MODULE 6-8

LENSES

(4) Before you begin this module, you should have studied the principles of refraction and ray tracing optical components.

1. Explain these terms, concepts or instruments. You should convey the same meaning as given in the text.

a. Converging lens

b. Diverging lens

c. Power of a lens

d. Focal length of a lens

e. Lens systems

f. Collimation

g. Spherical aberration

h. Chromatic aberration

i. Coma

j. Astigmatism

k. Distortion

l. Field curvature

m. AR coatings

n. Damage threshold

2. From a cross-sectional diagram, identify the following lenses:

a. Biconvex

b. Plano-convex

c. Convex meniscus

d. Biconcave

e. Plano-concave

f. Aspherical

g. Cylindrical

h. Fresnel

3. Experimentally determine the focal length of a concave lens and a convex lens.

4. Construct a simple beam expander using two positive lenses.

5. Illustrate spherical aberration in lenses and a way to read it.

6. Write an explanation of the effects of spherical aberration on a camera.

 

DISCUSSION

(6) Lenses are used to change the direction of rays of light. Figure 1 shows how lenses form images by refraction. The effect of a lens on light is embodied in Snell’s law—the law of refraction. This law states that, in passing from a rarer medium into a denser one, light is refracted toward the normal. In passing from a denser to a rarer medium, light is refracted away from the normal. The degree of bending or refracting is in accordance with the equation:

Fig. 1
Image formation by a simple lens

1. Converging lenses also are called "positive," "plus," or "convex" lenses. They are thicker in the middle than at the edges. As shown in Figure 3, they cause both parallel rays of light and diverging rays of light on the opposite side of the lens.

Fig. 2
Evolution of lenses from basic prism pairs

Fig. 3
Effect of positive (converging) lenses on plane and diverging spherical wave fronts

2. Diverging lenses are known as "negative," "minus," or "concave" lenses. They are thinner in the middle than at the edges. Figure 4 shows us that a diverging lens causes parallel rays of light to diverge or spread in opposite directions on the other side of the lens. If rays initially are diverging toward such a lens, they will be made to diverge even more strongly after they pass through the lens. (See Figure 4b.)

Fig. 4
Effect of negative (diverging) lenses on plane and diverging spherical wave fronts

(9) All lenses can be classified as converging or diverging. But further subdivisions of these two basic types can be made according to the curvature of the lens surfaces and the power of the lens. Spherical lenses are lenses whose surfaces are spherical in shape. Spherical lenses can be classified into the six sub-types shown in Figure 5. The biconvex lens—"i"—is used the most of these six.

Fig. 5
Types of simple spherical lenses

(11) The focal length of a "thin lens" is the distance the focal point is removed from the vertical centerline of the lens. The focal point F’ (Figure 6) of a positive lens is that point where parallel rays of light from the left are brought together. The focal point F on the left side of the positive lens is that point to which parallel rays, incident on the lens from right-to-left, would converge.

Fig. 6
Parallel rays of light brought to a focus by a positive thin lens

(12) The focal point F (Figure 7) of a negative lens is the point where parallel rays of light seem to originate. The focal point F ' on the right side of the lens is the point from which light would seem to diverge, after passing through the lens, if it were incident on the lens from right-to-left.

Fig. 7
Parallel rays of light incident on a negative thin lens diverge
as if coming from a point F on the left.

(13) Focal lengths are shown as f ' and f in Figures 6 and 7. For a thin lens, f and f ' are equal.

(16) Figure 8 shows how a cylindrical lens focuses incident parallel light onto a focal line rather than onto a focal point as in the spherical lens. Cylindrical lenses are used in optical data processing and in other applications where we don’t need the formation of an image in true dimensions. Cylindrical lenses sometimes are used also to image the rectangular mode shape of a pulsed nitrogen laser into a straight line. For example, they would be used in pumping an organic dye laser cell.

