Teaching Fourier optics through ray matrices

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 26 ( EUROPEAN JOURNAL OF PHYSICS oi: / /26/2/005 Teaching Fourier optics through ray matrices IMoreno 1,MMSánche-Lópe 1,CFerreira 2,JADavis 3 an F Mateos 1 1 Departamento e Ciencia y Tecnología e Materiales, Universia Miguel Hernáne, E03202 Elche, Spain 2 Departamento e Óptica, Universia e Valencia, E45100, Burjassot, Spain 3 Department o Physics, San Diego State University, San Diego, CA 92182, USA Receive 16 November 2004 Publishe 7 February 2005 Online at stacks.iop.org/ejp/26/261 Abstract In this work we examine the use o ray-transer matrices or teaching an or eriving some topics in a Fourier optics course, exploiting the mathematical simplicity o ray matrices compare to iraction integrals. A simple analysis o the physical meaning o the elements o the ray matrix provies a ast erivation o the conitions to obtain the optical Fourier transorm. We exten this erivation to ractional Fourier transorm optical systems, an erive the orer o the transorm rom the ray matrix. Some examples are provie to stress this point o view, both with classical an with grae inex lenses. This ormulation cannot replace the complete explanation o Fourier optics provie by the wave theory, but it is a complementary tool useul to simpliy many aspects o Fourier optics an to relate them to geometrical optics. 1. Introuction The eucation o physicists an engineers in the iels o optics, photonics an imaging is growing in importance because o their multiple applications [1]. Geometrical an Fourier optics are basic topics usually inclue in optics courses in physics or engineering egrees [2, 3]. The use o simple matrix algebra has been successully applie in the stuy o several optical topics [4] incluing geometrical systems (ray matrices, polariation optics (Jones matrices an transmission an relection properties o thin ilms an layere meia (ynamic matrices. These matrix methos are particularly useul when a large number o elements are consiere. Matrix algebra is wiely taught in mathematical courses in many scientiic an technical egrees an the introuction o optical topics base on this ormalism becomes amiliar to stuents. In this work, we use the ray-matrix ormalism or the stuy an erivation o Fourier optics. Although ray optics oes not provie an explanation o the wave phenomena, its simplicity makes it very useul or the analysis o certain aspects o moern optical elements /05/ $30.00 c 2005 IOP Publishing Lt Printe in the UK 261

2 262 I Moreno et al For instance, the stability conition or a laser resonator is usually analyse in terms o ray optics in many textbooks [4 6]. Fourier optics is a subject which exploits the wave nature o light, an thereore it is usually introuce by means o iraction integrals [7]. However, certain aspects o Fourier optics theory (such as the location o Fourier relate planes an the sie o the Fourier transorm can be erive in a simple way using ray optics [8]. In aition, ray matrices [9] or equivalent operators [10] have been employe in the escription o optical Fourier transorm systems. In this work, we introuce Fourier optics base on the ray-matrix ormalism. We use a general actoriation o the ray matrix closely relate to the Fourier transorm, which allows us to erive the Fresnel iraction equation in a very simple way. In the last ecae, the ractional Fourier transorm (FRFT the extension o the Fourier transorm to arbitrary orers has attracte a lot o interest in the optical community. Here we also present the application o the ray-matrix ormalism to analyse FRFT systems. We inclue several examples with classical reractive an with grae inex lenses. Although the propose analysis cannot replace the complete explanation o Fourier optics provie by wave theory, it is a useul complement to simpliy calculations an to relate Fourier an geometrical optics. This ormulation represents an easy an compact erivation o Fourier optics, useul or teaching the topic to unergrauate stuents in physics or engineering egrees. The outline o the paper is as ollows: in section 2 we review the main concepts o raymatrix optics, the matrices o the basic elements an their physical signiicance. In section 3 we introuce the connection between geometrical optics an Fourier optics, an we in the conitions on the ray matrix to obtain a lens system that perorms an optical Fourier transorm. We inclue the analysis o some typical examples stuie in texts evote to Fourier optics. Finally, in section 4 we apply the ray-matrix ormalism to the analysis o lens systems that perorm a ractional Fourier transorm. 2. Ray-matrix optics The ray-matrix ormalism applies to centre geometrical optical systems uner the paraxial approximation. In this approximation, optical rays are consiere to travel close to the optical axis (-axis in igure 1. A ray crossing a plane = 0 is escribe with two components, the height r( 0 an the angle r ( 0 at which it crosses the plane. Since the paraxial approximation inicates that the ray travels close to the -axis, r ollows the small angle approximation an can be consiere as the slope o the ray r = r/. For simplicity, here we eal with a one-imensional geometry (r. However, the ormulation is irectly applicable to usual lens systems because o their circular symmetry aroun the optical axis, or to anamorphic systems that can be actore into horiontal an vertical components. An optical system changes the position an the angle o the ray. An input ray with coorinates (r 1,r 1 at the input plane is change to an output ray with coorinates (r 2,r 2 at the output plane. In the paraxial approximation, these coorinates can be relate in the orm o a 2 2 ray matrix M as ( ( ( r2 A B r1 r 2 = C D r 1. (1 The most usual application or ray matrices is orming the image o an object. In this case, some important properties o the optical system are obtaine when any o the ABCD parameters vanish [10]. I A = 0, parallel rays at the input plane are ocuse at the output plane, which thereore eines the rear or image ocal plane o the optical system.

