An easy way to relate optical element motion to system pointing stability

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1 An eay way to relate optcal element moton to ytem pontng tablty J. H. Burge College o Optcal Scence Unverty o Arzona, Tucon, AZ 85721, USA jburge@optc.arzona.edu, ( ) ABSTRACT The optomechancal engneerng or mountng lene and mrror n magng ytem requently drven by the pontng or jtter requrement or the ytem. A mple et o rule wa developed that allow the engneer to quckly determne the couplng between moton o an optcal element and a change n the ytem lne o ght. Example are hown or cae o lene, mrror, and optcal ubytem. The dervaton o the tatonary pont or rotaton alo provded. Small rotaton o the ytem about th pont doe not caue mage moton. Keyword: Optcal algnment, optomechanc, pontng tablty, geometrcal optc 1. INTRODUCTION Optcal ytem can be qute complex, ung lene, mrror, and prm to create and relay optcal mage rom one pace to another. The tablty o the ytem lne o ght depend on the mechancal tablty o the component and the optcal entvty o the ytem. In general, tlt or decenter moton n an optcal element wll caue the mage to ht laterally. The entvty to moton o the optcal element uually determned ung computer mulaton n an optcal degn code. I done correctly, the computer mulaton wll provde accurate and complete data or the engneer, allowng the contructon o an error budget and complete tolerance analy. However, the computer-derved entvty may not provde the engneer wth nght that could be valuable or undertandng and reducng the entvte or or troublehootng n the eld. In th paper, a mple method preented or determnng thee entvte ung a ew hand calculaton. The relatonhp provde nght to the problem and can be evaluated on the ly to help wth engneerng tradeo or evaluatng ytem perormance. The couplng between moton o an optcal element and the ytem perormance mportant or everal type o ytem: 1. Sytem that requre tablty. The calbraton o a pectrograph requre table magng rom the lt to the detector. Optcal magng ytem or lthography requre accuracy n the mage placement o ten o nanometer. Boreght accuracy between optcal ytem requre tablty or each ytem. 2. Sytem entve to mage jtter. Vbraton o an optcal element wll caue the lne o ght or the ytem to jtter th can caue decreaed acuty or long expoure or lmt perormance o laer projector. Th paper provde analy o the rt-order behavor o optcal ytem, whch adequate to decrbe mall moton. The work preented here buld on well-known prncple o rt order ytem, whch are revewed n Secton 2. The eect o moton or len element, mrror, and optcal ytem derved n Secton 3. In general thee moton caue a change n angle and a lateral ht or the paraxal bundle o lght that ollow. The bac relatonhp that couple the angular and lateral perturbaton wth mage ht are derved n Secton 4. The relatonhp do not requre knowledge o the complete ytem, only the rt order properte o the element that perturbed, and the numercal aperture or the ytem. A heurtc explanaton and ome mple example are ncluded. Secton 5 gve an example and compare the reult wth a ray trace mulaton. Secton 6 dcue alternate applcaton o th work. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-1

