Aberration fields of a combination of plane symmetric systems

Transcription

1 Aberraton felds of a combnaton of pne symmetrc systems Lor B. Moore Anastaca M. vsc and Jose Sasan * College of Optcal Scences Unversty of Arzona 630 E. Unversty Boulevard Tucson AZ 857 USA * Correspondng author: ose.sasan@optcs.arzona.edu Abstract: By generalzng the wave aberraton functon to nclude pne symmetrc systems we descrbe the aberraton felds for a combnaton of pne symmetrc systems. The combned system aberraton coeffcents for the felds of sphercal aberraton coma astgmatsm defocus and dstorton depend on the ndvdual aberraton coeffcents and the orentatons of the ndvdual pne symmetrc component systems. The aberraton coeffcents can be used to calcute the locatons of the feld nodes for the dfferent types of aberraton ncludng coma astgmatsm defocus and dstorton. Ths work provdes an alternate vew for combnng aberratons n optcal systems. 008 Optcal Socety of Amerca OCIS codes: Aberraton expansons; Geometrc optcal desgn. References and lnks. R. A. Buchroeder Tlted component optcal systems Ph.D. dssertaton Unversty of Arzona Tucson Arzona R. V. Shack Aberraton theory OPTI 54 course notes College of Optcal Scences Unversty of Arzona Tucson Arzona. 3. K. P. Thompson Descrpton of the thrd-order optcal aberratons of near-crcur pupl optcal systems wthout symmetry J. Opt. Soc. Am. A K. P. Thompson Aberraton felds n tlted and decentered optcal systems Ph.D. dssertaton Unversty of Arzona Tucson Arzona J. M. Sasan Imagery of the Bteral Symmetrc Optcal System Ph.D. dssertaton Unversty of Arzona Tucson Arzona J. M. Sasan ow to approach the desgn of a bteral symmetrc optcal system Opt. Eng M. Andrews "Concatenaton of characterstc functons n amltonan optcs" J. Opt. Soc. Am G. Forbes "Concatenaton of restrcted characterstc functons" J. Opt. Soc. Am B. Stone and G. Forbes "Foundatons of frst-order yout for asymmetrc systems: an applcaton of amlton's methods" J. Opt. Soc. Am. A Introducton Optcal systems that do not have an axs of rotatonal symmetry have been and contnue to be of nterest n optcal desgn. In hs Ph. D. dssertaton Buchroeder [] descrbed a css of non-axally symmetrc systems called tlted component optcal systems constructed from axally-symmetrc components. These components may be tlted about ther nodal ponts n such a way that a partcur ray whch defnes the reference axs remans undevated. The aberratons of tlted component optcal systems then can be expressed usng Shack s [] vector aberraton functon where the feld vector s modfed by a dspcement term to account for each component tlt. th vector notaton the fnal system aberraton felds can be found and analyzed as shown by Thompson [34]. A pne symmetrc system s a system that has a pne of symmetry: that s one half of the system s a mrror mage of the other. Axally symmetrc and double-pne symmetrc systems belong to the css of pne symmetrc systems. Sasan [56] developed an aberraton functon for descrbng pne symmetrc optcal systems that uses a vector to defne the # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5655