Fig. 8
Cylindrical lens

(17) A third type of lens that’s classified by its shape is the Fresnel lens. The Fresnel lens looks different from the spherical and cylindrical lenses. A Fresnel lens is a disk-shaped optical component with almost no thickness. It focuses and collimates light by means of a series of concentric (like a bull’s-eye) grooves. Curvature of the grooves causes refraction of the incident beam, approximating that of a conventionally shaped lens as shown in Figure 9.

Fig. 9
Comparison of the shape of a normal lens to that of a Fresnel lens

(18) Figure 10 shows Fresnel lenses used as a collimator and as a collector. You can understand the effect of a Fresnel lens on light by again constructing the lens from many small prisms. The difference between conventional and Fresnel lenses is that the Fresnel lens has prisms laid out in a circular pattern in the same plane instead of being stacked together to form a regular lens.

Fig. 10
Action of Fresnel lens

(21) The lens of the eye is made of a transparent, flexible medium with attached muscle so that the lens can be made to change its shape (and consequently its focal length). The power of the lens thus is adjusted to focus objects located at different distances from the eye onto the back of the eye where the images eventually are interpreted by the brain. This changing of power is called accommodation of the eye. It’s shown in Figure 11.

Fig. 11
Accommodation of the eye

(24) The simple magnifier is a single lens that lets us see objects that are too small to be seen clearly by the unaided eye. To see a small object distinctly with the unaided eye, we have to place the object as near to the eye as possible. However, as we said in the last paragraph, the normal eye can focus clearly only on objects no closer than approximately 10 inches. The simple magnifier shown in Figure 12 is an "optical instrument" that, when focused on a nearby tiny object, produces a magnified, virtual image at a convenient distance for clear vision.

Fig. 12
Virtual image formed by positive lens acting as a magnifier

(25) A searchlight is an optical device that collects light from an intense "point" source and transmits this light as a nearly parallel, well-collimated beam. In many cases this is accomplished with parabolic mirrors. But in principle it can be done with a large lens as shown in Figure 13.

Fig. 13
Simple example of a searchlight

(26) The small source in Figure 13 is shown exaggerated in size for clarity. It’s made up of points 1, 2, 3. Since the source lies in the focal plane of the lens, each point on the source leads to a beam of parallel light leaving the lens. For example, point 1 gives the parallel bundle  ,  ',  ". Point 2 gives the parallel bundle ‚ , ‚ ', ‚ ". Point 3 gives the parallel bundle ƒ , ƒ ', ƒ ". You can see that, if the source becomes very small and point 2 and 3 move toward point 1, nearly all of the rays will be parallel to rays  ,  ',  ". The beam then is said to be well-collimated, progressing to the right with very little spreading or divergence.

(27) Now consider the applications shown in Figure 14. Figure 14a shows the essential geometry of a motion-picture projector, where the object is located a distance more than one focal length (but less than two focal lengths) from the lens and is projected as a real, inverted, magnified image. If the object is moved to exactly two focal lengths from the lens, a real inverted image of the same size is formed two focal lengths on the other side of the lens. This scheme for one-to-one reproduction is used in a copy machine as shown in Figure 14b.

Fig. 14
Other applications of the positive lens

(30) Concave and convex meniscus lenses are used to correct myopia (nearsightedness) and hyperopia (farsightedness). The concave and convex meniscus lenses are used instead of regular concave or convex lenses to enable the eye to scan a large field of view and yet always be the same distance from the meniscus lenses. Figure 15 illustrates the operation of a normal eye, the problem with the nearsighted and farsighted eye, and the correction introduced with the appropriate meniscus lenses.

Fig. 15
Vision defects and corrections

Uses of Lens Systems

(31) Some uses for the single lens have been mentioned already. Complex optical tasks generally require the use of two or more lenses in an optical system. One such well-known lens system is the telescope, shown in Figures 16 and 17. The Galilean telescope, shown in Figure 16, gives an upright image of a distant object.

Fig. 16
Galilean telescope

(34) If we assume the distant object to be an arrow with the arrowhead upward, the image is real and inverted, as shown. The real image, viewed by the eye through the eyepiece lens system, is seen as a magnified, virtual image as shown by the dotted lines. The image, compared with the original object, is inverted. But this is not important when we look at stars, for example.