2 2. FIRST ORDER OPTICS The analy n th paper ue well-known properte o rt order ytem, ncludng the repreentaton o the rt order behavor wth prncpal plane and the concept o the optcal nvarant. Thee are ummarzed here to dene the nomenclature ued n the paper. The concept are explaned more ully elewhere. 1, 2 The paraxal, or rt order behavor o an optcal ytem that ha ocal length can be repreented ung a ew cardnal pont and ome mple rule governng ray propagaton. An magng ytem can be reduced to two prncpal plane and two ocal pont, a hown n Fgure 1. Lght rom the object ncdent on the ront prncpal plane and t emerge rom the rear prncpal plane, whch n mage pace. The mappng between thee plane mple the ray heght y preerved rom one plane to the other and the angle u changed accordng to the mple relatonhp, y u' = u. Eq. 1 For ytem n ar, the rear ocal pont located one ocal length rom the rear prncpal plane and the ront ocal pont located one ocal length rom the ront ocal plane. h Object pace Image pace A F θ P P F A θ h z z d d A : object pont on ax A :mage o A P, P, prncpal pont F, F, ocal pont Fgure 1. Denton o cardnal pont to repreent the paraxal perormance o an optcal ytem An object located a dtance z rom the ront ocal pont wll cat an mage at z rom the rear ocal pont, obeyng the Newtonan mage relaton 2 z z' = Eq. 2 Lght ray rom an on-ax pont at nnty wll be parallel to the ax n object pace and they wll come to ocu at the rear ocal pont n mage pace. Converely, lght ray that go thought the ront ocal pont wll come out parallel to the ax n object pace. The nodal pont are dened a the pont wth unt angular magncaton. The angle o the lght come out the ame a t went n. The nodal pont and the prncpal pont concde or a ytem n ar. The poton o an o-ax mage can alway be ound ung th mple prncple, a hown n Fgure 1. The magncaton o the mage m, dened a the rato o mage heght to object heght, can then be calculated a m h' d' z+ Eq. 3 = = h d z ' + ' Whle the cardnal pont are ueul or determnng rt order properte o an optcal ytem the ray may not be phycal. It ueul to trace two real ray through the ytem a che ray and a margnal ray. The che ray tart at an o ax object pont and goe through the center o a urace o nteret, uually the top. The margnal ray tart at on object pont on ax and goe through the edge o the top, a hown n Fgure 2. For a ray o lght leavng urace, the ray heght and the angle wth repect to the ax are dened a y and θ repectvely a hown n Fgure 2. By conventon, the term wth bar over repreent the che ray and the un-barred term repreent the margnal ray. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-2

3 Margnal ray on ax ray that goe through edge o plane o nteret Che ray o ax ray that goe through center o the plane o nteret (typcally the top.) Plane o nteret y y Poton Margnal ray Che ray θ θ Fgure 2. Denton or electon and gn conventon or che and margnal ray. The eect o derent reractve ndce can be accommodated by denng the term u = n n θ. Th equvalent angle u wll be ued or ubequent analy n th paper. It well known that the value computed a I uy uy Eq. 4 nvarant. (The term I oten called the optcal nvarant.) In act, th quantty nvarant through the optcal ytem or any two lnearly ndependent ray. 3 Th mean that the value computed ung the ray heght and angle at urace wll have the ame value a computed or any other urace. 3. EFFECT OF OPTICAL ELEMENT MOTION I a mrror, len, or ubytem move ether laterally (perpendcular to the optcal ax) or n tlt, then the lne o ght or the optcal ytem wll be hted. For magng ytem, th wll reult n moton o the mage. For a laer projector, the eect wll be a devaton n the projected lght. Beore developng the relatonhp or the ytem, the mple eect due to element moton are analyzed n term o ther eect on the lght. We tart by denng a che ray that goe through the center o the ntal, unperturbed element. A the element perturbed, the output ray wll generally uer a combnaton o lateral ht and angular devaton. Thee are hown n Fgure 3. The mage ht due to thee perturbaton contant acro the eld, o we only need to know the ht n a ngle ray to determne the magntude o the eect. Intal on-ax ray α devated ray y Element moton : decenter α : tlt Central ray devaton y : lateral ht : change n angle Fgure 3. Tlt or lateral ht o an optcal element or ubytem wll generally caue a combnaton o angular devaton and lateral ht o a che ray. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-3

4 3.1 Eect o moton or a thn len The eect o len moton can be approxmated by gnorng the thckne o the element, treatng t a though all o the optcal power occur at the mddle o the len. (Although th an excellent approxmaton or mot cae, the complete analy gven below n Secton 3.4.) There no rt order eect o tltng the thn len a mall amount. Decenterng the len wll caue an angular devaton n the lght whch, to rt order, contant or every ray acro the len. The eect o tranlatng the len eay to ee graphcally or the central ray, a hown n Fgure 4. α =0 Fgure 4. Lateral tranlaton o a thn len by an amount caue the tranmtted lght to be devated by an angle. Tlt o the thn len ha no gncant eect. The eect o a thck len gven below ung a more general treatment nvolvng prncpal plane. For a len wth ocal length, hted by an amount, the ncdent central ray wll ntercept the hted len o t ax, but tll parallel to t ax. So the ray wll reract and go through the ocal pont. The angular ht can be determned graphcally (aumng the mall angle approxmaton) rom Fgure 4 or rom Eq. 1 a. Eq. 5 Note that th doe not depend on object or mage poton becaue we are only treatng the change n the central ray. Alo note that th value hold or negatve and potve lene. 3.2 Eect o moton or a mrror A powered mrror ha the ame entvty to lateral dplacement a the len above, and t alo very entve to tlt. Thee two cae are hown graphcally n Fgure 5. Upon relecton rom a mrror that tlted by an angle α, the relected lght wll have an angular change o θ α = 2α. Th ndependent o the ocal length o the mrror, o t apple or lat mrror a well a potvely and negatvely powered one. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-4