2 drecton of pne symmetry. The queston s: can the pne symmetrc formalsm be extended to a combnaton of pne symmetrc systems whch do not necessarly share the same orentaton for ther respectve pnes of symmetry? Usng the vector notaton developed by Thompson [34] we extend the pne symmetrc formalsm to a combnaton of pne symmetrc systems. The fnal or global system s not necessarly pne symmetrc. Instead t may not have any dentfable symmetry. The result of the work n ths paper s a more general theory for non-axally symmetrc systems than tlted component optcal systems gven that the component systems are not restrcted to beng axally symmetrc. For example ths theory can be appled to both systems comprsed of off-axs aspheres as long as a pne of symmetry can be defned and tlted pne symmetrc optcal components f they are tlted n the pne of symmetry. owever ths theory s not applcable to all asymmetrc systems. It cannot be appled to systems wth components that can not be smplfed as pne symmetrc. There are several methods to concatenate asymmetrc systems whch can be used to combne pne symmetrc systems. For example Andrews [7] Forbes [89] and Stone [9] have developed methods for the concatenaton of asymmetrc systems usng amltonan methods. Our concatenaton methodology dffers n that t s based n component system rotaton about an optcal axs ray. In addton to these methods modern commercal lens desgn programs are capable of analyzng combnatons of optcal systems. Our approach provdes an ntutve understandng of the combnaton of pne symmetrc systems that can gude a lens desgner usng commercal lens desgn soware. e show whch of the aberratons are dependent on the orentaton of the pne of symmetry. And we show that for these aberratons the combned system aberraton s smply the sum of the vectors denotng the drecton of the pne of symmetry of the subsystem weghted by the aberraton coeffcent for each subsystem. Conceptually t follows that a gven aberraton can be canceled by addng an equal and opposte amount of the aberraton f the orentaton of the pne of symmetry s chosen properly. It also follows that the aberraton cannot be canceled f the pne of symmetry orentatons are not chosen properly. To ad n ths conceptual understandng we have ncluded fgures for each of the aberratons and have grouped smr types of aberraton felds. For completeness we fnd the nodes of each of the aberraton felds. Ths provdes a conceptual example of how the aberraton felds for a combnaton of pne symmetrc systems contrbutes to the overall system aberratons. e buld on prevous work [34] n aberraton theory add graphcs to help crfy concepts and contrbute to the theory by ncludng pne symmetrc component systems. Specfcally the new contrbutons to the theory are: component system addton s specfed by angur dspcement rather than by component tlt resultng n an alternate vew of system concatenaton a wder css of optcal systems regardng the symmetry of the component systems can be treated 3 we pont out and dscuss the occurrence of lne nodes and 4 the graphcs contrbute to the understandng of vector aberraton theory.. Aberraton functon and felds The wave aberraton functon for a rotatonally symmetrc optcal system n vector form [] s where k m n k nm n n k m n s the feld vector and s the aperture vector as shown n Fg.. # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5656

3 Fg.. Conventons for the pne symmetry feld and aperture vectors. The aberraton functon can be modfed for a pne symmetrc system [5]: q p n m p q mn k k p q q n n m p n k where s a unt vector that specfes the drecton of pne symmetry. The ndces k m n p and q are nteger numbers. On the rght hand sde of Eq. the s represent the aberraton coeffcents and convey the magntude of a gven aberraton. Note that the frst subscrpt knp s the algebrac power of the feld vector and the second subscrpt mnq s the algebrac power of aperture vector. To combne several systems one can use a feld dspcement term σ for each of the tlted component systems as Thompson dd. e nstead combne several pne symmetrc systems usng the vector that ndcates the retve orentaton among each of the pne symmetrc systems. The aberraton functon descrbes the aberratons about the optcal axs. The center of the feld of vew the center of the aperture stop and the pupls le on the optcal axs ray []. Optcally the optcal axs ray s a straght lne; that s lookng at the optcal axs from mage space the axs appears lke a straght lne whle n realty t s a ray that s reflected refracted or dffracted by the system surfaces. In effect the system can be unfolded such that the optcal axs ray s a straght lne. The vector s perpendcur to the optcal axs ray. Parametrc expressons for the aberraton coeffcents of a pne symmetrc system are gven by Sasan [56]. Usng the notaton establshed by Sasan the aberraton functon for a sngle pne symmetrc system up to fourth-order s: z A pne symmetrc system Ext pupl pne Image pne α φ θ # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5657

4 The total or global aberraton functon of a number of pne symmetrc systems wth retve orentaton and wth aberraton coeffcents q p n q n m p n k s the sum of the ndvdual aberraton functons. The fact that the total aberraton s stll a sum of the ndvdual surface aberratons even when there are tlted or decentered components was dscussed by Buchroeder []. The sum to fourth-order s: Ths combnaton of pne symmetrc systems stll shares the same optcal axs ray. The retve orentaton of each of the systems s conveyed by the vectors. Fgure shows an example of possble orentatons of these vectors as seen lookng down the optcal axs ray. Fg.. Possble orentaton of vectors as seen lookng down the optcal axs. Notce that the vectors are all unt vectors n dfferent drectons. The optcal axs s n and out of the page at xy 00. To organze the aberratons n Eq. 4 we frst defne the varables lsted n Table. These varables bel each of the aberraton coeffcents by the aberraton type and feld dependence. Ths table also shows whch of the aberratons have no dependence on or are dependent on or. It shows how the aberraton coeffcents from each component are combned nto the system aberraton coeffcent. For the aberratons that depend on the combned system aberraton coeffcent s smply the sum of the aberraton coeffcent for the ndvdual components multpled by the vector denotng the orentaton of pne of symmetry. In general the pne of symmetry for each aberraton coeffcent of the combned system wll have a dfferent pne of symmetry so the combned vectors are redefned. To mnmze the combned system aberratons ether the ndvdual subcomponent aberraton coeffcents can be mnmzed or lke wth axally symmetrc systems the combned aberraton can be reduced by bancng aberratons wth equal but opposte amounts of the ndvdual component aberratons as long as the proper orentaton of the ndvdual pnes of symmetry s chosen. # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5658

