Introduction. THE OPTICAL ENGINEERING PROCESS. Engineering Support. Fundamental Optics

Transcription

1 Introduction The process o solving virtually any optical engineering problem can be broken down into two main steps. First, paraxial calculations (irst order) are made to determine critical parameters such as magniication, ocal length(s), clear aperture (diameter), and object and image position. These paraxial calculations are covered in the next section o this chapter. THE OPTICAL ENGINEERING PROCESS Optical Speciications Material Properties Second, actual components are chosen based on these paraxial values, and their actual perormance is evaluated with special attention paid to the eects o aberrations. A truly rigorous perormance analysis or all but the simplest optical systems generally requires computer ray tracing, but simple generalizations can be used, especially when the lens selection process is conined to a limited range o component shapes. In practice, the second step may reveal conlicts with design constraints, such as component size, cost, or product availability. System parameters may thereore require modiication. Because some o the terms used in this chapter may not be amiliar to all readers, a glossary o terms is provided in Deinition o Terms. Finally, it should be noted that the discussion in this chapter relates only to systems with uniorm illumination; optical systems or Gaussian beams are covered in. Engineering Support CVI Melles Griot maintains a sta o knowledgeable, experienced applications engineers at each o our acilities worldwide. The inormation given in this chapter is suicient to enable the user to select the most appropriate catalog lenses or the most commonly encountered applications. However, when additional optical engineering support is required, our applications engineers are available to provide assistance. Do not hesitate to contact us or help in product selection or to obtain more detailed speciications on CVI Melles Griot products. Determine basic system parameters, such as magniication and object/image distances Using paraxial ormulas and known parameters, solve or remaining values Pick lens components based on paraxially derived values Determine i chosen component values conlict with any basic system constraints Estimate perormance characteristics o system Determine i perormance characteristics meet original design goals.

2 Paraxial Formulas Sign Conventions The validity o the paraxial lens ormulas is dependent on adherence to the ollowing sign conventions: For lenses: (reer to igure.) s is = or object to let o H (the irst principal point) s is 4 or object to right o H Figure. h ront ocal point object v H H CA F s principal points rear ocal point Note location o object and image relative to ront and rear ocal points. = lens diameter CA = clear aperture (typically 90% o ) = eective ocal length (EFL) which may be positive (as shown) or negative. represents both FH and H F, assuming lens is surrounded by medium o index.0 m = s /s = h /h = magniication or conjugate ratio, said to be ininite i either s or s is ininite v = arcsin (CA/s) Sign conventions F For mirrors: When using the thin-lens approximation, simply reer to the let and right o the lens. is = or convex (diverging) mirrors is 4 or concave (converging) mirrors s is = or image to right o H (the second principal point) s is = or object to let o H s is 4 or image to let o H s is 4 or object to right o H m is = or an inverted image s is 4 or image to right o H m is 4 or an upright image s is = or image to let o H m is = or an inverted image m is 4 or an upright image s h = object height h = image height image s = object distance, positive or object (whether real or virtual) to the let o principal point H s = image distance (s and s are collectively called conjugate distances, with object and image in conjugate planes), positive or image (whether real or virtual) to the right o principal point H h.3 Optical Speciications Material Properties

3 Typically, the irst step in optical problem solving is to select a system ocal length based on constraints such as magniication or conjugate distances (object and image distance). The relationship among ocal length, object position, and image position is given by object F image = s + s (.) F Material Properties Optical Speciications This ormula is reerenced to igure. and the sign conventions given in Sign Conventions. By deinition, magniication is the ratio o image size to object size or This relationship can be used to recast the irst ormula into the ollowing orms: where (s=s ) is the approximate object-to-image distance. With a real lens o inite thickness, the image distance, object distance, and ocal length are all reerenced to the principal points, not to the physical center o the lens. By neglecting the distance between the lens principal points, known as the hiatus, s=s becomes the object-to-image distance. This simpliication, called the thin-lens approximation, can speed up calculation when dealing with simple optical systems. Example : Object outside Focal Point A -mm-high object is placed on the optical axis, 00 mm let o the let principal point o a LDX C ( = 50 mm). Where is the image ormed, and what is the magniication? (See igure..) or real image is 0.33 mm high and inverted..4 m s h = =. s h ( m s + = s ) ( m + ) sm = m + s+ s = m + + m sm ( + ) = s+ s s = s s = s = mm 66 m = s. 7 = = 033. s 00 (.) (.3) (.4) (.5) (.6) Figure. Example : Object inside Focal Point The same object is placed 30 mm let o the let principal point o the same lens. Where is the image ormed, and what is the magniication? (See igure.3.) s = s = 75 mm or virtual image is.5 mm high and upright. In this case, the lens is being used as a magniier, and the image can be viewed only back through the lens. Figure.3 Example 3: Object at Focal Point Example ( = 50 mm, s = 00 mm, s =66.7 mm) 75 m = s = = 5. s 30 image F F object Example ( = 50 mm, s = 30 mm, s = 475 mm) A -mm-high object is placed on the optical axis, 50 mm let o the irst principal point o an LDK C ( =450 mm). Where is the image ormed, and what is the magniication? (See igure.4.) = s s = 5 mm 5 m = s = = 05. s 50 or virtual image is 0.5 mm high and upright.