3 Teaching Fourier optics through ray matrices 263 r r, ( 0 r( 0 = 0 Figure 1. Ray coorinates. I D = 0, a point source at the input plane is converte into parallel rays at the output plane, an thereore the input plane is the ront or object ocal plane. I B = 0, any point source at the input plane ocuses at a corresponing point in the output plane, regarless o the input angle. Thereore, the output plane is the image o the input plane. I C = 0, the system is telescopic. When C 0, the system is ocal, C being the optical power. The ocal length o the system is eine as = 1/C. The basic matrices or esigning optical systems are the ree space propagation o a ray in a homogeneous meium, M P, an the passage through a spherical reractive thin lens, M L. These matrices are given by [3, 4] ( 1 M P ( = 0 1, (2 an ( 1 0 M L =, (3 1/ 1 where is the istance o propagation an is the ocal length o the thin lens. Here we also consier the case o seloc grin meia [3 6], which are eine by the inhomogeneous reractive inex istribution n(r = n 0 1 ω2 r 2, n 0 being the reractive inex on the axis (r = 0. The ray trajectories r( insie the seloc grin meia are sinusoial, ω being the angular requency o oscillation. The ray matrix escribing the seloc grin meia is given by the ollowing expression [4 6]: ( cos(ωl 1 M GRIN = ω sin(ωl, (4 ω sin(ωl cos(ωl where L is the length o the grin meium. A erivation o these ray matrices can be obtaine in many reerences [3 7]. Because the grin meia will be use in section 4 ealing with the ractional Fourier transorm, here we analyse them in etail. The istance L p = 2π/ω is eine as the pitch, an it is the istance corresponing to a complete oscillation o the rays. Depening on the length L, a grin lens has ierent properties, which are summarie in igure 2. The quarterpitch grin lens (L = L p /4 has a ray matrix with A = D = 0. Thereore, the ront an rear ocal planes are locate at the input an output planes respectively (igure 2(a. Rays

4 264 I Moreno et al Quarter pitch grin lens Hal pitch grin lens (a (b Full pitch grin lens (c Figure 2. Grin lenses with (a quarter pitch, (b hal pitch an (c ull pitch. (a r L r (b M r λ N = 0 Figure 3. Rays (continuous lines an waveronts (otte lines. (a A set o parallel rays is equivalent to a linear phase in the plane r. (b The eect o a lens is to introuce a quaratic phase actor. escribe a quarter oscillation, so parallel rays are ocuse on a point while a point source is transorme into a set o parallel rays. The hal-pitch grin lens (L = L p /2 has a ray matrix M GRIN = I where I is the ientity matrix. Because B = C = 0, the grin lens perorms imaging between the input an output planes, while simultaneously acting as a telescopic system. The minus sign inicates the inversion prouce in the output coorinates with respect to the input (igure 2(b. A similar situation occurs or the ull-pitch grin lens (L = L p, which has a ray matrix M GRIN = I. In this case there is no sign inversion (igure 2(c. 3. Fourier transorm properties an the ray matrix The previous ormalism is use in many texts to stuy geometrical optics. However, in general it is not use to teach Fourier transorming optical systems, which are usually treate using iraction integrals. However, the mathematical simplicity o ray matrices compare to iraction integrals makes them very useul or this purpose. The connection between ray an wave optics is the corresponence between a set o rays an the corresponing waveront, which is perpenicular to the ray irection [8, 10]. Figure 3 illustrates this connection. We consier a monochromatic illumination with wavelength λ. A set o tilte parallel rays is equivalent to a tilte plane waveront, i.e., a linear phase in the plane = 0 (igure 3(a. The waveront at the plane = 0 is escribe as the unction g(r = exp[ j2πr/p] where the perio p is ixe by the angle r. Regaring igure 3(a, points L an M are in phase (2π phase ierence i the istance MN is equal to the wavelength o the optical raiation. Then, assuming the paraxial approximation, the perio p

