1 August 000 Optcs Communcatons 18 000 1 10 www.elsever.comrlocateroptcom Transverse angular shft n the reflecton of lght beams Javer Alda ) Optcs Department. UnÕersty Complutense of Madrd, School of Optcs, AÕda. Arcos de Jalon srn, 8037 Madrd, Span Receved 5 January 000; receved n revsed form 6 Aprl 000; accepted 4 May 000 Abstract The reflecton of a lght beam onto a planar nterface can be out of the ncdence plane due to the msmatchng between the symmetres of the beam and the reflecton geometry. The state of polarzaton of the beam s also nvolved n the descrpton of the transverse angular shft. The moment characterzaton of laser beams s used, along wth the power expanson of the reflecton coeffcents and the geometrc factors, to fnd ths transverse angular shft. q 000 Elsever Scence B.V. All rghts reserved. 1. Introducton The reflecton of laser beams at delectrc and metallc nterfaces s a very common phenomenon n practcal systems handlng and transformng laser beams. The analyss of a delectrcrdelectrc nterw1,x restrcted face has been deeply treated by Porras to the plane of ncdence. He uses the characterstc parameters of a lght beam defned n terms of the moments of the beam. In ths paper we apply a smlar procedure to the calculaton of the transverse angular shft. Ths effect means a reflecton out of the plane of ncdence. The transverse angular shft was outlned and demonstrated n the 70 s for total nternal reflecton w3 5 x. Hugonn and Pett also appled a moment calculaton to obtan the dsplacement of the center of gravty of the beam. A few years ago, Nasalsk presented a characterzaton of the non-specular effects n the reflecton of lght beams wx 6. Hs treatment s based on a detaled ) Tel.: q34-91-394-6874; fax: q34-91-394-6885; e-mal: j.alda@fs.ucm.es analyss of the changes produced n the reflecton. Some of the results of our contrbuton wll be compared wth Nasalsk s calculaton. To characterze the beam we calculate the moments of the ntensty dstrbuton of ts plane wave decomposton. The transformaton of the moments by the reflecton s calculated by usng a Taylor expanson of the reflecton coeffcents and by takng nto account the geometry of the ncdence and the state of polarzaton of the ncomng beam. The reflected ampltude for a delectrcrmetallc plane nterface s obtaned usng the complex nature of the reflecton coeffcents. Once the moments of the reflected beam are obtaned, the angular shfts along the plane of ncdence and along a perpendcular drecton are calculated. The results show that an out-of-plane reflecton s obtaned when the symmetry of the beam does not match wth the symmetry of the reflecton geometry. It also depends on the state of polarzaton of the beam. Along the paper we wll assume that the beam s paraxal. Ths means that t extends wthn a small regon around the prncpal drecton of propagaton. 0030-4018r00r$ - see front matter q 000 Elsever Scence B.V. All rghts reserved. PII: S0030-4018 00 00788-4
( ) J. AldarOptcs Communcatons 18 000 1 10 In Secton we frst descrbe how the three dmensonal beam s transformed by a plane nterface. We calculate the transformaton by decomposng the beam nto plane waves and by usng the reflecton coeffcents. A detaled analyss of the geometry of the ncdence wll be necessary to correctly apply the reflecton coeffcents. Also the state of polarzaton wll play an mportant role n the calculaton. Secton 3 s devoted to the change of the moments characterzng the beam. A Taylor expanson untl second order s appled around the axs defned by the center of the ncomng beam. In Secton 4 we calculate the transverse angular shft wth the general equaton obtaned to transform the moments. Some cases are treated numercally n ths secton to show up the nfluence of the symmetry of the beam, ts orentaton, ts modal character, and the state of polarzaton. Fnally, Secton 5 summarzes the man conclusons of the paper.. Transformaton of the ampltude dstrbuton A gven laser beam can be descrbed by ts electrc feld. The vectoral character s relevant to fnd ts components along the ncdence and perpendcular planes. If we assume that the laser propagates along the Z-axs, the electrc feld at a gven