Introduction THE OPTICAL ENGINEERING PROCESS ENGINEERING SUPPORT

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1 Material Properties Optical Speciications Gaussian Beam Optics Introduction Even though several thousand dierent optical components are listed in this catalog, perorming a ew simple calculations will usually determine the appropriate optics or an application or, at the very least, narrow the list o choices. 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. 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 beginning on page.9. 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 Chapter, Gaussian Beam Optics. ENGINEERING SUPPORT 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 Melles Griot products. THE OPTICAL ENGINEERING PROCESS 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. Visit Us OnLine!

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 5 or object to right o H s is or image to right o H (the second principal point) s is 5 or image to let o H m is or an inverted image m is 5 or an upright image For mirrors: When using the thin-lens approximation, simply reer to the let and right o the lens. Figure. h ront ocal point object v H H F s principal points is or convex (diverging) mirrors is 5 or concave (converging) mirrors s is or object to let o H s is 5 or object to right o H s is 5 or image to right o H s is or image to let o H m is or an inverted image m is 5 or an upright image rear ocal point Note location o object and image relative to ront and rear ocal points. = lens diameter m = s /s = h /h = magniication or conjugate ratio, said to be ininite i either s or s is ininite v = arcsin (/s) h = object height h = image height Sign conventions F s 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 the principal point H = eective ocal length (EFL) which may be positive (as shown) or negative. represents both FH and H F, assuming lens to be surrounded by medium o index.0 h Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

3 Material Properties Optical Speciications Gaussian Beam Optics 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 = s +. s m = s = h. (.) s h This relationship can be used to recast the irst ormula into the ollowing orms: = m (s + s ) (m + ) = sm m + s + s = m + + m s (m + ) = s + s = 4 s s = 4 s s = 66.7 mm m = s = 66.7 s 00 = 0.33 (or real image is 0.33 mm high and inverted). (.) This ormula is reerenced to igure. and the sign conventions given on page.3. By deinition, magniication is the ratio o image size to object size or where (s + s ) is the approximate object-to-image distance. (.3) (.4) (.5) (.6) 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 0 LDX 03 ( = 50 mm). Where is the image ormed, and what is the magniication? (See igure..) object Figure. Figure.3 F F image Example ( = 50 mm, s = 00 mm, s = 66.7 mm) 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.) = 4 s s = 475 mm m = s 475 = = 4.5 s 30 (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. image F F object Example ( = 50 mm, s = 30 mm, s = 475 mm) Example 3: Object at Focal Point A -mm-high object is placed on the optical axis, 50 mm let o the irst principal point o an 0 LDK 09 ( = 50 mm). Where is the image ormed, and what is the magniication? (See igure.4.) = s s = 45 mm m = s 45 = = 40.5 s 50 (or virtual image is 0.5 mm high and upright)..4 Visit Us OnLine!

4 object F Figure.4 image Example 3 ( = 450 mm, s = 50 mm, s = 45 mm) 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 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 clear aperture (eective diameter). or -number =. NA = sin v = NA = (-number). F (.7) 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 highpower image. The other term used commonly in deining this cone angle is numerical aperture. Numerical aperture 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 (.8) (.9) 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 Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