4 A simple graphical method can also be used to determine paraxial image location and magniication. This graphical approach relies on two simple properties o an optical system. First, a ray that enters the system parallel to the optical axis crosses the optical axis at the ocal point. Second, a ray that enters the irst principal point o the system exits the system rom the second principal point parallel to its original direction (i.e., its exit angle with the optical axis is the same as its entrance angle). This method has been applied to the three previous examples illustrated in igures. through.4. Note that by using the thin-lens approximation, this second property reduces to the statement that a ray passing through the center o the lens is undeviated. F-NUMBER AND NUMERICAL APERTURE The paraxial calculations used to determine the necessary element diameter are based on the concepts o ocal ratio (-number or /#) and numerical aperture (NA). The -number is the ratio o the ocal length o the lens to its eective diameter, the clear aperture (CA). To visualize the -number, consider a lens with a positive ocal length illuminated uniormly with collimated light. The -number deines the angle o the cone o light leaving the lens which ultimately orms the image. This is an important concept when the throughput or light-gathering power o an optical system is critical, such as when ocusing light into a monochromator or projecting a high-power image. The other term used commonly in deining this cone angle is numerical aperture. The NA is the sine o the angle made by the marginal ray with the optical axis. By reerring to igure.5 and using simple trigonometry, it can be seen that and object F Figure.4 image -number =. CA NA = sinv = Example 3 ( = 450 mm, s = 50 mm, s = 45 mm) CA NA = ( -number). F (.7) (.8) (.9) CA Figure.5 principal surace F-number and numerical aperture Ray -numbers can also be deined or any arbitrary ray i its conjugate distance and the diameter at which it intersects the principal surace o the optical system are known. NOTE Because the sign convention given previously is not used universally in all optics texts, the reader may notice dierences in the paraxial ormulas. However, results will be correct as long as a consistent set o ormulas and sign conventions is used. v.5 Optical Speciications Material Properties

5 Material Properties Optical Speciications Imaging Properties o Lens Systems THE OPTICAL INVARIANT To understand the importance o the NA, consider its relation to magniication. Reerring to igure.6, CA NA (object side) = sinv = s CA NA (image side) = sinv = s which can be rearranged to show and CA = s sinv CA = s sinv leading to s sinv = v = NA. s sin NA s Since is simply the magniication o the system, s we arrive at m = NA NA. (.0) (.) (.) (.3) (.4) (.5) The magniication o the system is thereore equal to the ratio o the NAs on the object and image sides o the system. This powerul and useul result is completely independent o the speciics o the optical system, and it can oten be used to determine the optimum lens diameter in situations involving aperture constraints. When a lens or optical system is used to create an image o a source, it is natural to assume that, by increasing the diameter () o the lens, thereby increasing its CA, we will be able to collect more light and thereby produce a brighter image. However, because o the relationship between magniication and NA, there can be a theoretical limit beyond which increasing the diameter has no eect on light-collection eiciency or image brightness. Since the NA o a ray is given by CA/s, once a ocal length and magniication have been selected, the value o NA sets the value o CA. Thus, i one is dealing with a system in which the NA is constrained on either the object or image side, increasing the lens diameter beyond this value will increase system size and cost but will not improve perormance (i.e., throughput or image brightness). This concept is sometimes reerred to as the optical invariant. Example: System with Fixed Input NA Two very common applications o simple optics involve coupling light into an optical iber or into the entrance slit o a monochromator. Although these problems appear to be quite dierent, they both have the same limitation they have a ixed NA. For monochromators, this limit is usually expressed in terms o the -number. In addition to the ixed NA, they both have a ixed entrance pupil (image) size. Suppose it is necessary, using a singlet lens, to couple the output o an incandescent bulb with a ilament mm in diameter into an optical iber as shown in igure.7. Assume that the iber has a core diameter o 00 mm and an NA o 0.5, and that the design requires that the total distance rom the source to the iber be 0 mm. Which lenses are appropriate? By deinition, the magniication must be 0.. Letting s=s total 0 mm (using the thin-lens approximation), we can use equation.3, to determine that the ocal length is 9. mm. To determine the conjugate distances, s and s, we utilize equation.6, and ind that s = 00 mm and s =0 mm. We can now use the relationship NA = CA/s or NA =CA/s to derive CA, the optimum clear aperture (eective diameter) o the lens. With an image NA o 0.5 and an image distance (s ) o 0 mm, and ( m s + = s ), ( m + ) (see eq..3) s( m+ ) = s+ s, (see eq..6) 05. = CA 0 CA = 5 mm. Accomplishing this imaging task with a single lens thereore requires an optic with a 9.-mm ocal length and a 5-mm diameter. Using a larger diameter lens will not result in any greater system throughput because o the limited input NA o the optical iber. The singlet lenses in this catalog that meet these criteria are LPX C, which is plano-convex, and LDX C and LDX C, which are biconvex. SAMPLE CALCULATION To understand how to use this relationship between magniication and NA, consider the ollowing example..6 Making some simple calculations has reduced our choice o lenses to just three. The ollowing chapter,, discusses how to make a inal choice o lenses based on various perormance criteria.

