joined circle dip into the water. We can see that some water fills in between the space of these two circles. Now, again stick the left space with glue.
Now, take a handwritten notes register and read the sentences with the help of this lens. You will notice that word-to-word in each sentence will appear magnified.
Uses of Biconvex Lens
We use Biconvex lenses in our day-to-day life, as we discussed in the ‘how to make a biconvex lens’ section where this lens works as a magnifying glass to observe the small letters. Now, let’s see some more applications of biconvex lenses:
We talk about many imaging systems like microscopes, telescopes, binoculars, projectors, etc; however, all these systems use biconvex lenses for obtaining images.
A microscope uses biconvex lenses for imaging the things that are not visible to naked eye. For example, to determine the cellular structures of organs, germs, bacteria, and other microorganisms.
Telescopes use biconvex lenses to observe distant objects by their emission, electromagnetic radiations, and absorption. This helps to determine the temperature of the stars.
We can use biconvex lenses as burning glasses.
Biconvex lenses are found in the natural camera viz: the human eye, where they produce virtual images.
Biconvex lenses are positive lenses, and they are best-used for converging beams that are diverging in nature.
We find the applications of biconvex lens industries and also in image relays.
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