5 Material Properties Optical Speciications Gaussian Beam Optics Imaging Properties o Lens Systems THE OPTICAL INVARIANT To understand the importance o the numerical aperture, consider its relation to magniication. Reerring to igure.6, NA (object side) = sin v = s NA " (image side) = sin v = s which can be rearranged to show and = s sinv = s sinv leading to s s = sin v sinv = NA. NA" Since s 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 numerical apertures 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, we will be able to collect more light and thereby produce a brighter image. However, because o the relationship between magniication and numerical aperture, there can be a theoretical limit beyond which increasing the diameter has no eect on lightcollection eiciency or image brightness. Since the numerical aperture o a ray is given by /s, once a ocal length and magniication have been selected, the value o NA sets the value o. Thus, i one is dealing with a system in which the numerical aperture 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 numerical aperture. For monochromators, this limit is usually expressed in terms o the -number. In addition to the ixed numerical aperture, they both have a ixed entrance pupil (image) size. Suppose it is necessary, using a singlet lens rom this catalog, 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 a numerical aperture 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, = m (s + s ) (m + ) to determine that the ocal length is 9. mm. To determine the conjugate distances, s and s, we utilize equation.6, s (m + ) = s + s, and ind that s = 00 mm and s = 0 mm. We can now use the relationship NA = Ω/s or NA = Ω/s to derive Ω, the optimum clear aperture (eective diameter) o the lens. With an image numerical aperture o 0.5 and an image distance (s ) o 0 mm, 0.5 = 0 = 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 numerical aperture o the optical iber. The singlet lenses in this catalog that meet these criteria are 0 LPX 003, which is plano-convex, and 0 LDX 003 and 0 LDX 005, which are biconvex. SAMPLE CALCULATION To understand how to use this relationship between magniication and numerical aperture, consider the ollowing example. Making some simple calculations has reduced our choice o lenses to just three. Chapter, Gaussian Beam Optics, discusses how to make a inal choice o lenses based on various perormance criteria..6 Visit Us OnLine!

6 Figure.6 Figure.7 object side magniication = h" = 0. = 0.X h.0 ilament h = mm v s Numerical aperture and magniication s + s" = 0 mm NA = = 0.05 s s = 00 mm optical system = 9. mm = 5 mm 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 s v image side Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

7 Material Properties Optical Speciications Gaussian Beam Optics Lens Combination Formulas PARAXIAL 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. 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: = + 4. d This may be more amiliar in the orm s = = + 4 d. ( 4 d). + 4 d z = s 4. (.6) (.7) Notice that the ormula is symmetric with respect to 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 cases, COMBINATION SECONDARY PRINCIPAL-POINT LOCATION (.8) 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 how ar the secondary principal point o the inal (second) element is moved by being part o the combination: (.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. SYMBOLS d = combination ocal length (EFL), positive i combination inal ocal point alls to right o combination secondary principal point, negative otherwise. = ocal length (EFL) o irst element. = ocal length (EFL) o second element. = distance rom secondary principal point o irst element to primary principal point o second element (positive i primary principal point is to right o the secondary principal point, negative otherwise). s = distance rom secondary principal point o second element to inal combination ocal point (location o inal image or object at ininity to let), positive i the ocal point is to right o second element secondary principal point. z = distance to combination secondary principal point measured rom secondary principal point o second element, positive i combination secondary principal point is to right o secondary principal point o second element. Note: These paraxial ormulas apply to coaxial combinations o both thick and thin lenses immersed in any luid with reractive index independent o position. They assume that light propagates rom let to right through an optical system..8 Visit Us OnLine!

8 Figure.8 Figure.9 INDIVIDUAL ELEMENT st element COMBINATION elements SUBSYSTEM d d nd element z, rom ormula 3rd element combination secondary principal plane (to ind combination primary principal plane, apply procedure to reversed combination resulting rom end-to-end rotation) subsystem secondary principal plane n- elements nth element to be added to complete the system d z, rom ormula system secondary COMPLETE SYSTEM principal plane principal planes not crossed system primary principal plane (secondary principal plane located by z ormula or reversed system) lens combinations or systems may exhibit crossed principal planes; single lenses cannot SUBSYSTEM principal planes internal but crossed n- elements Generalization rom combinations to systems 3 4 d subsystem secondary principal plane nth element to be added to complete the system subsystem primary principal plane 3 4 d>0 d<0 Extreme meniscus-orm lenses with external principal planes (drawing not to scale) Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

9 d s z <0 z<0 d s Material Properties Optical Speciications Gaussian Beam Optics H Figure. Achromatic combinations: Air-spaced lens combinations can be made nearly achromatic, even though both elements are made o the same material. Achieving achromatism requires that, in the thin-lens approximation, d = ( + ). ocal plane H H combination secondary principal plane Figure.0 Positive lenses separated by distance greater than + : is negative, while both s and z are positive. Lens symmetry is not required. d 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. combination secondary principal plane combination ocus Figure. Telephoto combination: The most important characteristic o the telephoto 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 0 LDX 89 and 0 LDK 0, with d = 78. mm, will yield s =.0 m, = 5. m, and z = 43. m. t c n H H s s Figure.3 Condenser coniguration: A pair o identical plano-convex lenses have their convex vertices in 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. t c n.0 Visit Us OnLine!