5 α α Fgure 5. Lateral tranlaton o a mrror caue the tranmtted lght to be devated by and angle. Tlt o the mrror by an angle α caue angular devaton o α = 2 α. A lnear combnaton o tranlaton and rotaton or a mrror gve + 2α. There one combnaton o tranlaton and rotaton whch ha no eect. Th occur when the mrror tlt equal to the rato o the dplacement over 2. Th o coure equvalent to rotatng the mrror about t center o curvature. Eq Eect o moton o a plane parallel plate The tlt o a plane parallel plate, uch a a wndow, wth thckne t by an amount α caue a lateral ht y a hown n Fgure 6. O coure the lateral tranlaton o a plane parallel plate ha no optcal eect. α Plane parallel plate thckne t ndex n y α t n y n ( 1 ) t α or n = Fgure 6. Tlt o a plane parallel plate caue a lateral devaton that depend on the thckne and reractve ndex o the plate. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-5

6 3.4 Eect o moton o an optcal ytem A thck len or any general et o powered optcal element can be treated ung prncpal plane a decrbed above. The eect o moton o uch an optcal ytem more complcated, but t can be calculated drectly. I we decompoe general moton nto a combnaton o pure lateral tranlaton and rotaton about the ront prncpal pont P, the relatonhp can be obtaned graphcally, a n Fgure 7. Pure tranlaton Pure rotaton about ront prncpal pont Sytem ax P P y = 0 ( = eectve ocal length) y α PP' θ = 0 α P P y PP (PP = dtance between prncpal pont) Fgure 7. General relatonhp or the moton o an optcal ytem repreented by prncpal plane. Rotaton about a general pont C can be evaluated a a lnear combnaton o tranlaton and rotaton about the prncpal pont. Conder rotatng an angle α C about a general pont C, whch located dtance CP rom the ront prncpal pont a hown n Fgure 8. For mall angle, the rotaton about th pont caue an eectve ytem tranlaton o = α C CP. The combned eect o rotaton and tranlaton caue the central ray to devate by y αc PP' CP. αc Eq. 7 α c P P ε(d ) C y d Fgure 8. Relatonhp or the moton o an optcal ytem repreented by prncpal plane a t rotated about an arbtrary pont C. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-6