Fig. 17
Astronomical or celestial telescope

(36) One use of a telescope of this type in laser work is for the function of beam expansion. As shown in Figure 18, if a beam of light from the laser is used to illuminate the eyepiece and therefore is sent through the telescope "backward," it will emerge from the objective lens larger in diameter than when it entered and the rays will be somewhat more collimated or parallel.

Fig. 18
Principle of beam collimation and expansion

(37) As shown in Figure 19, there are two different ways to produce beam expansion or beam reduction. The first method (Figure 19a) uses a convex lens followed by another convex lens. Note that the focal point F₁ of lens L₁ falls on the focal point F₂ of lens L₂. This beam expander/collimator often is called the Keplerian beam expander. It and the one shown in Figure l9b have names derived from alleged inventors of refracting telescopes that closely resemble the beam expanders.

Fig. 19
Two common types of beam expanders/collimators

(40) For either version of the two-lens beam expander, we can define the lateral magnification as

(41) Optics texts show that the lateral magnification of the beam expander is given simply by the ratio of the focal lengths,

(42) We also can show that, to a good approximation, the divergence angle of the expanded beam is related to the divergence angle of the incident unexpanded beam by,

(45) Another use of lens systems is in the compound microscope. The microscope is very similar to the telescope except that the objective lens now has a much smaller focal length. So it has a definite focal point close to the front lens. In the compound microscope shown in Figure 20, the objective lens forms an enlarged image I₁ of the small object at O.

Fig. 20
Compound microscope

(46) The eye—viewing the image through the eyepiece—sees a much enlarged image according to the principles of the simple magnifier. Since the apparent size of the image is larger than the actual size of the object, the angle subtended at the eye is larger. So, there is magnification. It’s interesting to note that placing a short-focal-length lens in the front of a telescope changes it to a microscope. This is shown in Figure 21.

Fig. 21
Converting a telescope into a microscope

Problems in Lenses

(48) Aberration can be defined as a faulty image formation caused by failure of all the incident light rays from a point on the source to focus as the corresponding point image in the image plane. The types of defects or aberrations in the image that we will discuss are classified as:

Chromatic aberration		occurs in first order for light that is not monochromatic
Spherical aberration Coma Astigmatism Distortion Field curvature		third-order "primary" or "Seidel" aberrations

(53) Although a mathematical description of these effects is not appropriate here, we will discuss briefly the five Seidel aberrations and the unrelated effect called "chromatic aberration." We also should note that modern lens-design software that runs on microcomputers is accurate to fifth order. It takes into account even higher-order aberrations than those studied by Seidel. Much of the task of modern optical engineers is minimizing these third-, fifth- and higher-order effects to produce exceptionally high-quality images, while keeping the size, weight and cost of the optical system within reason.

(55) Some photographic emulsions are most sensitive to light at the blue end of the spectrum. A lens with chromatic aberration may form an image that appears sharp to the eye, but it produces a blurred, out-of-focus image on the film negative. Figure 22 shows chromatic aberration. In the figure, white rays from a point object are split up inside the lens and come to separate foci that depend for their position on wavelength. Wavelengths range from about 4200 angstrom units at the blue end to 7500 angstrom units for the red end of the spectrum.

Fig. 22
Chromatic aberration

(56) The best correction for chromatic aberration is the use of a negative lens, made of a glass with a different index of refraction, in combination with a positive lens. The projection system shown in Figure 23 uses lenses, taken together, to correct for chromatic aberration.

Fig. 23
Achromatic projection lens system

(57) You saw in Figure 14a that a single biconvex lens can be used as a projection lens. In practice, however, a projection lens consists of a pair of lenses that are corrected for chromatic aberration. Since the projected image is generally large compared to the object, aberrations (defects in perfect focus) will be quite noticeable. Figure 23 shows that a magnified image will be formed on a distant screen if a well-illuminated object or real image is placed in the focal plane of a projection lens.