5 Table. Coeffcent and vector defntons Unform Pston: up Defocus: d 0000 Defocus from Constant Astgmatsm: 000 dca Constant Astgmatsm: caca Constant Coma: cc cc Quadratc Dstorton I: qdiqdi 00 Sphercal Aberraton: sa Feld Curvature: fc 000 Feld Dspcement: fd fd 000 Magnfcaton: m 00 Magnfcaton from Anamorphsm: 0 ma Anamorphsm: aa 0 Lnear Astgmatsm: 0 Quadratc Dstorton II: qdiiqdii 0 Lnear Coma: lc 300 Cubc Dstorton: cd 300 Lnear Pston: lplp 000 Quadratc Pston I: qpi 0000 Quadratc Pston IIa: 000 qpiia Quadratc Pston IIb: qpiibqpiib 000 Feld Tlt: 00 Cubc Pston: cpcp 3000 Quadratc Astgmatsm: 00 Quartc Pston: qp In Table the terms constant unform lnear quadratc and cubc descrbe the algebrac power of the feld dependence frst subscrpt. e chose the name unform pston for the aberraton that would typcally be called constant pston n ths paper so that the subscrpt would be dfferent from cubc pston whch uses cp. Each tme the drecton unt vector occurs n Table t s always normalzed to one. For example n the constant coma case we have: cc Therefore the coeffcent accounts for the entre weght such as n the constant coma term: cc Multplyng Eq. 5 and Eq. 6 results n the constant coma equaton n Table. # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5659

6 In order to better llustrate the feld dependence of the aberraton functon n Eq. 4 we group the aberraton coeffcents by ther dependence on the aperture vector. As shown n Table ths groups the aberraton functon nto sx aberraton felds: sphercal aberraton coma astgmatsm defocus dstorton and pston. Table. Aberraton felds of a combnaton of pne symmetrc systems Feld of Pston up lp lp qpi qpiia qpiib qpiib Feld of Dstorton fd fd m qdi qdi qdii qdii cd Feld of Defocus d dca fc { } Feld of Astgmatsm ma cp cp a a * qp { } ca ca Feld of Coma { } Feld of Sphercal Aberraton sa cc cc lc Some of the terms n Table nvolve the product of vectors such as or. Ths operaton named vector multplcaton by Shack [] s dfferent from both a vector dot product and a vector cross product. If we have two vectors A and B expressed as then the vector product snαˆ cosαˆ sn βˆ cos βˆ A a exp α a B bexp β b AB s defned as: AB abexp ˆ α β ab sn α β ˆ cos α β The conugate of a vector s a reflecton of the vector across the y-axs: B * sn β ˆ cos β ˆ b sn βˆ cos βˆ bexp β b The followng vector denttes were also used to generate Table [3]: A B AC [ A A B C A BC] * and A BC AB C For an expnaton of vector multplcaton see Thompson [3]. The vector denttes shown n Eqs. and are used to splt the anamorphc dstorton and each of the astgmatsm terms nto a combnaton of two aberratons. Ths s dscussed n the appendx. The aberratons dscussed here are to the fourth-order of approxmaton as dscussed by Sasan [5] and do not nclude hgher-order terms. For example terms lke cylndrcal feld curvature are not consdered. The use of hgher-order terms would mpact the nature of the aberraton felds and ther pont and nodal lnes where they vansh. # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5660