z plane s gven by ts ampltude dstrbuton n that plane, C Ž x, y., and by ts polarzaton state. In ths paper we have dropped the tme dependence because we are prmarly nterested n the spatal and angular characterzaton. The ampltude dstrbuton s expanded as a sum of plane waves whose ampltudes are gven by the Fourer transform of the ampltude feld dstrbuton, FŽ j,h.. Ths angular spectral dstrbuton, the angular spectrum, also carres nformaton of the polarzaton state of the beam. If the beam propagates around ts man drecton nsde a paraxal angular regon, the magntude of the plane wave decomposton wthn a homogeneous delectrc s expressed as follows, FŽ j,h, z. ½ 5 p 1 sfž j,h. exp z 1y l Ž j qh., l Ž 1. where l s the wavelength wthn the homogeneous delectrc meda, and j and h are the spatal frequences along the X and Y drectons respectvely. The drecton of propagaton for each plane wave s gven by the untary vector ˆ Is lj,lh,( 1y Ž lj qlh.. Ž. ž / The nterface s characterzed by the ndex of refracton n both sdes of the nterface and by ts untary normal vector N, ˆ ˆNs Ž ysn 0,0,ycos 0., Ž 3. where 0 s the angle of ncdence of the center of the ncomng beam. In the reflecton, the ampltude of each plane wave changes accordng wth the value of the reflecton coeffcent, r. Dependng on the polarzaton state of the ncomng wavefront, t wll be necessary to use both the parallel and the perpendcular reflecton coeffcents, or ether one. These coeffcents are gven by the Fresnel formulae n terms of the ndexes of refracton and the angle of ncdence of the ncomng plane wave wx 7. The ncdence angle for each plane wave s gven as: Ž j,h. y1 scos Ž NPI ˆ ˆ.. To obtan the reflected plane wave for each ncomng component t wll necessary: Ž. to apply the vectoral reflecton law to know the drecton of the reflected plane wave, and Ž. to multply the ncdent ampltude dstrbuton by the reflecton coeffcent to obtan the ampltude of the reflected plane wave. The vectoral form of the reflecton law wx 8 provdes the followng expresson for the reflected vector: ˆ Rs ljcos ysn ( 1ylj ylh,lh, ž 0 0 ( ylj sn 0ycos 0 1ylj ylh. 4 Once the drecton of the reflecton s known by usng the prevous equaton, t s possble to transform the ncomng plane wave nto the reflected one by changng the coordnate system and modfyng the ampltude by means of the reflecton coeffcents. The reference system for the ncdent beam was chosen n such a way that Eq. Ž. 1 descrbes the beam Ž see Fg. 1.. Ths means that the Z axs represents the straght lne where the center of gravty of the /
( ) J. AldarOptcs Communcatons 18 000 1 10 3 The electrc feld vector for any ncomng component of the plane wave decomposton can be wrtten as, F Ž j,h. sf Ž j,h. uˆ qf Ž j,h. uˆ x x y y sf Ž j,h. uˆ qf Ž j,h. u ˆ. Ž 5. 5 5 H H Fg. 1. Geometry of the coordnate systems for the descrpton of the ncdent and reflected beams. The coordnate system for the reflecton s the mrror mage of the coordnate system for the ncdent beam. beam les. Then, the frst order moments of the ntensty dstrbuton of the angular spectrum are zero for the ncdent beam. Besdes, the reference system s algned wth the plane of ncdence. The XZ plane defnes the man plane of ncdence,.e., the plane of ncdence for the center of the beam. Ths choce means that the projecton on the XZ plane of the electrc feld of the plane wave that propagates wthn ths plane s the parallel component for the center of the beam. The reflected beam s descrbed wth a reference system that s the mrror reflecton of the one used for the ncdent beam. Therefore, the Z X axs remans close to the man drecton of propagaton of the beam, and X X and Y X represent the coordnates on the transversal plane. The actual drecton of the propagaton of the reflected beam s gven by the frst order moments of ts plane-wave decomposton. The departure of ths actual drecton from the Z X axs defnes the angular shfts. The ncdent and the reflected reference systems have opposte characters: one of them s dextro and the other s levo. However, ths can be taken nto account f necessary. The relaton between the ncdent and reflected reference systems makes the coordnate transformaton very X