(58) Rays that come from an object on the axis of a lens and pass through a portion of the lens near the center come to a focus at a certain point on the axis of the lens. Rays from this same object point that pass through the lens near the edges of the lens should come to a focus at the same point, if a good image is to be formed. In practice, because of spherical aberration, the marginal rays tend to come to a different point than do the "center" rays. The difference between these focal points is caused by spherical aberration of the lens. Longitudinal spherical aberration occurs when the marginal and paraxial rays passing through a positive lens fail to meet at a common focal point.

Fig. 24
Spherical aberration

(59) Spherical aberration is shown in Figure 24 for a bundle of parallel rays, all originating from an on-axis point of a distant object. Spherical aberration can be corrected with a diaphragm in front of or behind the lens that cuts out the marginal ray. Another possible correction occurs with a specially ground meniscus lens.

(60) This type of aberration is restricted to "off-axis" image points. In some respects we can consider it as spherical aberration of an oblique bundle of rays. Since the lens has no symmetry about a line passing through the center of the lens and an off-axis point in the image, the effects of coma are complex and unsymmetrical. Coma is shown by Figure 25. (No hard rule for elimination of coma can be laid down.) It might help to cover part of the lens with a diaphragm. If this fails the only choice is to discard the faulty lens.

Fig. 25
Coma

(61) Astigmatism is the inability of a lens to bring all the rays from a vertical plane to the same point as the rays from a horizontal plane. This problem is shown is Figure 26. Astigmatism can be improved with special lens shapes and by reducing the lens aperture, thus eliminating the offending marginal rays.

Fig. 26
Astigmatism

(62) Distortion is the magnification of image points near the optical axis more (or less) than the magnification of image points at off-axis positions. Distortion causes parallel lines to bow and images to lose their proportion. When axial magnification is greater than off-axis magnification, the lens has positive or barrel distortion. When axial magnification is less, it has negative or pincushion distortion. The effects of these types of distortion are shown in Figure 27 for a uniform and parallel grid image.

Fig. 27
Effects of distortion on squared patterns

(63) Distortion does not depend on lens aperture. It can be corrected only with specially ground lenses or filter optics.

(65) To a good approximation, for spherical lenses, the image "surface" is paraboloidal in shape. This condition is unacceptable, for example, in the case of a 35-mm slide projector, where the object is a plane (the slide) and the image usually also is focused on a plane (the screen). If just a simple, spherical lens were used to project the image, the center and the edge of the screen could not be focused simultaneously.

Fig. 28
Field curvature. Object in flat image plane focuses into curved image surface.
Not drawn to scale.

(66) To minimize this problem, combinations of positive and negative lenses usually are used. A typical high-quality 16-mm movie projector lens is shown in Figure 29. The negative elements of the lens combine with the positive focusing lenses to form a so-called "field flattener," that greatly minimizes the curvature of field.

Fig. 29
High-quality projector compound lens containing field-fattening elements. (Hecht)

Reducing Aberrations

(68) Dramatic reduction in some aberrations can be achieved by simply turning the lens around (method #1). Reducing the aperture or "stopping down" a system usually will cause a significant reduction in aberrations (except curvature of field), but with a concurrent loss in image intensity.

Optical Coatings

(69) An uncoated lens with an index of refraction of 1.50 will reflect about 4% of incident light, normal to each surface. In some applications, particularly where a large number of optical elements lie in the optical path, such loss may be unacceptable. To reduce reflective losses, optical elements generally are coated with one or more layers of transparent material. Figure 30 shows the effect of coating an optical substrate with an antireflection (AR) coating.

Fig. 30
Ray trace of a single layer AR coating. The thickness t of the coating is chosen such that the rays reflecting from interfaces (1) and (2) are 180° out of phase.