7 3. Graphcal vew of aberraton feld components All of the components of the aberraton felds n graphcal form are shown n Tables 3 and 4 except for pston whch s a phase error that does not affect the mage. These are smr to those presented by Thompson for tlted component systems but they correspond to a combnaton of pne symmetrc systems whch ncludes tlted component systems. In our graphcal representaton anamorphsm s descrbed wth respect to the average mage sze. Astgmatsm s descrbed wth respect to the medal surface; see Appendx. Table 3. Aberraton feld components that contrbute to dstorton mappng errors shown n both grd and vector plot forms. In the grd form the dotted lnes show the nomnal mappng postons of a square grd. The sold lne shows the dstorton of the square grd. In the vector form the vectors show the magntude and drecton of the mappng dstorton. All vectors used n creatng these graphs are pontng to the rght. Feld Dspcement Magnfcaton Anamorphsm Quadratc Dstorton I Quadratc Dstorton II: Cubc Dstorton: # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 566

8 Table 4. Aberraton feld components. The feld of defocus s represented by crcles that convey the sze of the defocused mage across the feld. Astgmatsm s represented by a proecton of the astgmatc focal lnes. Coma s represented by a collecton of crcles. Each crcle represents a fxed magntude of the aperture vector thus the collecton of crcles show the magntude and orentaton of the aberraton n the feld. Sphercal aberraton s represented by crcles that show the magntude of the aberraton. All vectors used n creatng these graphs are pontng to the rght. Color onlne: Red denotes locatons n the feld where the focus poston s before the mage pne Defocus Feld Tlt Feld Curvature Constant Astgmatsm Lnear Astgmatsm Quadratc Astgmatsm Constant Coma Lnear Coma Sphercal Aberraton 4. Aberraton feld nodes Snce each of the aberraton felds lsted n Table s the sum of multple terms each wth a dfferent feld dependence t s possble that these terms sum to zero at certan locatons n the feld. These specal locatons are called feld nodes and geometrcally can be ponts lnes or crcles. Followng the work of Thompson [34] for tlted component systems we fnd the nodes for a combnaton of pne symmetrc systems. In many cases the locaton of the nodes s smr but because we use dfferent vector defntons than Thompson the equatons used to fnd the nodes are not always the same. To fnd the nodes n the feld of vew we treat each aberraton feld ndependently. Notce that the pupl dependence for each feld n Table has been factored out. Ths leads to a term n brackets that depends only on the feld vector pne of symmetry vectors and the # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 566

9 aberraton coeffcents. In order to fnd the locaton of the nodes we set ths bracketed term equal to zero and solve for as a functon of the aberraton coeffcents and. 4. Sphercal aberraton The feld of sphercal aberraton s unform; t does not vary as a functon of the feld of vew so there are no feld nodes. 4. Coma The feld of coma can be lnear or constant wth respect to the feld of vew: Thus there can be one feld node: 4.3 Astgmatsm cc cc 0. 3 lc. 4 cc cc lc The magntude of the astgmatsm can be unform lnear or quadratc as a functon of the feld of vew. Ths feld can have one or two nodes where the astgmatsm vanshes. Shack [] was the frst to recognze that the feld of astgmatsm could have two nodes and he called ths case bnodal astgmatsm. In the presence of constant lnear and quadratc astgmatsm the node postons satsfy the equaton: caca 0. 5 The locatons of the nodes are at the two feld ponts that can be found solvng Eq. 5 and are gven by: ca ± ca. 6 4 There are some specal cases where the locaton of the node s smplfed: If there s no constant or lnear astgmatsm ca 0 and 0 there s one feld node at the optcal axs ray 0. hen there s no constant astgmatsm 0 there are two nodes: ca 0 and. 7 3 If there s no lnear astgmatsm 0 then there are two nodes at: ±. 8 ca ca 4 If there s no quadratc astgmatsm 0 the feld of astgmatsm s lnear n feld and there s only one node located at: ca ca. 9 # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5663

10 4.4 Defocus The nodes of defocus can be pont nodes lne nodes or crcle nodes. A sngle pont node s a crcle node of radus zero. The defocus nodes must satsfy the followng equaton: d 0 dca fc. 0 Note that the astgmatsm terms n the feld of defocus descrbe the defocus requred to move from the mage pne to the medal astgmatc surface. A lne node wll occur f the quadratc astgmatsm term s banced by the feld curvature term fc. As an example f the defocus also bances the defocus from constant astgmatsm then the lne node satsfes the followng equatons: d dca 0 or 0. Thus the lne node occurs when the feld vector s perpendcur to the vector. If the two defocus terms do not cancel then d dca. 3 The lne node wll sh and t no longer crosses through the center of the feld of vew. It s possble to sh the node outsde the feld of vew. An example of a lne node s plotted n Fg. 3. Fg. 3. A lne node n the feld of defocus from a combnaton of feld tlts. Color onlne: Red s focused before the mage pne. owever n general the nodes from defocus wll be crcur. The values of the coeffcents and the drectons of the vectors wll determne the locaton of the crcle and where t s centered. Fgure 4 shows an example crcur node n the feld of defocus. Fg. 4. A crcur node n the feld of defocus from a combnaton of feld curvature and constant defocus. Color onlne: Red s focused before the mage pne. If the feld tlt s banced by the lnear astgmatsm then the crcur node s centered on the on-axs feld pont. The radus of ths crcur node s derved as follows: d 0 dca fc d dca fc 4 # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5664