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 lensrelated 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 sinß = n sinß (.0) where ß is the angle o incidence, ß 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 wavelength l d Reraction o light at a dielectric boundary APPLICATION NOTE Technical Assistance Detailed perormance analysis o an optical system is accomplished using computerized ray-tracing sotware. Melles Griot applications engineers have the capability to provide a ray-tracing analysis o simple catalog components 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 Melles Griot oice. Alternately, a database containing prescription inormation or most o the components listed in this catalog is available on the catalog CD-ROM. I you would like to obtain a copy o this database, please contact your Melles Griot representative. For analysis o more complex optical systems, or the design o totally custom lenses, Melles Griot Optical Systems, located in Rochester, New York, can supply the necessary support. This group specializes in the design and abrication o high-precision, multielement lens systems. For more inormation about their capabilities, please call your Melles Griot representative. v v Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

11 Material Properties Optical Speciications Gaussian Beam Optics Even though tools or 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: 3 5 sin v = v 4 v /3! + v /5! 4 v /7! + v /9! 4... The irst approximation we can make is to replace all sine unctions with their arguments (i.e., replace sin ß with ß 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 sinß = ß is reasonably valid or ß 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 sinß ß. 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 computers, this was excessively tedious and time consuming. Seidel addressed this issue by developing a method o calculating aberrations resulting rom the ß 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. 7 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. aberration-ree lens u paraxial ocal plane longitudinal spherical aberration LSA F F TSA transverse spherical aberration Figure.5 Spherical aberration o a plano-convex lens (.) 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. Visit Us OnLine!

12 third-order, monochromatic, spherical aberration o a plano-convex lens used at ininite conjugate ratio can be estimated by spot size due to spherical aberration = (.) /# 3 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 quality 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 0 LAO series which produce minimal spherical aberration when properly used. object point Figure.6 optical axis tangential plane tangential image (ocal line) principal ray sagittal plane optical system Astigmatism represented by sectional views 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 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. paraxial ocal plane sagittal image (ocal line) Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

13 Material Properties Optical Speciications Gaussian Beam Optics 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. 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 there are two separate astigmatic ocal suraces that 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. S P,O Figure.8 Figure.9 points on lens positive transverse coma ocal plane Positive transverse coma spherical ocal surace Field curvature corresponding points on S S 60 Figure.7 Imaging an o-axis point source by a lens with positive transverse coma.4 Visit Us OnLine!

14 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. 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. CHROMATIC ABERRATION The aberrations previously described are purely a unction o the shape o the lens suraces, and can be observed with monochromatic light. There are, however, other aberrations that arise when these optics are used to transorm light containing multiple wavelengths. OBJECT PINCUSHION DISTORTION BARREL DISTORTION The index o reraction o a material is a unction o wavelength. Known as dispersion, this is discussed in Chapter 4, 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. Variations o Aberrations with Aperture, Field Angle, and Image Height Aberration Lateral Spherical Longitudinal Spherical Coma Astigmatism Field Curvature Distortion Chromatic white light ray Aperture blue light ray blue ocal point red light ray red ocal point Figure.0 Pincushion and barrel distortion Figure. Longitudinal chromatic aberration (Ω) Ω 3 Ω Ω Ω Ω Field Angle (ß) ß ß ß ß 3 Image Height (y) y y y y 3 longitudinal chromatic aberration Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

15 Material Properties Optical Speciications Gaussian Beam Optics 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. aperture Figure. Lateral color red light ray blue light ray ocal plane lateral color 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 on page.6 give sample values that can be used to estimate the perormance o other catalog achromats..6 Visit Us OnLine!