smple: Ž x, y X. sž x, y., and n the spatal-frequency X doman Ž j,h X. sž j,h.. In ths equaton uˆ5 and uˆh are two untary vectors along the parallel and perpendcular drectons respectvely. On the other hand, uˆx and uˆy are the untary vectors along the axes of reference of the transversal plane XY. In general uˆx and uˆ5 do not concde. Therefore, the decomposton n parallel and perpendcular drecton becomes a key ssue that s analyzed as follow. For every plane wave of the angular spectrum t s possble to defne a local ncdence plane. Ths plane s defned by the ncdent and normal vectors ŽEqs. Ž. and Ž 3... The cross product of these two vectors defnes a vector perpendcular to the local ncdence plane. Ths vector s gven as zsi=n. ˆ ˆ The projecton of ths vector on the transversal plane, XY, defnes the drecton of the perpendcular component of the feld. Ths drecton s gven by the Õx and Õy components of the vector z that can be wrtten as, Õx sylhcos 0, Ž 6. ( y 0 0 Õ sljcos y 1y lj y lh sn. 7 By usng these two components we defne an angle, b, that represents the local rotaton of the parallel and perpendcular drectons wth respect to the man parallel and perpendcular drectons. These man drectons concde wth the X and Y drectons respectvely. The angle wll be gven as bsytan y1 Õ rõ. Ž 8. x y The reflected angular spectrum can be calculated by locally rotatng the ncdent angular spectrum to meet the actual parallel and perpendcular drectons. After ths rotaton we apply the reflecton coeffcents, and fnally the rotaton s undone to retreve
4 ( ) J. AldarOptcs Communcatons 18 000 1 10 the correct orentaton. Ths transformaton can be wrtten n matrcal form as follows, Fx r cosb ysnb r5 0 cosb snb Fx r s ž / ž / ž 0 r / ž / y H Fy ž / F snb cosb ysnb cosb r cos b q r sn b Ž r y r. snbcosb 5 H 5 H Fx s. ž ž Ž r y r. snbcosb r sn b q r cos b / Fy / 5 H 5 H Ž 9. The last = matrx of the prevous equaton can be denoted as the reflecton matrx. Ths result concdes wth Eq. Ž 61. of Ref. wx 5. As we stated at the begnnng of ths paper, ths analyss s vald for lght beams whose angular extenson s wthn the paraxal range. Ths condton s also related wth the transversalty condton for lght beams. Ths means that once the paraxal condton can be appled then the component of the electrc feld along the propagaton drecton at a gven pont of a curved wavefront can be neglected. Therefore, wthn the paraxal approach, the electrc vector les on the transversal plane XY. 3. Transformaton of the moments The moments of the ntensty dstrbuton of the angular spectrum, m, j, are defned by the followng ntegral, HH m s < FŽ j,h. < j j h k dj dh. Ž 10. j,k The angular man drectons of propagaton and the components of the dvergence tensor are gven by w9,10x ux slm1,0rm 0,0, Ž 11. uy slm0,1rm 0,0, Ž 1. xx,0 0,0 Ž 1,0 0,0. u s 4l m rm y m rm, 13 uxys4l m1,1rm0,0y m1,0 m0,1 r m 0,0, Ž 14. yy 0, 0,0 Ž 0,1 0,0. u s 4l m rm y m rm, 15 both for the ncdent and reflected beam, as a functon of the moments of the angular spectrum of the beam. A superscrpt or r wll be added to dstngush between the ncdent and the reflected beam respectvely. For the ncdent beam, the electrc feld s gven by Eq. Ž. 5 wth respect to an orthogonal reference system on the transversal plane. At ths pont t s necessary to ntroduce a parameter descrbng the state of polarzaton of the ncomng wave. Ths parameter s the followng rato Fx Ž j,h. xž j,h. s. Ž 16. F Ž j,h. y Nasalsk defnes t n terms of the parallel and perpendcular drectons. However, n our case t s more nterestng to use the rato between the x and y components. Ths parameter s complex n nature < < a and wll be expressed as x s x e. Wthn the paraxal approach, as Nasalsk states, the polarzaton rato can be consdered constant for the whole beam, xž j,h. fx 0. If ths condton s not fulflled then the dependence of x wth respect to Ž j,h. must be taken nto account. By usng the rato of polarzaton the squared modulus of the ncdent plane wave s calculated as < F < s < x < q 1 < F <, Ž 17. y and therefore there s a relaton between the moments of the total ntensty and the moments of the ntensty of the component along the drecton Y. Ths relaton s, m, y j,k mj,k s, 18 < x < 0 q1 where the superndex, y means the moment of the component y of the ncdent beam. We assume that x0 s constant on the transversal plane. After some calculaton usng Eq. Ž. 9 the square modulus of the reflected dstrbuton, < F r <, can be wrtten as, r 4 4 ½ 5 H < F < s < r < cos bq< r < sn b ž / q < r < q< r < sn bcos b < x < 5 H 0 < < 4 < < 4 q r sn bq r cos b 5 H ž 5 H / ž 5 H / q < r < q< r < sn bcos b < r < y< r < =snbcosb< x < cosa < F <, Ž 19. where a s the phase of x. 0 5 0 y