7 For the cae where the mage located a dtance d rom the rear prncpal pont, the mage moton ε due to the optcal ytem rotatng an amount α c about pont C determned by the combned eect: CP ε αc PP' + αc d'. Eq. 8 The eect o tltng a real plano-convex thck len makng an mage rom nnty wa examned. A 50 mm dameter len, 200 mm ocal length wth 6.2 mm center thckne wa ued. The ront prncpal pont P le on the vertex o the curved urace and PP 2.1 mm. We rotate about the mddle o CP = -3.1 mm, and we evaluate the moton at ocu o d =. Evaluatng Eq. 8, the entvty o mage moton to len rotaton only 1 µm/mrad. Th entvty wa corroborated by drect mulaton ung Zemax 4 ray trace otware. Th entvty very mall ndeed. A 1 µm mage ht caued by 1 mrad len tlt correpond to an angle o only mrad n object pace, o the eect only 0.5% o the len tlt. The entvty to len decenter 1 µm/µm, whch much more mportant. The relatonhp above can be ued to determne the tatonary pont or rotaton. The tatonary pont dened uch that mall ytem rotaton about th pont doe not caue mage moton. The tatonary ound by ettng Eq. 8 to zero and olvng or CP tatonary, the poton o the tatonary pont relatve to the ront prncpal pont: CP tatonary = PP '. d ' Eq. 9 Evaluatng Equaton 9 or ome nteretng cae: For a thn len, PP = 0, and the tatonary pont occur at CP = 0, rotatng about the prncpal pont. For an object at nnty, d =, o CP tatonary = -PP, whch mean that the tatonary pont occur at the rear prncpal pont. Th prncple ued or ndng the prncpal pont wth a nodal lde 5. The tatonary pont depend not only on the optcal ytem, but alo on the object and mage poton. A real bconvex thck len operatng at 1:1 conjugate ha t tatonary pont n the mddle. (CP = -0.5 PP and d = 2.) 4. COUPLING OF OPTICAL ELEMENT MOTION TO SYSTEM LINE OF SIGHT The relatonhp between optcal element moton and the lght are gven above. The reultng eect on mage moton n an optcal ytem derved here ung the optcal nvarant. Th provde a hortcut, analogou to ung energy conervaton or mechancal problem. The oluton depend only on the rt order ray heght and lope at the element o nteret and the numercal aperture (or ocal rato) at the mage. There no need to perorm analy through the ntermedate optc. The optcal nvarant ued here dened o the che ray goe through the center o the element o nteret, not the ytem aperture top. Alo, the optcal ytem ollowng element ncluded, but any optc that precede th element are gnored. Th done o that the eld angle at element can be taken a the mall angle due to the element malgnment. (For the on-ax object pont, there no eld angle or the precedng optc.) Table 1 how the evaluaton o the optcal nvarant at urace and at the nal mage. The ollowng denton, hown n Fgure 9, have been appled: D = dameter o on-ax beam at urace (= 2 y ) NA = u = equvalent numercal aperture or the lght ater element change n angular devaton or lght mmedately ater element y change n lateral devaton or lght mmedately ater element NA numercal aperture or ytem F n workng ocal rato or ytem ε = reultng mage ht. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-7

8 Table 1. Evaluaton o optcal nvarant at urace and at nal mage plane At urace At mage plane N u : margnal ray angle u = NA = numercal aperture at u N = NA =1/2F n y : margnal ray heght y = D / 2 dened by beam ootprnt 0 u : che ray angle y : che ray heght I uy uy : optcal nvarant u = θ due to element moton u N y = y due to element moton N D NA y 2 y = y ε, mage moton NA ε = ε F 2 n Lght rom pont on ax, Bundle dened by aperture Element NA and F n baed on ytem ocu Image ht ε O-ax lght gnored Beam ootprnt on element Nomnal margnal ray at element u = NA y Perturbed central ray rom element u = θ D y = y Fgure 9. Denton or ytem analy relatng mage ht to element moton Equatng the value or the optcal nvarant n Table 1, the ollowng oluton obtaned: NA ε = F D y NA n Eq. 10 where and y gve the devaton o the central ray a decrbed above, and a calculated n Secton 3. Th reult appear to depend on the ze o the top, but t doe not. The value o D, F n, NA, and NA all depend on the top ze, but the quantty F n D and NA /NA normalze out the top ze. I the top reduced by a actor o two then D would reduce by a actor o two and F n would ncreae by th ame actor o the product unchanged. Lkewe, both NA and NA would decreae by th ame actor o the rato unchanged. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-8