6 Figure.6 Figure.7 CA CA object side v s Numerical aperture and magniication magniication = h = 0. = 0.! h.0 ilament h = mm s + s = 0 mm CA NA = = 0.05 s s = 00 mm optical system = 9. mm CA = 5 mm s v image side CA NA = = 0.5 s iber core h = 0. mm s = 0 mm Optical system geometry or ocusing the output o an incandescent bulb into an optical iber.7 Optical Speciications Material Properties

7 Lens Combination Formulas Many optical tasks require several lenses in order to achieve an acceptable level o perormance. One possible approach to lens combinations is to consider each image ormed by each lens as the object or the next lens and so on. This is a valid approach, but it is time consuming and unnecessary. Symbols Optical Speciications Material Properties It is much simpler to calculate the eective (combined) ocal length and principal-point locations and then use these results in any subsequent paraxial calculations (see igure.8). They can even be used in the optical invariant calculations described in the preceding section. EFFECTIVE FOCAL LENGTH The ollowing ormulas show how to calculate the eective ocal length and principal-point locations or a combination o any two arbitrary components. The approach or more than two lenses is very simple: Calculate the values or the irst two elements, then perorm the same calculation or this combination with the next lens. This is continued until all lenses in the system are accounted or. The expression or the combination ocal length is the same whether lens separation distances are large or small and whether and are positive or negative: =. (.6) + d This may be more amiliar in the orm d = +. (.7) Notice that the ormula is symmetric with respect to the interchange o the lenses (end-or-end rotation o the combination) at constant d. The next two ormulas are not. COMBINATION FOCAL-POINT LOCATION For all values o,, and d, the location o the ocal point o the combined system (s ), measured rom the secondary principal point o the second lens (H ), is given by s = ( d). (.8) + d This can be shown by setting s =d4 (see igure.8a), and solving = + s s c = combination ocal length (EFL), positive i combination inal ocal point alls to the right o the combination secondary principal point, negative otherwise (see igure.8c). = ocal length o the irst element (see igure.8a). = ocal length o the second element. d = distance rom the secondary principal point o the irst element to the primary principal point o the second element, positive i the primary principal point is to the right o the secondary principal point, negative otherwise (see igure.8b). s =distance rom the primary principal point o the irst element to the inal combination ocal point (location o the inal image or an object at ininity to the right o both lenses), positive i the ocal point is to let o the irst element s primary principal point (see igure.8d). s =distance rom the secondary principal point o the second element to the inal combination ocal point (location o the inal image or an object at ininity to the let o both lenses), positive i the ocal point is to the right o the second element s secondary principal point (see igure.8b). z H = distance to the combination primary principal point measured rom the primary principal point o the irst element, positive i the combination secondary principal point is to the right o secondary principal point o second element (see igure.8d). z H =distance to the combination secondary principal point measured rom the secondary principal point o the second element, positive i the combination secondary principal point is to the right o the secondary principal point o the second element (see igure.8c). Note: These paraxial ormulas apply to coaxial combinations o both thick and thin lenses immersed in air or any other luid with reractive index independent o position. They assume that light propagates rom let to right through an optical system. or s..8

8 COMBINATION SECONDARY PRINCIPAL-POINT LOCATION Because the thin-lens approximation is obviously highly invalid or most combinations, the ability to determine the location o the secondary principal point is vital or accurate determination o d when another element is added. The simplest ormula or this calculates the distance rom the secondary principal point o the inal (second) element to the secondary principal point o the combination (see igure.8b): z s =. (.9) COMBINATION EXAMPLES It is possible or a lens combination or system to exhibit principal planes that are ar removed rom the system. When such systems are themselves combined, negative values o d may occur. Probably the simplest example o a negative d-value situation is shown in igure.9. Meniscus lenses with steep suraces have external principal planes. When two o these lenses are brought into contact, a negative value o d can occur. Other combined-lens examples are shown in igures.0 through.3. (a) (b) Figure.8 lens lens and lens H H H H d H H s = d4 Lens combination ocal length and principal planes s lens combination (c) (d) c 3 4 z H H c d>0 3 4 d<0 H c z H c Figure.9 Extreme meniscus-orm lenses with external principal planes (drawing not to scale) lens combination.9 Optical Speciications Material Properties

9 d s z <0 z<0 d s Optical Speciications ocal plane combination secondary principal plane Figure.0 Positive lenses separated by distance greater than = : is negative and both s and z are positive. Lens symmetry is not required. H H H combination secondary principal plane combination ocus Figure. Telephoto combination: The most important characteristic o the telephoto lens is that the EFL, and hence the image size, can be made much larger than the distance rom the irst lens surace to the image would suggest by using a positive lens ollowed by a negative lens (but not necessarily the lens shapes shown in the igure). For example, is positive and = 4 /. Then is negative or d less than /, ininite or d = / (Galilean telescope or beam expander), and positive or d larger than /. To make the example even more speciic, catalog lenses LDX C and LDK C, with d = 78. mm, will yield s =.0 m, = 5. m, and z = 43. m. t c n t c n d Material Properties Figure. Achromatic combinations: Air-spaced lens combinations can be made nearly achromatic, even though both elements are made rom the same material. Achieving achromatism requires that, in the thin-lens approximation, This is the basis or Huygens and Ramsden eyepieces. This approximation is adequate or most thick-lens situations. The signs o,, and d are unrestricted, but d must have a value that guarantees the existence o an air space. Element shapes are unrestricted and can be chosen to compensate or other aberrations..0 ( + ) d =. H H s s Figure.3 Condenser coniguration: The convex vertices o a pair o identical plano-convex lenses are on contact. (The lenses could also be plano aspheres.) Because d = 0, = / = /, / = s, and z = 0. The secondary principal point o the second element and the secondary principal point o the combination coincide at H, at depth t c /n beneath the vertex o the plano surace o the second element, where t c is the element center thickness and n is the reractive index o the element. By symmetry, the primary principal point o the combination is similarly located in the irst element. Combination conjugate distances must be measured rom these points.