( ) J. AldarOptcs Communcatons 18 000 1 10 5 As we can see, the result nvolves not only the reflecton coeffcents Ž r, r. 5 H, but also the geometry of the ncdence Ž b., and the polarzaton state of the beam Ž x. 0. These dependences can be obtaned as a functon of the spatal frequences Ž j,h.. In the paraxal regme the values of the reflecton coeffcents for the whole beam are around the values of the coeffcent for the center of the beam,.e., jsh s0, that ncdes at an angle 0. Then, after expandng n powers, we wrte the reflecton coeffcent as, 00 10 01 1 0 11 r Ž j,h. sr qr jqr hq r j qr jh 0 qr h q PPP, Ž 0. where the superscrpts ndcate the order of the dervaton wth respect to the varables Ž j,h..be- sdes, the dervatves must be evaluated at jshs0. In the prevous equaton we dropped the subscrpts 5 and H because the expanson s done for both coeffcents. After some calculus t s found that, wthn our reference system, the frst dervatve of r wth respect to h s zero, both for the parallel and the perpendcular drectons. Ths fact s a consequence of the geometry of the problem. Gven an ncdence angle, 0, f we take another ncdence drecton out of the man plane of ncdence Žplane XZ., the reflecton coeffcent wll vary n a symmetrc way wth respect to the value obtaned nsde the man ncdence plane. Therefore, the dependence of r wth respect to h reaches a statonary pont at hs0. Then, wthn the second order approach gven n the prevous equaton, there are only four coeffcents dfferent from zero. These coeffcents are: r 00, r 10, r 0, and r 0. Ths expanson s also referred n Nasalsk s paper n Eqs. Ž 7.a. and Ž 7.b.. However, he uses an exponental form for the reflecton coeffcents that s more adapted to hs formalsm. The stuaton of the local plane of ncdence and the decomposton onto the parallel and perpendcular local drectons are characterzed by the angle b. Ths angle was defned n the prevous secton ŽEq. Ž.. 8. The trgonometrc functons that appear n Eq. Ž 19. can be also expanded untl second order as follows, cos 4 bs1y L h, Ž 1. sn 4 bsl h, Ž. cosbsnbsylh, Ž 3. cos bsn bsl h, Ž 4. where L s lrtan 0. These equatons also provde another restrcton to the expanson. The angle of ncdence must be large enough to allow the approach. For example, f the ncdence s normal, the angle b vares from yprtopr nterchangng the role of the parallel and perpendcular components along the way. In ths case the approxmaton clearly fals. Therefore, we must assure that the transverse extenson of the beam s small compared wth the actual value of the ncdence angle. The transverse extenson of the ncdent beam s characterzed by the parameter uyy. Then, ths condton can be wrt- ten as uyy < 0. Ž 5. Also the polarzaton rato could be expanded n powers of the spatal frequences f necessary. At ths pont, and by usng these prevous equatons t s possble to calculate the moments of the reflected beam n terms of the moments of the ncdent beam. After applyng the prevous expansons for r 5, r H, and the trgonometrc functons, the moments of the reflected beam are as follows 1 N r mj,ks Ý a m, Ž 6 k,l jql,kqm. < x < 0 q1 lqms0 where the coeffcents are gven by < 00 < < < < 00 a s r x q r <, Ž 7. 0,0 5 0 H Ž H H H H. ž / 00 10 ) 00) 10 a s r r qr r < x < 1,0 5 5 5 5 0 q r 00 r 10 ) qr 00 ) r 10, Ž 8. < 00 < < 00 a sy r y r < Lx < < cosa, Ž 9. 0,1 5 H 0 a s r r qr r q< r < < x < 1 00 0 ) 00) 0 10,0 5 5 5 5 5 0 q r r qr r q< r <, Ž 30. 1 00 0 ) 00) 0 10 H H H H H 00 10 ) 00) 10 1,1 Ž 5 5 5 5. a s r r qr r Ž H H H H. 0 ž 00 10 ) 00) 10 y r r qr r Lx < < cosa, Ž 31. a s r r qr r ql < r < 1 00 0 ) 00) 0 00 0, 5 5 5 5 H / ž < < < < 5 H / y< r < < x < q r r qr r 00 1 00 0 ) 00) 0 5 0 H H H H 00 00 ql r y r. 3