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 sin v 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 q = (r + r ). (.3) (r 4 r ) Figure.3 Figure.3 shows the transverse and longitudinal spherical aberration 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. Best-orm shapes are used in Melles Griot laser-line-ocusing singlet lenses. 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 exact transverse spherical aberration (TSA) SHAPE FACTOR (q) exact longitudinal spherical aberration (LSA) Aberrations o positive singlets at ininite conjugate ratio as a unction o shape 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 nearminimum 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. Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

17 Material Properties Optical Speciications Gaussian Beam Optics 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 (0 LPX 03) 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 (0 LAO 04), 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 plano-convex 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 (0 LDX 07), 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 0 LPX 08. 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 0 LPX 03 0 LAO 04 Figure.4 Single-element plano-convex lens compared with a two-element achromat The 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 0 LAO 037 lenses. Once again, spherical aberration is not evident, even in the /.7 ray..8 Visit Us OnLine!

18 ray -numbers SYMMETRIC BICONVEX LENS IDENTICAL PLANO-CONVEX LENSES LDX 07 0 LPX 08 IDENTICAL ACHROMATS 0 LAO 037 Figure.5 Three possible systems or use at the unit conjugate ratio paraxial image plane Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

19 Material Properties Optical Speciications Gaussian Beam Optics Diraction Eects In all light beams, some energy is spread outside the region predicted by rectilinear propagation. This eect, known as diraction, is a undamental and inescapable physical phenomenon. Diraction can be understood by considering the wave nature o light. Huygen s principle (igure.6) states that each point on a propagating waveront is an emitter o secondary wavelets. The combined ocus o these expanding wavelets orms the propagating wave. 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. Huygen s 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. Consequently, Fresnel diraction is rarely important in most optical setups. Fraunhoer diraction, however, is oten very important. 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 arield 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. 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 0.6 l d = =. l /#. N.A. 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 =.44 l /# (.4) where d is the diameter o the ocused spot produced rom planewave 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 = I x 0 Figure.6 secondary wavelets waveront aperture J (x) x where I 0 = peak irradiance in image J (x) = x ( 4) Huygen s principle J (x) = Bessel unction o the irst kind o order unity x = n= πd sin v l n+ n4 x (n 4 )!n! n4 where l = wavelength D= aperture diameter v = angular radius rom pattern maximum. some light diracted into this region waveront (.5) This useul ormula shows the ar-ield irradiance distribution rom a uniormly illuminated circular aperture o diameter, D..0 Visit Us OnLine!

20 AIRY DISC DIAMETER =.44 l /# Figure.7 Center o a typical diraction pattern or a circular aperture 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 where I = I x 0 I 0 x = Energy Distribution in the Diraction Pattern o a Circular or Slit Aperture Ring or Band sin x x Central Maximum First Dark First Bright Second Dark Second Bright Third Dark Third Bright Fourth Dark Fourth Bright Fith Dark Note: Position variable (x) is deined in the text. = peak irradiance in image pw sin v l where l = wavelength w = slit width v = angular deviation rom pattern maximum. Position (x) 0.0.p.64p.3p.68p 3.4p 3.70p 4.4p 4.7p 5.4p (.6) Circular Aperture Relative Intensity (I x /I 0 ) ENERGY DISTRIBUTION TABLE The table below 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 sin v = x pd l sin v = 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 tan (v) (.9) where s is the secondary conjugate distance. This last result is oten seen in a dierent orm, namely the diraction-limited spot-size equation. For a circular lens that was stated at the outset o this section: d =.44 l /# Energy in Ring (%) Position (x) p.43p.00p.46p 3.00p 3.47p 4.00p 4.48p 5.00p Slit Aperture Relative Intensity (I x /I 0 ) (.7) where D is the aperture diameter. For a slit aperture, this relationship is given by (.8) (see.4) This value represents the smallest spot size that can be achieved by an optical system with a circular aperture o a given -number. Energy in Band (%) Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