(71) A simple analysis shows that, for light striking the surfaces at normal incidence, the minimum thickness t of the coating should be given by

(72) Maximum cancellation occurs if the two reflected waves are of the same amplitude. If we assume that little absorption occurs in the coating, the reflected rays will be equal in amplitude if,

(73) For a typical case of a flint (n₂ = 1.72) glass lens surrounded by air, the ideal coating index should be

(75) A single layer of MgF₂ can greatly reduce reflections from a surface, as shown in Figure 31. Note that the effectiveness of the AR coating depends dramatically on the index of refraction of the substrate material.

Fig. 31
Reflectance of coated surface of index n_g when coated with a
quarter wavelength of magnesium fluoride (index = 1.38). (Meiles Griot)

(76) Figure 32 shows the effect of wavelength on the reflectance from an optical surface with a single AR coating layer. Although this coating is optimized for 550 nm, it’s surprisingly insensitive to incident wavelength. The effectiveness of MgF₂ as an antireflection coating is increased dramatically with increasing refractive index of the component material (see Figure 31). This means that, for use on high-index materials, there is often little point in using more complex coatings.

Fig. 32
Reflectance curve for MgF₂ on BK 7 optical glass. (Melles Griot)

(78) Particularly when used with high-energy pulsed lasers, you must consider the damage threshold of optical coatings. The probability of damage occurring to the AR coating depends on a number of factors including: the coating and substrate material, defects or contaminants in the coating or substrate, polarization of the incident light, wavelength of the light, and peak value of the electric field caused by the incident light. Lens manufacturers usually will specify the nominal damage threshold for coated and uncoated optics that they supply. Modern AR coatings can be very resistant to laser-induced damage. For example, a hard single-layer oxide coating typically will withstand 5 joules/cm² in 1-nsec pulses from an Nd:YAG laser irradiating the surface at an incident angle of 45°. This represents an irradiance of 5 GW/cm².

1. Explain the following terms, concepts, or instruments. Your explanations should convey the same meanings given in the text.

a. Converging lens

b. Diverging lens

c. Power of a lens

d. Focal length of a lens

e. Lens systems

f. Collimation

g. Spherical aberration

h. Chromatic aberration

i. Coma

j. Astigmatism

k. Distortion

2. From the given cross-sectional diagrams identify the following lenses:

a. ________________ b. ________________

c. ________________ d. ________________

e. ________________ f. ________________

g. ________________ h. ________________

3. Explain, in your own words, the procedure for experimentally determining the focal length of a concave and a convex lens.

4. When working with high-power lasers, why is it necessary to use a beam expander made of a negative and a positive lens rather than one made from two positive lenses? Use accurately drawn ray diagrams to help explain your answer.

5. Using an accurately drawn diagram illustrate the difference between spherical aberration and chromatic aberration.

6. Depth of field is the range of distances over which a camera is in satisfactory focus. Use the concept of spherical aberration to explain why a camera has better depth of field when the lens aperture is small than it has when the lens aperture is large.

7. A 1.00-watt argon laser with an exit-beam diameter of 2.00 mm and a full-angle beam divergence of 1.20 mrad is shined onto a target 3.00 km away. Neglecting atmospheric loss, what is the average irradiance on the target? The output beam of the laser is directed into a Galilean beam expander comprised of a negative lens (f = –7.00 mm) and a positive lens (f = +25.0 cm). Assume that the overall transmission of the bearn expander/collimator is 92% and that it is focused properly. Calculate the average irradiance on the distant target.

8. The reflectance from a transparent surface is given by

where T is the fractional amount of light reflected from the interface and n is the index of refraction of the transparent medium. (The other side of the interface is assumed to have an index = 1.00.) For a borosilicate crown glass lens, illuminated at normal incidence with a red HeNe laser beam, what is the reflective loss (in percent) per surface?

9. What is the thinnest coating of MgF₂ that can be applied to a borosilicate crown glass lens designed to work at normal incidence with a red HeNe laser beam? After the coating is applied, what is the approximate reflective loss in percent per surface? (See Fig. 31.)

10. Contact three vendors regarding the purchase of the following lens:

Type: Plano-convex

Material: Any high-quality optical glass transparent in the visible.

Focal length +20 cm.

Diameter: At least 2 cm.