10 Perormance Factors Ater paraxial ormulas have been used to select values or component ocal length(s) and diameter(s), the inal step is to select actual lenses. As in any engineering problem, this selection process involves a number o tradeos, including perormance, cost, weight, and environmental actors. The perormance o real optical systems is limited by several actors, including lens aberrations and light diraction. The magnitude o these eects can be calculated with relative ease. Numerous other actors, such as lens manuacturing tolerances and component alignment, impact the perormance o an optical system. Although these are not considered explicitly in the ollowing discussion, it should be kept in mind that i calculations indicate that a lens system only just meets the desired perormance criteria, in practice it may all short o this perormance as a result o other actors. In critical applications, it is generally better to select a lens whose calculated perormance is signiicantly better than needed. DIFFRACTION Diraction, a natural property o light arising rom its wave nature, poses a undamental limitation on any optical system. Diraction is always present, although its eects may be masked i the system has signiicant aberrations. When an optical system is essentially ree rom aberrations, its perormance is limited solely by diraction, and it is reerred to as diraction limited. In calculating diraction, we simply need to know the ocal length(s) and aperture diameter(s); we do not consider other lens-related actors such as shape or index o reraction. Since diraction increases with increasing -number, and aberrations decrease with increasing -number, determining optimum system perormance oten involves inding a point where the combination o these actors has a minimum eect. ABERRATIONS To determine the precise perormance o a lens system, we can trace the path o light rays through it, using Snell s law at each optical interace to determine the subsequent ray direction. This process, called ray tracing, is usually accomplished on a computer. When this process is completed, it is typically ound that not all the rays pass through the points or positions predicted by paraxial theory. These deviations rom ideal imaging are called lens aberrations. The direction o a light ray ater reraction at the interace between two homogeneous, isotropic media o diering index o reraction is given by Snell s law: n sinv = n sinv (.0) where v is the angle o incidence, v is the angle o reraction, and both angles are measured rom the surace normal as shown in igure.4. material index n material index n Figure.4 APPLICATION NOTE wavelength l v Reraction o light at a dielectric boundary Technical Assistance Detailed perormance analysis o an optical system is accomplished by using computerized ray-tracing sotware. CVI Melles Griot applications engineers are able to provide a ray-tracing analysis o simple catalog-component systems. I you need assistance in determining the perormance o your optical system, or in selecting optimum components or your particular application, please contact your nearest CVI Melles Griot oice. v. Optical Speciications Material Properties

11 Material Properties Optical Speciications Even though tools or the precise analysis o an optical system are becoming easier to use and are readily available, it is still quite useul to have a method or quickly estimating lens perormance. This not only saves time in the initial stages o system speciication, but can also help achieve a better starting point or any urther computer optimization. The irst step in developing these rough guidelines is to realize that the sine unctions in Snell s law can be expanded in an ininite Taylor series: sin v = v v / 3! + v / 5! v / 7! + v / 9!... (.) The irst approximation we can make is to replace all the sine unctions with their arguments (i.e., replace sinv with v itsel and so on). This is called irst-order or paraxial theory because only the irst terms o the sine expansions are used. Design o any optical system generally starts with this approximation using the paraxial ormulas. The assumption that sinv = v is reasonably valid or v close to zero (i.e., high -number lenses). With more highly curved suraces (and particularly marginal rays), paraxial theory yields increasingly large deviations rom real perormance because sinv v. These deviations are known as aberrations. Because a perect optical system (one without any aberrations) would orm its image at the point and to the size indicated by paraxial theory, aberrations are really a measure o how the image diers rom the paraxial prediction. As already stated, exact ray tracing is the only rigorous way to analyze real lens suraces. Beore the advent o electronic computers, this was excessively tedious and time consuming. Seidel* addressed this issue by developing a method o calculating aberrations resulting rom the v 3 /3! term. The resultant third-order lens aberrations are thereore called Seidel aberrations. To simpliy these calculations, Seidel put the aberrations o an optical system into several dierent classiications. In monochromatic light they are spherical aberration, astigmatism, ield curvature, coma, and distortion. In polychromatic light there are also chromatic aberration and lateral color. Seidel developed methods to approximate each o these aberrations without actually tracing large numbers o rays using all the terms in the sine expansions. In actual practice, aberrations occur in combinations rather than alone. This system o classiying them, which makes analysis much simpler, gives a good description o optical system image quality. In act, even in the era o powerul ray-tracing sotware, Seidel s ormula or spherical aberration is still widely used. 9 SPHERICAL ABERRATION Figure.5 illustrates how an aberration-ree lens ocuses incoming collimated light. All rays pass through the ocal point F. The lower igure shows the situation more typically encountered in single lenses. The arther rom the optical axis the ray enters the lens, the nearer to the lens it ocuses (crosses the optical axis). The distance along the optical axis between the intercept o the rays that are nearly on the optical axis (paraxial rays) and the rays that go through the edge o the lens (marginal rays) is called longitudinal spherical aberration (LSA). The height at which these rays intercept the paraxial ocal plane is called transverse spherical aberration (TSA). These quantities are related by TSA = LSA#tan(u ). Spherical aberration is dependent on lens shape, orientation, and conjugate ratio, as well as on the index o reraction o the materials present. Parameters or choosing the best lens shape and orientation or a given task are presented later in this chapter. However, the third aberration-ree lens u paraxial ocal plane longitudinal spherical aberration LSA F F TSA transverse spherical aberration (.) * Ludwig von Seidel, Figure.5 Spherical aberration o a plano-convex lens