21 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. 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 chapter. Material Properties Optical Speciications Gaussian Beam Optics 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 to determine the ocused spot size rom such a beam are discussed in Chapter, Gaussian Beam Optics. Furthermore, NORMALIZED PATTERN IRRADIANCE (y) CIRCULAR APERTURE slit aperture 9.0% within irst bright ring 83.9% in Airy disc POSITION IN IMAGE PLANE (x) 90.3% in central maximum 95.0% within the two adjoining subsidiary maxima circular aperture y = c J (x) x x n+ where J (x) = x ( 4 ) (n 4 )!n! n= Note : J (x) is the Bessel unction n 4 o the irst kind o order unity. p x = D sinv l l = wavelength n 4 D = aperture diameter v = angular radius rom pattern maximum y s sin x p =, where x = w sin x v l l = wavelength w = slit width v = angular deviation direction o pattern maximum SLIT APERTURE Figure.8 Fraunhoer diraction pattern o a singlet slit superimposed on the Fraunhoer diraction pattern o a circular aperture. Visit Us OnLine!

22 Lens Selection Having discussed the most important actors that aect a lens or a lens system s perormance, we will now address the practical matter o selecting the optimum catalog components or a particular task. The ollowing useul relationships are important to keep in mind throughout the selection process: $ Diraction-limited spot size =.44 /# $ Approximate on-axis spot size o a plano-convex lens at the ininite conjugate resulting rom spherical aberration = $ Optical invariant = Example : Collimating an Incandescent Source Produce a collimated beam rom a quartz halogen bulb having a -mm-square ilament. Collect the maximum amount o light possible and produce a beam with the lowest possible divergence angle. This problem, illustrated in igure.9, involves the typical tradeo between light-collection eiciency and resolution (where a beam is being collimated rather than ocused, resolution is deined by beam divergence). To collect more light, it is necessary to work at a low -number, but because o aberrations, higher resolution (lower divergence angle) will be achieved by working at a higher -number. In terms o resolution, the irst thing to realize is that the minimum divergence angle (in radians) that can be achieved using any lens system is the source size divided by system ocal length. An o-axis ray (rom the edge o the source) entering the irst principal point o the system exits the second principal point at the same angle. Thereore, increasing system ocal length improves this limiting divergence because the source appears smaller. An optic that can produce a spot size o mm when ocusing a perectly collimated beam is thereore required. Since source size is inherently limited, it is pointless to strive or better resolution. This level o resolution can be achieved easily with a plano-convex lens. Figure.9 m = NA NA". Collimating an incandescent source /# 3 While angular divergence decreases with increasing ocal length, spherical aberration o a plano-convex lens increases with increasing ocal length. To determine the appropriate ocal length, set the spherical aberration ormula or a plano-convex lens equal to the source (spot) size: /# = mm. This ensures a lens that meets the minimum perormance needed. To select a ocal length, make an arbitrary -number choice. As can be seen rom the relationship, as we lower the -number (increase collection eiciency), we decrease the ocal length, which will worsen the resultant divergence angle (minimum divergence = mm/). In this example, we will accept / collection eiciency, which gives us a ocal length o about 0 mm. For / operation we would need a minimum diameter o 60 mm. The 0 LPX 09 its this speciication exactly. Beam divergence would be about 8 mrad. Finally, we need to veriy that we are not operating below the theoretical diraction limit. In this example, the numbers (-mm spot size) indicate that we are not, since diraction-limited spot size =.44! 0.5 mm! =.44 mm. Example : Coupling an Incandescent Source into a Fiber On pages.6 and.7 we considered a system in which the output o an incandescent bulb with a ilament o mm in diameter was to be coupled into an optical iber with a core diameter o 00 µm and a numerical aperture o 0.5. From the optical invariant and other constraints given in the problem, we determined that system ocal length is 9. mm, diameter = 5 mm, s = 00 mm, s = 0 mm, NA = 0.5, and NA = 0.05 (or / and /0). The singlet lenses that match these speciications are the plano-convex 0 LPX 003 or biconvex lenses 0 LDX 003 and 0 LDX 005. The closest achromat would be the 0 LAO 00. v min = source size v min (see eq..) Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