Coating: Single or multilayer, "tuned" for use with the ruby laser line.

Obtain as much information as you can regarding the lenses, surface and bulk material properties, aberrations, AR coating efficiency, coating damage threshold, etc., and of course, cost.

If you were planning to use this lens to focus a 5 joule ruby laser (pulse width = 250 m sec), expanded beam diameter = 2.0 cm, onto a workpiece, which lens would you select? Are there other aspects of your application that you need to know to decide?

Optical bench or table with optical mounts

2 Positive lenses (one short focal length < 5 cm, one long focal length 720 cm)

2 Negative lenses (one short focal length < 5 cm, one long focal length 720 cm)

Ground glass screen

HeNe laser

2 Optical stops (1 center stop, 1 edge stop)

Cross-line object

Illuminating lamp

Metric ruler or metric caliper

Determining the Focal Length of a Lens

(80) Set up the optical bench as shown in Figure 33. Place a convex (positive) lens of unknown focal length in the lens holder on the optical bench. Point the optical bench at some distant bright object, for instance, a lamp in a long corridor. The distance should be not less than 50 meters. Also place a ground glass screen in its holder on the optical bench and form an image of the distant object produced by the lens on this screen. The difference between the readings of the lens holder and the ground glass screen holder will be the focal length of the lens. You might have to make a number of independent readings and average them for accuracy.

Fig. 33
Arrangement to determine focal length

(81) Determination of focal length of a concave (negative) lens takes a little more effort. You first must set up the meter bench as before and slide the ground glass screen until the bright object is in focus in front of a convex lens. Write down this position of the screen (S₁). Then place the concave lens of unknown focal length between the convex lens and this position. This is shown in Figure 34.

Fig. 34
Determining focal length of a concave lens

As before, take a number of independent readings for accuracy.

The Celestial Telescope Used as a Beam Expander

(84) Shine a laser beam "backward" through the "telescope." Make any adjustments necessary to produce an expanded, collimated output beam. Record the focal lengths of the lenses and the diameters of the input and output beams. Repeat for the Galilean beam expander/collimator shown in Figure l9b.

Demonstration of Spherical Aberration and One Correction

(86) Place the lamp, the cross-line object, the convex lens, and the ground glass screen on the optical bench or table as shown in Figure 35.

Fig. 35
Arrangement to demonstrate spherical aberration

(87) Place the "center-of-lens" stop—shown in Figure 36¾ in front of the lens and move the screen until the image of the cross line is sharply in focus.

Fig. 36
Center-of-lens stop

(89) Move the ground glass screen back until the image is once again in focus. The amount of spherical aberration the lens has is indicated by the distance between these two focal points. Spherical aberration normally is corrected by using an edge-of-lens stop or a diaphragm so that only the center portion of the lens is used.

Fig. 37
Edge-of-lens stop

(90) Repeat the above procedure with a larger-diameter lens. Compare the spherical aberration of the two lenses.

Hecht, E. Optics. 2nd edition. Addison Wesley, 1987.

Jacobs, Donald H. Fundamentals of Optical Engineering. McGraw-Hill Book Co., 1943.

Jenkins, Francis A., and Harvey E. White. Fundamentals of Optics. 4th edition. McGraw-Hill Book Company, 1976.

Kingslake, Rudolf. Applied Optics and Optical Engineering. Volume III. Academic Press, Inc., 1965.

The Newport Corporation Catalog No. 100. Newport Corporation, 18235
Mt. Baldy Circle, Fountain Valley, CA 92728-8020, (714) 963-9811.

Optics Guide 3. Melles Griot, 1770 Kettering Street, Irvine, CA 92714, (714) 261-5600.

Oriel Corporation Catalog Vol III. Oriel Corporation, 250 Long Beach Blvd., Stratford, CT 06497-0872, (203) 377-8282.

Pedrotti, Frank L., and Leno S. Pedrotti. Introduction to Optics. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1987.

Pollock, H. W. Applied Physics. 2nd edition. Prentice-Hall, 1971.