12 order, monochromatic, spherical aberration o a plano-convex lens used at ininite conjugate ratio can be estimated by spot size due to spherical aberration = /# Theoretically, the simplest way to eliminate or reduce spherical aberration is to make the lens surace(s) with a varying radius o curvature (i.e., an aspheric surace) designed to exactly compensate or the act that sin v v at larger angles. In practice, however, most lenses with high surace accuracy are manuactured by grinding and polishing techniques that naturally produce spherical or cylindrical suraces. The manuacture o aspheric suraces is more complex, and it is diicult to produce a lens o suicient surace accuracy to eliminate spherical aberration completely. Fortunately, these aberrations can be virtually eliminated, or a chosen set o conditions, by combining the eects o two or more spherical (or cylindrical) suraces. In general, simple positive lenses have undercorrected spherical aberration, and negative lenses usually have overcorrected spherical aberration. By combining a positive lens made rom low-index glass with a negative lens made rom high-index glass, it is possible to produce a combination in which the spherical aberrations cancel but the ocusing powers do not. The simplest examples o this are cemented doublets, such as the LAO series which produce minimal spherical aberration when properly used. object point Figure.6 optical axis tangential plane (.3) ASTIGMATISM When an o-axis object is ocused by a spherical lens, the natural asymmetry leads to astigmatism. The system appears to have two dierent ocal lengths. As shown in igure.6, the plane containing both optical axis and object point is called the tangential plane. Rays that lie in this plane are called tangential, or meridional, rays. Rays not in this plane are reerred to as skew rays. The chie, or principal, ray goes rom the object point through the center o the aperture o the lens system. The plane perpendicular to the tangential plane that contains the principal ray is called the sagittal or radial plane. The igure illustrates that tangential rays rom the object come to a ocus closer to the lens than do rays in the sagittal plane. When the image is evaluated at the tangential conjugate, we see a line in the sagittal direction. A line in the tangential direction is ormed at the sagittal conjugate. Between these conjugates, the image is either an elliptical or a circular blur. Astigmatism is deined as the separation o these conjugates. The amount o astigmatism in a lens depends on lens shape only when there is an aperture in the system that is not in contact with the lens itsel. (In all optical systems there is an aperture or stop, although in many cases it is simply the clear aperture o the lens element itsel.) Astigmatism strongly depends on the conjugate ratio. tangential image (ocal line) principal ray sagittal plane optical system Astigmatism represented by sectional views paraxial ocal plane sagittal image (ocal line).3 Optical Speciications Material Properties

13 COMA In spherical lenses, dierent parts o the lens surace exhibit dierent degrees o magniication. This gives rise to an aberration known as coma. As shown in igure.7, each concentric zone o a lens orms a ring-shaped image called a comatic circle. This causes blurring in the image plane (surace) o o-axis object points. An o-axis object point is not a sharp image point, but it appears as a characteristic comet-like lare. Even i spherical aberration is corrected and the lens brings all rays to a sharp ocus on axis, a lens may still exhibit coma o axis. See igure.8. As with spherical aberration, correction can be achieved by using multiple suraces. Alternatively, a sharper image may be produced by judiciously placing an aperture, or stop, in an optical system to eliminate the more marginal rays. Figure.8 positive transverse coma ocal plane Positive transverse coma Optical Speciications FIELD CURVATURE Even in the absence o astigmatism, there is a tendency o optical systems to image better on curved suraces than on lat planes. This eect is called ield curvature (see igure.9). In the presence o astigmatism, this problem is compounded because two separate astigmatic ocal suraces correspond to the tangential and sagittal conjugates. Field curvature varies with the square o ield angle or the square o image height. Thereore, by reducing the ield angle by one-hal, it is possible to reduce the blur rom ield curvature to a value o 0.5 o its original size. Positive lens elements usually have inward curving ields, and negative lenses have outward curving ields. Field curvature can thus be corrected to some extent by combining positive and negative lens elements. Figure.9 spherical ocal surace Field curvature Material Properties S P,O points on lens corresponding points on S S 60 Figure.7 Imaging an o-axis point source by a lens with positive transverse coma.4