23 Material Properties Optical Speciications Gaussian Beam Optics We can immediately reject the biconvex lenses because o spherical aberration. We can estimate the perormance o the 0 LPX 003 on the ocusing side by using our spherical aberration ormula: (0) spot size = = 84 mm. 3 We will ignore, or the moment, that we are not working at the ininite conjugate. This is slightly smaller than the 00-µm spot size we re trying to achieve. However, since we are not working at ininite conjugate, the spot size will be larger than given by our simple calculation. This lens is thereore likely to be marginal in this situation, especially i we consider chromatic aberration. A better choice is the achromat. Although a computer ray trace would be required to determine its exact perormance, it is virtually certain to provide adequate perormance. Example 3: Symmetric Fiber-to-Fiber Coupling Couple an optical iber with an 8-µm core and a 0.5 numerical aperture into another iber with the same characteristics. Assume a wavelength o 0.5 µm. This problem, illustrated in igure.30, is essentially a : imaging situation. We want to collect and ocus at a numerical aperture o 0.5 or /3.3, and we need a lens with an 8-µm spot size at this -number. Based on the lens combination discussion on page.8, our most likely setup is either a pair o identical plano-convex lenses or achromats, aced ront to ront. To determine the necessary ocal length or a plano-convex lens, we again use the spherical aberration estimate ormula: = mm. This ormula yields a ocal length o 4.3 mm and a minimum diameter o.3 mm. The 0 LPX 43 meets these criteria. The biggest problem with utilizing these tiny, short ocal length lenses is the practical considerations o handling, mounting, and positioning them. Since using a pair o longer ocal length singlets would result in unacceptable perormance, the next step might be to use a pair o the slightly longer ocal length, larger achromats, such as the 0 LAO 00. The perormance data, given on page.6, shows that this combination does provide the required 8-mm spot diameter. Because airly small spot sizes are being considered here, it is important to make sure that the system is not being asked to work below the diraction limit:.44! 0.5 mm! 3.3 = 4 mm. Since this is hal the spot size caused by aberrations, it can be saely assumed that diraction will not play a signiicant role here. An entirely dierent approach to a iber-coupling task such as this would be a pair o spherical ball lenses (06 LMS series), listed on page 5.5, or one o the gradient-index lenses (06 LGT series), listed on page 5.9. s = s"= Figure.30 Symmetric iber-to-iber coupling.4 Visit Us OnLine!

24 Example 4: Diraction-Limited Perormance Determine at what -number a plano-convex lens being used at an ininite conjugate ratio with 0.5-mm wavelength light becomes diraction limited (i.e., the eects o diraction exceed those caused by aberration). To solve this problem, set the equations or diraction-limited spot size and third-order spherical aberration equal to each other. The result depends upon ocal length, since aberrations scale with ocal length, while diraction is solely dependent upon -number. Substituting some common ocal lengths into this ormula, we get /8.6 at = 00 mm, /7. at = 50 mm, and /4.8 at = 0 mm m /# = 0.067!! m! 3 /# or /4 /# = (54.9! ). When working with these ocal lengths (and under the conditions previously stated), we can assume essentially diraction-limited perormance above these -numbers. Keep in mind, however, that this treatment does not take into account manuacturing tolerances or chromatic aberration, which will be present in polychromatic applications. MELLES GRIOT LENS DATABASE A database containing prescription inormation or most o the optical components listed in this catalog is included in the Melles Griot catalog on CD-ROM. This database, in a Zemax ormat, acilitates the determination o Spot size Prescription inormation Waveront distortion. Please contact our sales department or your ree Melles Griot Catalog on CD-ROM: Phone: / (949) FAX: (949) mglit@irvine.mellesgriot.com Non-US customers should contact the nearest Melles Griot oice (see back cover). Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