11 d dca. 5 fc Note that the quantty n parenthess must be negatve or else the feld radus where the crcur node exsts wll be magnary and there wll not be a crcur node at all. 4.5 Dstorton Dstorton s purely a mappng error whch does not cause blurred mages. Therefore t s possble to correct dstorton wth post-processng of the mage. There are stll applcatons where post-processng s not possble or dstorton should be mnmzed to reduce the error n the post-processng. For ths reason we descrbe the types of nodes found n the feld of dstorton. The dstorton nodes satsfy the followng equaton: fd fd * 0 m ma a a qdii qdii qdi qdi cd. 6 Table 5 shows some sample dstorton vector plots wth two three and four nodes and lst the contrbutng coeffcent amounts. Table 5. Some dstorton plots showng 3 and 4 nodes. The surface maps represent the magntude of the dstorton. The ponts where the surfaces meet the pne are the nodes. In the vector plots the vectors represent both the magntude and drecton of the mappng dstorton. The shadng represents the magntude of the dstorton. Darker shades have less dstorton. All vectors used n creatng these graphs are pontng to the rght. nodes 3 nodes 4 nodes fd -0.5; qdi - a -; cd fd ; qdi - qdii qdii As wth the feld of defocus t s also possble to get crcur nodes and lne nodes wth the feld of dstorton. For example f all other terms cancel except feld dspcement and quadratc dstorton I then the feld of dstorton smplfes to { }. 7 fd fd qdi qdi Ths leads to a crcur node shown n Table 6 wth a radus derved as follows: # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5665

12 fd fd 0 qdiqdi 8 fd. 9 Lke wth the case of crcur defocus nodes fd and qdi must have opposte sgns or the radus of the crcle node wll be magnary and there wll not be a node n the feld. Another crcur node can be created from magnfcaton and cubc dstorton. Ths set also has a node on axs. If: 0 m ma cd then there s one node on axs and a crcur node shown n Table 6 at qdi m ma cd 30 m ma. 3 cd Agan the term m ma and cd must have dfferent sgns for there to be a node. A lne node n the dstorton feld may be found when there s no cubc dstorton. One smple example of a lne node s the case of quadratc dstorton II. Addtonally by addng an equal but opposte amount of quadratc dstorton I to quadratc dstorton II n effect qdi qdii and qdi qdii a lne node wll be created. A lne node may also be found by addng complementary amounts of magnfcaton and anamorphsm. It s also possble to get a lne node and a pont node usng a combnaton of quadratc dstorton II and magnfcaton. Table 6. Some dstorton plots showng lne and crcur nodes. The surface maps represent the magntude of the dstorton. In the vector plots the vectors represent both the magntude and drecton of the mappng dstorton. The shadng represents the magntude of the dstorton. Darker shades have less dstorton. All vectors used n creatng these graphs are pontng to the rght. a Crcur dstorton node b Crcur dstorton node c Lne dstorton node wth on-axs node wth on-axs node fd -; qdi m ma -; cd m ma -0.75; qdii # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5666