6 ( ) J. AldarOptcs Communcatons 18 000 1 10 The angular characterstcs of the reflected beam can be obtaned from the results of Eq. Ž 6.. These angular parameters were defned n Eqs. Ž 11. Ž 15.. 4. Transverse angular shft One of the most nterestng consequences of the calculaton presented prevously s that t predcts a reflecton outsde of the plane of ncdence. Transwx 6 verse angular shfts have been already descrbed and demonstrated wx 3 by analyzng the three dmensonal propagaton of the electromagnetc feld along the nterface of two delectrc meda w11 13 x. In the present paper ths result s obtaned by calculatng the characterstc parameters of the reflected beam n three dmensons, extendng the Porras analyss who calculated nsde the plane of ncdence w1, x. Based on the transformaton of plane waves by means of the reflecton coeffcents, our approach allows to descrbe both the delectrcrdelectrc nterface partal reflecton, and the delectrcrmetallc nterface reflecton. The transverse angular shft s gven by Eq. Ž 1.. To better calculate t, we wll assume that the ncdent beam s algned wth the Z axs. Ths means that m1,0 sm 0,1 s0, and therefore ux suy s0. After some substtutons usng Eqs. Ž 7. Ž 3. and Eqs. r 13 15, we fnd the followng results for uy wthn the zero, frst, and second order of approxmaton n the calculaton of the moments of the reflected beam Ž Ns0,1, and respectvely.. u r Ž 0. su s0, Ž 33. y y 1 r uy Ž 1. s Ž a1,0uxyqa0,1u yy., Ž 34. 4l uy r Ž. 0 uxy uyy m,1 m1, m 0,3 a1,0 q a0,1 q a,0 q a1,1 q a 0, 4l 4l m0,0 m0,0 m0,0 sl. a q a uxx q a uxy q a uyy 0,0,0 1,1 0, 4l 4l 4l Ž 35. Fg.. Plot of the a coeffcents, normalzed to the factor Ž< x < q1., j 0, as a functon of the ncdence angle 0. The plots are for the case of an ArrAl plane nterface. Each plot contans several cases of lnear or ellptc polarzatons.
( ) J. AldarOptcs Communcatons 18 000 1 10 7 The zeroth order approach produces a null transverse shft. For the frst order approach we already fnd a non null transverse shft that depends on the elements of the dvergence tensor. The dependence nvolves hgher order moments Ž untl 3rd order. f the approach s extended to the nd order. Eq. Ž 34. can be compared wth Eq. Ž 40. of the Nasalsk s paper. Although Nasalsk splts the calculaton nto the parallel and perpendcular components, the dependence wth the angular dvergence s also found when the Raylegh range dstance used n hs calculatons s wrtten n terms of the dvergence. The second order approach ŽEq. Ž 35.. nvolves the same knd of dependence but n a lttle more complcated fashon. In the present paper t s possble to fnd the transverse angular shft wthout the parallel and perpendcular decomposton. Actually, the geometry of the ncdence and the state of polarzaton are ncluded n the a coeffcents through the Taylor, j expanson used to obtan the coeffcents n terms of L Žsee Eqs. Ž 1. Ž 4.. and the rato of polarzaton, x. 4.1. Numercal examples In ths secton we perform some numercal calculaton of the transverse angular shft n the second order approxmaton for some few typcal examples. These examples nvolve crcular and ellptc Gaussan beams, Hermte Gaussan modes, and several states of polarzaton. The calculaton wavelength s ls633 nm, and the nterfaces are ArrAl Žn Al s 1.44q5.3., ArrAg Ž n s0.q3.44. Ag, and external and nternal reflecton n an ArrGlass Žn s 1.5. nterface. The transformaton of the moments depends on the value of the a, j coeffcents. They are related wth the values of the refracton ndexes nvolved n Fg. 3. Varaton of the longtudnal and transverse angular shft as a functon of the angle of ncdence and the ellptcty of a Gaussan beam for the parallel and perpendcular lnear polarzatons. The axes of the ellptcal ntensty dstrbuton are orented at 458 wth respect to the plane of ncdence. The value of the orthogonal Gaussan dvergences along the axes of the ellpse are: u1 s 0 mrad, and u s 0, 13, and 10 mrad as ndcated n the fgure. The drecton of the axs descrbed by u1 s along the frst-thrd quadrants of the XY plane. The nterface s between two delectrc meda of ndex 1.5 and 1 Ž nternal reflecton..