25 Material Properties Optical Speciications Gaussian Beam Optics Spot Size In general, the perormance o a lens or lens system in a speciic circumstance should be determined by an exact trigonometric ray trace. Melles Griot applications engineers can supply ray-trace data or particular lenses and systems o catalog components on request. However, or certain situations, some simple guidelines can be used or lens selection. The optimum working conditions or some o the lenses in this catalog have already been presented. The ollowing tables give some quantitative results or a variety o simple and compound lens systems that can be constructed rom standard catalog optics. In interpreting these tables, remember that these theoretical values obtained rom computer ray tracing consider only the eects o ideal geometric optics. Eects o manuacturing tolerances have not been considered. Furthermore, remember that using more than one element provides a higher degree o correction but makes alignment more diicult. When actually choosing a lens or a lens system, it is important to note the tolerances and speciications clearly described or each Melles Griot lens in the product listings. The tables give spot size or a variety o lenses used at several dierent -numbers. All the tables are or on-axis, uniormly illuminated, collimated input light at 63.8 nm. They assume that the lens is acing in the direction that produces a minimum spot size. When the spot size caused by aberrations is smaller or equal to the diraction-limited spot size, the notation DL appears next to the entry. The shorter ocal length lenses produce smaller spot sizes because aberrations increase linearly as a lens is scaled up. Focal Length = 0 mm / /3 /5 /0 0 LDX LPX LAO (DL) *Diraction-limited perormance is indicated by DL. Focal Length = 60 mm Spot Size (µm)* (DL) 5 (DL) 4 5 (DL) 8 (DL) 5 (DL) The eect on spot size caused by spherical aberration is strongly dependent on -number. For a plano-convex singlet, spherical aberration is inversely dependent on the cube o the -number. For doublets, this relationship can be even higher. On the other hand, the spot size caused by diraction increases linearly with -number. Thus, or some lens types, spot size at irst decreases and then increases with -number, meaning that there is some optimum perormance point where both aberrations and diraction combine to orm a minimum. Unortunately, these results cannot be generalized to situations where the lenses are used o axis. This is particularly true o the achromat/aplanatic meniscus lens combinations because their perormance degrades rapidly o axis. Focal Length = 30 mm / /3 /5 /0 0 LPX LAO (DL) *Diraction-limited perormance is indicated by DL. Spot Size (µm)* Spot Size (µm)* 80 8 (DL) 5 (DL) 0 LAO 059 & 0 LAM (DL) 8 (DL) 5 (DL) 0 LDX 3 0 LPX 7 0 LAO LAO 6 & 0 LAM 6 / /3 /5 / (DL) (DL) (DL) 6 5 (DL) 8 (DL) 5 (DL) *Diraction-limited perormance is indicated by DL..6 Visit Us OnLine!