13 5. Summary Ths paper adds to the theory of non-axally symmetrc systems. Specfcally we extended the pne symmetrc vector aberraton functon to determne the aberraton felds for a combnaton of pne symmetrc systems that do not necessarly share the same orentaton for ther respectve pnes of symmetry. Noteworthy s that the system combnaton s carred out by rotatons about the optcal axs ray of each system component. Ths paper provdes mathematcal expressons for the resultng aberraton felds: sphercal aberraton coma astgmatsm defocus and dstorton. To help the conceptual understandng of the aberratons we defned and plotted the ndvdual aberraton terms that contrbute to each feld. In addton the paper furthers the concept of feld nodes by usng the equatons for the aberraton felds to calcute and llustrate the locatons of the feld nodes whch may be pont nodes lne nodes or crcle nodes dependng on the aberraton feld. Although ths theory s applcable only to the range of asymmetrc systems that can be consdered pne symmetrc t s n prncple more general than the prevous theores that apply only to axally symmetrc system components. Appendx A The anamorphc dstorton and astgmatsm terms n Eq. 4 can each be splt n to two terms. For the case of anamorphc dstorton the splt terms change the reference to an average magnfcaton. For the case of astgmatsm the splt terms change the reference to the medal astgmatc surface.. Anamorphsm Anamorphc dstorton s gven by: Ths equaton can be splt nto: * 0 0 * ma a a. 33 The frst term n Eq. 33 represents a magnfcaton change and the second term represents a mappng change n two orthogonal drectons. It s the second term that s used n Table 3. Thus n Table 3 anamorphsm s descrbed wth respect to the average magnfcaton rather than as the usual anamorphc one-drectonal mappng stretch. Ths s shown graphcally n Fg. 5. Fg. 5. Anamorphsm can be descrbed as an average magnfcaton center fgure plus an anamorphc term fgure on the rght. # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5667

14 . Astgmatsm For the case of astgmatsm each of the terms n Eq. 4 reted to the feld of astgmatsm can be splt nto two aberraton components. For example the term that descrbes lnear astgmatsm s splt nto: 0 The frst term of the splt. 34 has the same functonal form as feld tlt. Ths locates the medal astgmatc surface on a tlted pne. In Table the feld of astgmatsm s descrbed from the medal astgmatc surface by the second term of the splt. Fgure 6 shows the components of the feld of astgmatsm constant lnear and quadratc astgmatsm and ther reton to the mage pne as expressed n Eq. 4. The surfaces shown are the locus of the astgmatc focal lnes. The medal surface s shown n blue onlne. In contrast Fg. 7 shows the astgmatc surfaces wth respect to the medal surface as mathematcally represented n Table and graphcally shown n Table 4. Ths representaton s used by Thompson [34]. Fg. 6. Astgmatc surfaces and ther retonshp accordng to the astgmatsm terms n Eq. 4. Defocus from the mage pne s n the Z drecton. Color onlne: The medal surface s shown n blue The surfaces n Fgs. 6 and 7 are the locus of the astgmatc lne mages. These surfaces are called sagttal and tangental astgmatc surfaces for the case of quadratc astgmatsm. owever because the orentaton of the lne mages n lnear astgmatsm depart from the radal symmetry of quadratc astgmatsm the terms sagttal and tangental are not qute approprate for descrbng the lne mages of lnear astgmatsm. Instead we wll refer to them wth the more general term lne mage astgmatc surfaces [4]. As shown n Fg. 7 for the case of lnear astgmatsm the lne mage astgmatc surfaces are along a cone and astgmatc lnes wth the same orentaton are located along a lne. For example note the same orentaton of the astgmatc lnes along the dashed red or green lnes n Fg. 7 for lnear astgmatsm. # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5668

15 Fg. 7. Locaton and orentaton of the astgmatc surfaces for the lnear constant and quadratc astgmatsm wth respect to the medal surface and as mathematcally represented n Table. Defocus from the medal surface s n the Z drecton. The dashed lnes n the lnear astgmatsm fgure hghlght the locatons of the sagttal and tangental foc. Color onlne: The medal surface s blue. The tangental foc are red. The sagttal foc are green. 3. Transverse ray aberratons The transverse ray aberraton vector ε was used to make some of the fgures n ths paper. Table 7 provdes the transverse ray aberratons derved from the standard retonshp ε nu' 35 where n s the ndex of refracton and u s the margnal ray slope n mage space. # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5669

16 Feld of Dstorton Feld of Defocus Feld of Astgmatsm Feld of Coma Feld of Sphercal Aberraton nu' ε fd Table 7. Transverse ray aberratons fd m ma aa qdi qdii qdii cd qdi nu' ε { } nu' ε nu' ε nu' ε 4 d dca { * * * } ca ca * fc cc cc cc lc sa The followng vector denttes were used: a a a a 38 a a a 39 and 4 40 where a * s any vector etc that does not depend on. # $5.00 USD Receved 30 May 008; revsed 9 Jul 008; accepted 30 Aug 008; publshed 9 Sep 008 C 008 OSA 9 September 008 / Vol. 6 No. 0 / OPTICS EXPRESS 5670