14 DISTORTION The image ield not only may have curvature but may also be distorted. The image o an o-axis point may be ormed at a location on this surace other than that predicted by the simple paraxial equations. This distortion is dierent rom coma (where rays rom an o-axis point ail to meet perectly in the image plane). Distortion means that even i a perect o-axis point image is ormed, its location on the image plane is not correct. Furthermore, the amount o distortion usually increases with increasing image height. The eect o this can be seen as two dierent kinds o distortion: pincushion and barrel (see igure.0). Distortion does not lower system resolution; it simply means that the image shape does not correspond exactly to the shape o the object. Distortion is a separation o the actual image point rom the paraxially predicted location on the image plane and can be expressed either as an absolute value or as a percentage o the paraxial image height. It should be apparent that a lens or lens system has opposite types o distortion depending on whether it is used orward or backward. This means that i a lens were used to make a photograph, and then used in reverse to project it, there would be no distortion in the inal screen image. Also, perectly symmetrical optical systems at : magniication have no distortion or coma. Figure.0 OBJECT PINCUSHION DISTORTION Pincushion and barrel distortion BARREL DISTORTION CHROMATIC ABERRATION The aberrations previously described are purely a unction o the shape o the lens suraces, and they can be observed with monochromatic light. Other aberrations, however, arise when these optics are used to transorm light containing multiple wavelengths. The index o reraction o a material is a unction o wavelength. Known as dispersion, this is discussed in Material Properties. From Snell s law (see equation.0), it can be seen that light rays o dierent wavelengths or colors will be reracted at dierent angles since the index is not a constant. Figure. shows the result when polychromatic collimated light is incident on a positive lens element. Because the index o reraction is higher or shorter wavelengths, these are ocused closer to the lens than the longer wavelengths. Longitudinal chromatic aberration is deined as the axial distance rom the nearest to the arthest ocal point. As in the case o spherical aberration, positive and negative elements have opposite signs o chromatic aberration. Once again, by combining elements o nearly opposite aberration to orm a doublet, chromatic aberration can be partially corrected. It is necessary to use two glasses with dierent dispersion characteristics, so that the weaker negative element can balance the aberration o the stronger, positive element. white light ray Figure. blue light ray blue ocal point red light ray red ocal point Longitudinal chromatic aberration Variations o Aberrations with Aperture, Field Angle, and Image Height longitudinal chromatic aberration Aperture Field Angle Image Height Aberration () (v) (y) Lateral Spherical 3 Longitudinal Spherical Coma v y Astigmatism v y Field Curvature v y Distortion v 3 y 3 Chromatic.5 Optical Speciications Material Properties

15 LATERAL COLOR Lateral color is the dierence in image height between blue and red rays. Figure. shows the chie ray o an optical system consisting o a simple positive lens and a separate aperture. Because o the change in index with wavelength, blue light is reracted more strongly than red light, which is why rays intercept the image plane at dierent heights. Stated simply, magniication depends on color. Lateral color is very dependent on system stop location. For many optical systems, the third-order term is all that may be needed to quantiy aberrations. However, in highly corrected systems or in those having large apertures or a large angular ield o view, third-order theory is inadequate. In these cases, exact ray tracing is absolutely essential. APPLICATION NOTE Achromatic Doublets Are Superior to Simple Lenses Because achromatic doublets correct or spherical as well as chromatic aberration, they are oten superior to simple lenses or ocusing collimated light or collimating point sources, even in purely monochromatic light. Although there is no simple ormula that can be used to estimate the spot size o a doublet, the tables in Spot Size give sample values that can be used to estimate the perormance o catalog achromatic doublets. blue light ray red light ray lateral color Optical Speciications aperture ocal plane Figure. Lateral Color Material Properties.6

16 Lens Shape Aberrations described in the preceding section are highly dependent on application, lens shape, and material o the lens (or, more exactly, its index o reraction). The singlet shape that minimizes spherical aberration at a given conjugate ratio is called best-orm. The criterion or best-orm at any conjugate ratio is that the marginal rays are equally reracted at each o the lens/air interaces. This minimizes the eect o sinv v. It is also the criterion or minimum surace-relectance loss. Another beneit is that absolute coma is nearly minimized or best-orm shape, at both ininite and unit conjugate ratios. To urther explore the dependence o aberrations on lens shape, it is helpul to make use o the Coddington shape actor, q, deined as ( r + r) q = ( r r). Figure.3 shows the transverse and longitudinal spherical aberrations o a singlet lens as a unction o the shape actor, q. In this particular instance, the lens has a ocal length o 00 mm, operates at /5, has an index o reraction o.587 (BK7 at the mercury green line, 546. nm), and is being operated at the ininite conjugate ratio. It is also assumed that the lens itsel is the aperture stop. An asymmetric shape that corresponds to a q-value o about or this material and wavelength is the best singlet shape or on-axis imaging. It is important to note that the best-orm shape is dependent on reractive index. For example, with a high-index material, such as silicon, the best-orm lens or the ininite conjugate ratio is a meniscus shape. ABERRATIONS IN MILLIMETERS Figure exact transverse spherical aberration (TSA) (.4) At ininite conjugate with a typical glass singlet, the plano-convex shape (q = ), with convex side toward the ininite conjugate, perorms nearly as well as the best-orm lens. Because a plano-convex lens costs much less to manuacture than an asymmetric biconvex singlet, these lenses are quite popular. Furthermore, this lens shape exhibits near-minimum total transverse aberration and near-zero coma when used o axis, thus enhancing its utility. For imaging at unit magniication (s = s = ), a similar analysis would show that a symmetric biconvex lens is the best shape. Not only is spherical aberration minimized, but coma, distortion, and lateral chromatic aberration exactly cancel each other out. These results are true regardless o material index or wavelength, which explains the utility o symmetric convex lenses, as well as symmetrical optical systems in general. However, i a remote stop is present, these aberrations may not cancel each other quite as well. For wide-ield applications, the best-orm shape is deinitely not the optimum singlet shape, especially at the ininite conjugate ratio, since it yields maximum ield curvature. The ideal shape is determined by the situation and may require rigorous ray-tracing analysis. It is possible to achieve much better correction in an optical system by using more than one element. The cases o an ininite conjugate ratio system and a unit conjugate ratio system are discussed in the ollowing section SHAPE FACTOR (q) exact longitudinal spherical aberration (LSA) Aberrations o positive singlets at ininite conjugate ratio as a unction o shape.7 Optical Speciications Material Properties