9 4.1 Eect o angular devaton For almot all cae, the eect o angular devaton o the lght wll domnate the mage ht. Snce the total ht a lnear um, we can look eparately at the eect o angle by mply ettng the other term, y to zero. The reultng relaton very mple and t requre only knowledge o D, the ze o the beam ootprnt, and the nal ocal rato. Some example or the cae o angular devaton are For a ytem whch contan only a ngle len or mrror n tranlaton, the product FD n mply equal to the ocal length, gvng the expected reult that ε = θ. Th make the relatonhp eay to remember! For a len or mrror element wth ocal length movng laterally an amount, the mage ht wll be D F ε = F =. Eq. 11 n n D Th mple reult gve the magncaton o mage moton to lateral moton mply the rato o the ytem workng ocal rato to the eectve -number o the element, dened by the rato the ocal length to the dameter o the on ax beam ootprnt. (Note that th relatonhp doe not depend on the numercal aperture o the lght that llumnatng the element, nor doe t depend on the ze o the element.) For a mrror that rotated by a mall angle α, the relatonhp reduced to ε = α. 2FD n Eq. 12 So the lever arm whch relate mage moton to rotaton o the mrror mply 2F n D, whch exactly the value one would expect or a mrror that n the convergng beam n mage pace wth no other element between the mrror and the mage. Th doe not depend on the mrror ocal length, and t work or lat. Be careul when evaluatng the beam ootprnt or tlted mrror. The value or D hould be determned by the ze o the bundle o lght, not the elongated projecton on the tlted mrror. The mple relatonhp n Eq. 12 hould make ene to omebody who work n the area o optcal tetng. The tlt o an optc uually meaured n rnge. The eect o the tlt on the ntererogram doe not depend on whch optc caued the tlt, but only by the number o rnge ntroduced. Gven the ame number o tlt rnge, the tlt angle o a mrror wth 10 cm dameter beam ootprnt would be twce that o a 20 cm mrror. 4.2 Eect o pure tranlaton The eect o pure tranlaton or the lght evaluated ung Eq. 10, where the angular devaton et to zero. The lght rom the element wll generally orm an mage, ether vrtual or real. The optc between th element and the nal ocu can be thought o a relayng th mage. The magncaton o th relay can be calculated a the rato o the numercal aperture, exactly a Eq. 10. Some nteretng concluon are ( ) t n 1 NA The eect o tltng a wndow ε α whch doe not depend on the beam ootprnt. n NA A tlted wndow n collmated pace ha no eect (NA = 0). For lene, the beam tranlaton eect due to len tlt typcally only a ew percent a large a the eect on angular devaton caued by element decenter. I th mall eect to be evaluated, the analy mut nclude the poton o the pvot pont relatve to the prncpal pont to nclude the angular eect. Both term n Eq. 7 mut be ncluded n the analy. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-9

10 5. EXAMPLE FOR AN IMAGING SYSTEM Thee relatonhp were corroborated by drect mulaton o an magng ytem ung Zemax 4 optcal degn code. A Cooke trplet that provded wth the program wa modeled and perturbed. A layout o the optcal ytem and the key parameter are gven n Fgure 10. Parameter or Cooke trplet Sytem 50 mm EFL 10 mm Entrance pupl 30 eld o vew Fn = 5 (0.1 NA) Len1 Focal length 1: 33.7 mm Beam ootprnt D1 10 mm NA Len2 Focal length 2: mm Beam ootprnt D2 7.8 mm NA Len 3 Focal length 3: 24.3 mm Beam ootprnt D3 8.6 mm NA3 0.1 Fgure 10. Cooke trplet ued or evaluaton o mage moton. A the ndvdual element were perturbed n Zemax 4, the mage moton and the degradaton o the mage qualty were evaluated. Frt, a large ht hown to llutrate the concept. Len 2 wa hted by 1 mm and all mage were een to ht by 2.3 mm a predcted ung Eq. 11 wth 2 /D 2 = 2.15 or the /5 ytem. Th large malgnment alo caued gncant aberraton that vared omewhat wth eld poton, but the mage ht wa contant acro the eld, a een n Fgure 11. Sytem layout wth decentered len Image plane Len 2 hted 1 mm Fgure 11. Image ht acro the 35 mm ocal plane due to a 1 mm decenter or Len 2. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-10

11 The lene were then hted by maller amount to allow a quanttatve comparon. The relatonhp developed above were evaluated and compared wth the drect mulaton or all 3 len element. The mulaton matche the analytcal predcton extremely well, a een n Fgure 12. Th graph how mulated mage poton wth the rm mage ze repreented wth the error bar Len 3 Len 1 mage ht n µm len ht n µm Fgure 12. Comparon o analytc predcton o mage ht (old lne) wth computer mulaton or element decenter. The data pont rom the mulaton are repreented wth error bar zed accordng to the rm mage blur on ax. Len 2 The eect o tltng the lene wa alo evaluated and compared wth the Zemax 4 mulaton, hown n Fgure 13. In ome cae, the aberraton are larger than the mage tlt. In order to make a good comparon, the mage poton wa dened by the che ray, not by the centrod. 60 mage ht n µm Len 3 Len 2-20 Len len tlt n mrad Fgure 13. Comparon o analytc predcton o mage ht (old lne) wth computer mulaton or tlted element. The data pont rom the mulaton are repreented wth error bar zed accordng to the rm radu o the on ax pont pread uncton. Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-11