8 ( ) J. AldarOptcs Communcatons 18 000 1 10 the nterface, n, n t, the angle of ncdence, 0, and the state of polarzaton of the ncomng beam defned by x 0. Fg. represents the values of the aj,k coeffcents ŽEqs. Ž 7. Ž 3.. as a functon of the angle of ncdence, 0, for an ArrAl plane nterface. Actually, the plots nclude the normalzaton factor 1rŽ< x < q 1. 0 that allows the comparson between dfferent states of polarzatons. Although the numercal values of these coeffcents are small, they are multpled by the values of the moments of the ncdent beam to obtan the moments of the reflected beam. The moments are calculated n the spatalfrequency doman and they can reach very large values producng a non neglgble angular shft. As we wll see n the cases presented below, the numercal value of the transverse angular shft s, n certan cases, comparable to the longtudnal angular shft obtaned wthn the plane of ncdence. The symmetry of the ncdent beam s a crtcal characterstc to enhance the transverse angular shft. Ths fact s shown n Fg. 3 where we plotted the angular shfts for ellptc Gaussan beams wth dfferent rato between axes. The ellptcal ntensty dstrbutons are orented havng ther axs 458 apart from the plane of ncdence. The case s an nternal reflecton for a delectrcrdelectrc Ž n rn s 1r1.5. nterface. The beams are lnearly polarzed along the parallel and perpendcular man drectons. These drectons concde wth the X and Y axes respectvely. The plots on the left correspond wth the longtudnal angular shft, u r. The plots on the rght x are for the transverse angular shft, uy r. We can see that u r s about twce larger than u r. The transverse x angular shft ncreases wth the ellptcty of the ncomng beam, beng neglgble wth respect to ux r for crcularly symmetrc beams. Our results for ux r y t Fg. 4. Varaton of the longtudnal and transverse angular shfts for several Gaussan Hermte modes. The orentaton of the axes of the ellpse s 458 wth respect to the plane of ncdence Ž as n Fg. 3.. The Gaussan dvergences of the mode Ž 0,0. are u1 s 0 mrad and u s 6.7 mrad. The modes represented are Ž 1,1., Ž,., and Ž 3,3., as shown n the fgure. The states of polarzaton are lnear along the parallel and perpendcular man drectons as ndcated n the fgures. The nterface s ArrAg.
( ) J. AldarOptcs Communcatons 18 000 1 10 9 concde very well wth those presented n Fg. of Ref. wx 1 takng nto account that n our case the dvergence s 0 mrad. The sgns also change due to the opposte orentaton of the X axs. Beyond the lmt angle and wthn our formalsm the a, j coeffcents untl second order are: a0,0s < x < q1, a sa sa sa s0, and 0 1,0 0,1,0 1,1 a s < x < q 1 L. Ž 36. 0, 0 Therefore, after some substtutons, the transverse angular shft for total nternal reflecton s gven by m 0,3rm r 3 0,0 uy Ž. s4l. Ž 37. 4tan qu 0 yy Ths equaton means that f the beam s symmetrc wth respect to the XZ plane then m 0,3 s0, and there s not transverse angular shft. Therefore the contrbuton of the asymmetry of the beam to the effect would be reported by ths equaton. The beams used n our examples are all symmetrc wth respect to the XZ. Therefore, the calculated transverse angular shft s zero after the lmt angle Ž see Fg. 3.. The multmodal character of the beam also affects the value of the transverse angular shft. In Fg. 4 we plotted the value of uy r calculated for several Her- mte Gaussan modes ncdng onto an ArrAg nterface. The fgure shows up that hgher modes mples hgher angular shfts. Both ux r and uy r are plotted n ths fgure for lnear polarzed Hermte Gaussan beams wth ellptc shape orented at 458 and orthogonal Gaussan dvergences of 0 mrad and 6.7 mrad. The state of polarzaton s lnear pontng along the parallel and the perpendcular drectons. In Fg. 5 we have calculated the longtudnal and transverse angular shft for a Gaussan beam wth crcular shape and dvergence of 0 mrad for an Fg. 5. Varaton of the longtudnal and transverse angular shfts as a functon of the angle of ncdence for a crcularly symmetrc Gaussan beam havng a dvergence of 0 mrad. For the fgures on the top the polarzaton s lnear orented at dfferent angles: 08 Ž parallel., 458, and 90 8 Ž perpendcular.. At the bottom, the fgures corresponds wth ellptcal polarzaton states havng < x < 0 s1. The nterface s ArrGlass Ž ns1.5..