17 Material Properties Optical Speciications Lens Combinations INFINITE CONJUGATE RATIO As shown in the previous discussion, the best-orm singlet lens or use at ininite conjugate ratios is generally nearly plano-convex. Figure.4 shows a plano-convex lens (LPX C) with incoming collimated light at a wavelength o 546. nm. This drawing, including the rays traced through it, is shown to exact scale. The marginal ray (ray -number.5) strikes the paraxial ocal plane signiicantly o the optical axis. This situation can be improved by using a two-element system. The second part o the igure shows a precision achromat (LAO ), which consists o a positive low-index (crown glass) element cemented to a negative meniscus high-index (lint glass) element. This is drawn to the same scale as the plano-convex lens. No spherical aberration can be discerned in the lens. O course, not all o the rays pass exactly through the paraxial ocal point; however, in this case, the departure is measured in micrometers, rather than in millimeters, as in the case o the planoconvex lens. Additionally, chromatic aberration (not shown) is much better corrected in the doublet. Even though these lenses are known as achromatic doublets, it is important to remember that even with monochromatic light the doublet s perormance is superior. Figure.4 also shows the -number at which singlet perormance becomes unacceptable. The ray with -number 7.5 practically intercepts the paraxial ocal point, and the /3.8 ray is airly close. This useul drawing, which can be scaled to it a plano-convex lens o any ocal length, can be used to estimate the magnitude o its spherical aberration, although lens thickness aects results slightly. UNIT CONJUGATE RATIO Figure.5 shows three possible systems or use at the unit conjugate ratio. All are shown to the same scale and using the same ray -numbers with a light wavelength o 546. nm. The irst system is a symmetric biconvex lens (LDX C), the best-orm singlet in this application. Clearly, signiicant spherical aberration is present in this lens at /.7. Not until /3.3 does the ray closely approach the paraxial ocus. A dramatic improvement in perormance is gained by using two identical plano-convex lenses with convex suraces acing and nearly in contact. Those shown in igure.5 are both LPX C. The combination o these two lenses yields almost exactly the same ocal length as the biconvex lens. To understand why this coniguration improves perormance so dramatically, consider that i the biconvex lens were split down the middle, we would have two identical plano-convex lenses, each working at an ininite conjugate ratio, but with the convex surace toward the ocus. This orientation is opposite to that shown to be optimum or this shape lens. On the other hand, i these lenses are reversed, we have the system just described but with a better correction o the spherical aberration. ray -numbers PLANO-CONVEX LENS ACHROMAT paraxial image plane LPX C LAO Figure.4 Single-element plano-convex lens compared with a two-element achromat Previous examples indicate that an achromat is superior in perormance to a singlet when used at the ininite conjugate ratio and at low -numbers. Since the unit conjugate case can be thought o as two lenses, each working at the ininite conjugate ratio, the next step is to replace the plano-convex singlets with achromats, yielding a our-element system. The third part o igure.5 shows a system composed o two LAO lenses. Once again, spherical aberration is not evident, even in the /.7 ray..8

18 Figure.5 ray -numbers SYMMETRIC BICONVEX LENS IDENTICAL PLANO-CONVEX LENSES LDX C LPX C IDENTICAL ACHROMATS LAO Three possible systems or use at the unit conjugate ratio paraxial image plane.9 Optical Speciications Material Properties

19 Optical Speciications Diraction Eects In all light beams, some energy is spread outside the region predicted by geometric propagation. This eect, known as diraction, is a undamental and inescapable physical phenomenon. Diraction can be understood by considering the wave nature o light. Huygens principle (igure.6) states that each point on a propagating waveront is an emitter o secondary wavelets. The propagating wave is then the envelope o these expanding wavelets. Intererence between the secondary wavelets gives rise to a ringe pattern that rapidly decreases in intensity with increasing angle rom the initial direction o propagation. Huygens principle nicely describes diraction, but rigorous explanation demands a detailed study o wave theory. Diraction eects are traditionally classiied into either Fresnel or Fraunhoer types. Fresnel diraction is primarily concerned with what happens to light in the immediate neighborhood o a diracting object or aperture. It is thus only o concern when the illumination source is close to this aperture or object, known as the near ield. Consequently, Fresnel diraction is rarely important in most classical optical setups, but it becomes very important in such applications as digital optics, iber optics, and near-ield microscopy. Fraunhoer diraction, however, is oten important even in simple optical systems. This is the light-spreading eect o an aperture when the aperture (or object) is illuminated with an ininite source (plane-wave illumination) and the light is sensed at an ininite distance (ar-ield) rom this aperture. From these overly simple deinitions, one might assume that Fraunhoer diraction is important only in optical systems with ininite conjugate, whereas Fresnel diraction equations should be considered at inite conjugate ratios. Not so. A lens or lens system o inite positive ocal length with plane-wave input maps the ar-ield diraction pattern o its aperture onto the ocal plane; thereore, it is Fraunhoer diraction that determines the limiting perormance o optical systems. More generally, at any conjugate ratio, ar-ield angles are transormed into spatial displacements in the image plane. CIRCULAR APERTURE Fraunhoer diraction at a circular aperture dictates the undamental limits o perormance or circular lenses. It is important to remember that the spot size, caused by diraction, o a circular lens is d =.44l(/#) where d is the diameter o the ocused spot produced rom plane-wave illumination and l is the wavelength o light being ocused. Notice that it is the -number o the lens, not its absolute diameter, that determines this limiting spot size. The diraction pattern resulting rom a uniormly illuminated circular aperture actually consists o a central bright region, known as the Airy disc (see igure.7), which is surrounded by a number o much ainter rings. Each ring is separated by a circle o zero intensity. The irradiance distribution in this pattern can be described by I where I x = 0 J( x) x I0 = peak irradiance in the image J ( x)= Bessel unction o the irst kind o order unity x ( ) = n= x n ( n )! n! n n+ (.5) (.6) Material Properties secondary wavelets waveront some light diracted into this region waveront AIRY DISC DIAMETER =.44 l /# Figure.7 Center o a typical diraction pattern or a circular aperture aperture Figure.6 Huygens principle.0