12 6. FURTHER DISCUSSION There are ome ytem varaton that requre ome modcaton to thee relaton. A mple modcaton allow calculaton o lne o ght tablty or nnte conjugate ytem, and a change o varable allow calculaton o pupl tablty. Alo, thee relaton can be ued to determne mage blur due to lope aberraton acro the pupl rom optcal urace rregularte. The relatonhp are derved above or magng ytem where the lght come to ocu wth a nte -number. The analy work the ame or nnte conjugate ytem where the angular change n lne o ght θ LOS can be determned by replacng ε wth θlos D 0 where D 0 the beam dameter n collmated pace. Snce the optcal F n nvarant wa ued or the dervaton, θ LOS can be evaluated equally well n object pace or mage pace a long a the approprate beam ze D 0 ued. In cae where the pupl ht mportant, the ame relatonhp can be appled where the pupl mage tranerred rom one pace to another. Whether the pupl mage vrtual or real, t can be tranerred accordng to the lnear magng relatonhp and the ht o th pupl mage due to element moton can be calculated n the ame way a the mage ht above. When perormng thee calculaton, a top or the pupl magng optc mut be aumed, and the value or F n and NA mut be taken approprately or th cae. A decrbed above, the oluton doe not depend on the choce o the top. Another applcaton o the relatonhp can be ued to determne the couplng between waveront aberraton acro the pupl and mage blur or an optcal ytem. A waveront aberraton, pobly due to an optcal urace mperecton, wll caue the lght to propagate wth lope error acro the beam ootprnt. When thee ray ntercept the ocal plane, they wll be hted rom the deal ocu due to the aberraton n the ame way that the central ray wa ound to ht. Applyng the method above, the mappng between waveront lope aberraton and mage error ε ( ρ, ρ ) = FD θ ( ρ, ρ ) x y n x y Eq. 13 where ε(ρ x, ρ y ) repreent mage aberraton (ρ x, ρ y ) repreent the waveront lope varaton acro the beam ootprnt. (ρ x, ρ y ) are the pupl coordnate over the beam ootprnt wth dameter D. F n the nal ytem ocal rato For example, a 20 mm /10 len wth 2 µm PV atgmatm n one urace wll caue waveront varaton o 1 µm PV, whch ha lope varaton o ±100 µrad. I th ued n an optcal ytem where the beam ootprnt ee the ull urace, the mage degradaton due to the atgmatm n the len would gve a pont pread uncton wth 20 µm radu. 7. CONCLUSION An eay method o relatng mage moton to tlt and decenter o optcal element wa developed. Th drectly applcable to ytem that requre mage tablty, uch a pectrograph, laer projector, photonc coupler, and ytem that requre boreght. Thee relatonhp can be appled or any element n an optcal ytem wthout requrng a complete mulaton o the ytem only a ew mple parameter mut be known. Thee ame relatonhp can be appled to determne mage degradaton rom urace rregularte. REFERENCES 1. W. J. Smth, Modern Optcal Engneerng, 3rd Edton.,(McGraw-Hll, 2000). 2. J. E. Grevenkamp, Feld Gude to Geometrcal Optc, (SPIE Pre, 2004). 3. V. N. Mahajan, Optcal Imagng and Aberraton, Part 1. Ray Geometrcal Optc, pp (SPIE Pre 1998). 4. Zemax Development Corporaton, Bellevue WA, ( 5. V. J. Doherty, P. D. Chapnk, Precon evaluaton o len ytem ung a nodal lde/mtf optcal bench, n Advanced Optcal Manuacturng and Tetng II, Proc. SPIE 1591, pp (1991). Current Development In Len Degn And Optcal Engneerng VII Proc. SPIE 6288 (2006) 62880I-12