10 ( ) J. AldarOptcs Communcatons 18 000 1 10 ArrGlass Ž n s 1.5. ncdence. The plots n the top of the fgure correspond wth lnear polarzaton at 08 Ž parallel man drecton along the X axs., 458, and 908 Ž perpendcular man drecton along the Y axs.. In these cases the transverse angular shft s about two orders of magntude smaller than ux r. The bot- tom of the fgure represents several cases of ellptc polarzed lght. We fx < x < 0 s 1. Now the longtudnal angular shft s about ten tmes smaller than for the lnear polarzaton case, and t does not depend on the phase of x 0. 5. Conclusons In ths paper we have proposed a method for calculatng the angular moments of reflected threedmensonal laser beams wthn the paraxal approach. Ths method can be appled to delectrcrdelectrc and delectrcrmetallc reflectons. Ths study completes prevous contrbutons restrcted to the plane of ncdence. The method uses a power expanson of the reflecton coeffcents and apples t to the three-dmensonal plane-wave decomposton of the ncdent beam. The geometry of the ncdence and the local change of the plane of ncdence s ncorporated to the calculaton. Its dependence wth respect to j and h s also expanded untl second order. The polarzaton state of the beam s assumed to be constant wthn the paraxal approach. As a result of the method, we focus on the transverse angular shft. Ths effect has been descrbed prevously wthn a more pure electromagnetc framework. The same effect s obtaned here wthn the moment characterzaton of laser beams. The actual value of the transverse shft depends on the matchng of the symmetry of the beam and the geometry of the ncdence. It also depends on the multmodal character of the beam and the state of polarzaton of the ncomng beam. Acknowledgements Ths work was prmarly developed durng a stay of the author at the Gnzton Laboratory of the Stanford Unversty. The author s deeply grateful to the program Becas del Amo of the Unversty Complutense of Madrd, for fnancng the stay as a Vstng Scholar. I also want to thank Prof. Anthony Segman for hs support durng the stay. References wx 1 M.A. Porras, Opt. Commun. 131 Ž 1996. 13. wx M.A. Porras, Opt. Commun. 135 Ž 1997. 369. wx 3 C. Imbert, Phys. Rev. D 5 Ž 197. 787. wx 4 O. Costa de Beauregard, C. Imbert, Y. Levy, Phys. Rev. D 15 Ž 1977. 3553. wx 5 J.P. Hugonn, R. Pett, J. Optcs 8 Ž 1977. 73 Ž n French.. wx 6 W. Nasalsk, J. Opt. Soc. Am. A 13 Ž 1996. 17. wx 7 E. Hecht. Optcs, Addson Wesley, 1998. wx 8 W.T. Welford, Aberratons of optcal systems, Adam Hlger, 1968. wx 9 M.A. Porras, J. Alda, E. Bernabeu, Appl. Opt. 31 Ž 199. 6389. w10x J. Alda, J. Alonso, E. Bernabeu, J. Opt. Soc. Am. A 14 Ž 1997. 737. w11x F. Falco, T. Tamr, J. Opt. Soc. Am. A 7 Ž 1990. 185. w1x J-J Greffet, C. Baylard, Opt. Commun. 93 Ž 199. 71. w13x B.R. Horowtz, T. Tamr, J. Opt. Soc. Am. 61 Ž 199. 586.