26 Aberration Balancing To improve system perormance, optical designers make sure that the total aberration contribution rom all suraces taken together sums to nearly zero. Normally, such a process requires computerized analysis and optimization. However, there are some simple guidelines that can be used to achieve this with lenses available in this catalog. This approach can yield systems that operate at a much lower -number than can usually be achieved with simple lenses. Speciically, we will examine how to null the spherical aberration rom two or more lenses in collimated, monochromatic light. Thus, this technique will be most useul or laser beam ocusing and expanding. Figure.3 shows the third-order longitudinal spherical aberration coeicients or our o the most common positive and negative lens shapes when used with parallel, monochromatic incident light. The plano-convex and plano-concave lenses both show minimum spherical aberration when oriented with their curved surace acing the incident parallel beam. All other conigurations exhibit larger amounts o spherical aberration. With these lens types, it is now possible to show how various systems can be corrected or spherical aberration. A two-element laser beam expander is a good starting example. In this case, two lenses are separated by a distance which is the sum o their ocal lengths, so that the overall system ocal length is ininite. This system will not ocus incoming collimated light, but it will change the beam diameter. By deinition, each o the lenses is operating at the same -number. The equation or longitudinal spherical aberration shows that or two lenses with the same -number, aberration varies directly with the ocal lengths o the lenses. The sign o the aberration is the same as ocal length. Thus, it should be possible to correct the spherical Figure.3 positive lenses negative lenses aberration coeicient (k) plano-convex (reversed) 0 LPX plano-concave (reversed) 0 LPK symmetric-convex 0 LDX symmetric-concave 0 LDK longitudinal spherical aberration (3rd order) = k /# plano-convex (normal) 0 LPX plano-concave (normal) 0 LPK Third-order longitudinal spherical aberration o typical lens shapes aberration o this Galilean-type beam expander, which consists o a positive ocal length objective and a negative diverging lens. I a plano-convex lens o ocal length oriented in the normal direction is combined with a plano-concave lens o ocal length oriented in its reverse direction, the total spherical aberration o the system is LSA = 0.7 /# /# Ater setting this equal to zero, we obtain.069 = = (.30) To make the magnitude o aberration contributions o the two elements equal so they will cancel out, and thus correct the system, select the ocal length o the positive element to be 3.93 times that o the negative element. Figure.3 shows a beam-expander system made up o catalog elements, in which the ocal length ratio is 4:. This simple system is corrected to about /6 wavelength at 63.8 nm, even though the objective is operating at /4 with a 0-mm aperture diameter. This is remarkably good waveront correction or such a simple system; one would normally assume that a doublet objective would be needed and a complex diverging lens as well. This analysis does not take into account manuacturing tolerances. A beam expander o lower magniication can also be derived rom this inormation. I a symmetric-convex objective is used together with a reversed plano-concave diverging lens, the aberration coeicients are in the ratio o.069/0.403 =.65. Figure.3 shows a system o catalog lenses that provides a magniication o Fundamental Optics Gaussian Beam Optics Optical Speciications Material Properties Visit Us Online!

27 Material Properties Optical Speciications Gaussian Beam Optics a) CORRECTED 4!BEAM EXPANDER = 40 mm 0-mm diameter plano-concave 0 LPK 00 b) CORRECTED.7x BEAM EXPANDER = 40 mm 0-mm diameter plano-concave 0 LPK 00 = 80 mm.4-mm diameter plano-convex 0 LPX 49 c) SPHERICALLY CORRECTED 5-mm EFL /.0 OBJECTIVE = 45 mm 5-mm diameter plano-concave 0 LPK 003 = 54 mm 3-mm diameter symmetric-convex 0 LDX 9 = 50 mm () 7-mm diameter plano-convex 0 LPX 08.7 (the closest possible given the available ocal lengths). The maximum waveront error in this case is only /4 wave, even though the objective is working at /3.3. The relatively ast speed o these objectives is a great advantage in minimizing the length o these beam expanders. They would be particularly useul with Nd:YAG and argon-ion lasers, which tend to have large output beam diameters. These same principles can be utilized to create high numerical aperture objectives that might be used as laser ocusing lenses. Figure.3 shows an objective consisting o an initial negative element, ollowed by two identical plano-convex positive elements. Again, all o the elements operate at the same -number, so that their aberration contributions are proportional to their ocal lengths. To obtain zero total spherical aberration rom this coniguration, we must satisy = 0 or = Thereore, a corrected system should result i the ocal length o the negative element is just about hal that o each o the positive lenses. In this case, = 45 mm and = 50 mm yield a total system ocal length o about 5 mm and an -number o approximately /. This objective, corrected to /6 wave, has the additional advantage o a very long working distance. UV OPTICS Figure.3 balancing Combining catalog lenses or aberration The material presented in this section is based on the work o John F. Forkner. Melles Griot now oers a selection o UV optics ranging rom 93 to 355 nm. See Chapter 6, UV Optics, or details..8 Visit Us OnLine!