20 and D x = p sinv l where l = wavelength D = aperture diameter v = angular radius rom the pattern maximum. This useul ormula shows the ar-ield irradiance distribution rom a uniormly illuminated circular aperture o diameter D. SLIT APERTURE A slit aperture, which is mathematically simpler, is useul in relation to cylindrical optical elements. The irradiance distribution in the diraction pattern o a uniormly illuminated slit aperture is described by I x = sinx I0 x APPLICATION NOTE Rayleigh Criterion In imaging applications, spatial resolution is ultimately limited by diraction. Calculating the maximum possible spatial resolution o an optical system requires an arbitrary deinition o what is meant by resolving two eatures. In the Rayleigh criterion, it is assumed that two separate point sources can be resolved when the center o the Airy disc rom one overlaps the irst dark ring in the diraction pattern o the second. In this case, the smallest resolvable distance, d, is 0.6l d = =.l( /# ) NA where I0 = peak irradiance in image pwsin v x = l where l = wavelength w = slit width v = angular deviation rom pattern maximum. (.7) ENERGY DISTRIBUTION TABLE The accompanying table shows the major eatures o pure (unaberrated) Fraunhoer diraction patterns o circular and slit apertures. The table shows the position, relative intensity, and percentage o total pattern energy corresponding to each ring or band. It is especially convenient to characterize positions in either pattern with the same variable x. This variable is related to ield angle in the circular aperture case by l sinv = x pd where D is the aperture diameter. For a slit aperture, this relationship is given by l sinv = x pw where w is the slit width, p has its usual meaning, and D, w, and l are all in the same units (preerably millimeters). Linear instead o angular ield positions are simply ound rom r=s tanv where s is the secondary conjugate distance. This last result is oten seen in a dierent orm, namely the diraction-limited spot-size equation, which, or a circular lens is d = 44. ( /#) This value represents the smallest spot size that can be achieved by an optical system with a circular aperture o a given -number, and it is the diameter o the irst dark ring, where the intensity has dropped to zero. The graph in igure.8 shows the orm o both circular and slit aperture diraction patterns when plotted on the same normalized scale. Aperture diameter is equal to slit width so that patterns between x-values and angular deviations in the ar-ield are the same. GAUSSIAN BEAMS Apodization, or nonuniormity o aperture irradiance, alters diraction patterns. I pupil irradiance is nonuniorm, the ormulas and results given previously do not apply. This is important to remember because most laser-based optical systems do not have uniorm pupil irradiance. The output beam o a laser operating in the TEM 00 mode has a smooth Gaussian irradiance proile. Formulas used to determine the ocused spot size rom such a beam are discussed in. Furthermore, when dealing with Gaussian beams, the location o the ocused spot also departs rom that predicted by the paraxial equations given in this chapter. This is also detailed in. (.8) (.9) l (see eq..5). Optical Speciications Material Properties

21 Energy Distribution in the Diraction Pattern o a Circular or Slit Aperture Circular Aperture Slit Aperture Relative Energy Relative Energy Position Intensity in Ring Position Intensity in Band Ring or Band (x) (I x /I 0 ) (%) (x) (I x /I 0 ) (%) Central Maximum First Dark.p p 0.0 First Bright.64p p Second Dark.3p p 0.0 Second Bright.68p p Third Dark 3.4p p 0.0 Third Bright 3.70p p Fourth Dark 4.4p p 0.0 Fourth Bright 4.7p p Fith Dark 5.4p p 0.0 Note: Position variable (x) is deined in the text. CIRCULAR APERTURE Optical Speciications Material Properties NORMALIZED PATTERN IRRADIANCE (y) slit aperture 9.0% within irst bright ring 83.9% in Airy disc POSITION IN IMAGE PLANE (x) circular aperture 90.3% in central maximum 95.0% within the two adjoining subsidiary maxima SLIT APERTURE Figure.8 Fraunhoer diraction pattern o a singlet slit superimposed on the Fraunhoer diraction pattern o a circular aperture.