5 Teaching Fourier optics through ray matrices 265 is equal to the raction λ/r. Thereore, we can conclue that the set o parallel rays crossing the plane = 0 with inclination r is equivalent to a phase istribution g(r given by ] g(r = exp [j2π rr. (5 λ In aition, the action o a lens is to transorm a plane wave into a spherical wave whose centre is its rear ocal point (igure 3(b. It is well known that the transmittance t l (r o a lens o ocal length is a quaratic phase actor [7] given by t l (r = exp [ j πr2 λ ], (6 which causes the change in the waveront. Equations (5 an (6 provie the relation between ray an wave optics necessary to relate ray matrices to Fourier optics. They will be use in the ollowing to erive the scale o the Fourier transorm an the phase istribution at the output plane when a perect Fourier transorm is not obtaine Exact optical Fourier transorm (A = D = 0 We want to in the conitions on the ray matrix that lea to a Fourier transorm. convenient to rewrite the general matrix equation (1as r 2 = Ar 1 + Br 1, r 2 = Cr 1 + Dr 1. It is (7a (7b We use the two ollowing properties o the generalie Fourier transorm [11]: (1 The Fourier transorm o a elta unction is a linear phase in the requency space: FT{δ(r a} =exp[ j2πau]. (8a (2 The Fourier transorm o a linear phase is a elta unction in the requency space: FT{exp[ j2πra]} =δ(u a. (8b Here a is a constant, u is the spatial requency an FT stans or the Fourier transorm operation, eine as G(u = FT{g(r} = + g(r exp[ j2πru]r. (9 In terms o optical raiation, the elta unction is equivalent to a point source, while the linear phase correspons to a tilte set o parallel rays (equation (5. Consequently, the irst conition (8a states that a point in the input plane leas to a constant angle in the output plane an can be easily satisie in equation (7b by making the matrix element D = 0. The secon conition (8b states that a constant angle in the input plane leas to a point in the output plane, an can be easily satisie in equation (7a by making the matrix element A = 0. Thereore, the ray matrix o a Fourier transorming system must have parameters A = D = 0. The exact Fourier transorm is perorme between the ront an rear ocal planes o the optical system. In the usual case o lenses in air, the ray matrix is unimoular (AD BC = 1 an the Fourier transorming system can be written as ( 0 B M FT =, (10 1/B 0

6 266 I Moreno et al r 1 =a FT system r 2 = a/b Figure 4. Relation between the location o a point source in the input plane an the angle o the plane wave at the output plane. where B is equal to the ocal length s o the system. Thereore, an optical system which has a ray matrix in the orm o equation (10, prouces an optical Fourier transorm between its input an output planes. We note that i D is not equal to ero, then the output plane will have a quaratic phase that will not be etecte in intensity, as will be shown in section 2.3. The sie o the optical Fourier transorm can be easily erive rom equation (10. We consier a isplace point source at the input, i.e. g 1 (r 1 = δ(r 1 a. Input rays can be consiere to have ixe coorinate r 1 = a an variable angle r 1 (igure 4. The output rays are given by ( r2 r 2 = ( ( ( 0 B a Br 1/B 0 r 1 = 1. (11 a/b The output angle r 2 is constant (equal to a/b, an thereore they are parallel rays. The equivalent phase istribution at the plane = 2 is given, rom equation (5, by [ g 2 (r 2 = exp j2π r 2r 2 ] [ = exp j2π r ] 2a. (12 λ λb The comparison o equations (12 an (8a leas to the relation between the spatial requency u o the input signal an the spatial coorinate r 2 at the plane = 2, u = r 2 λb = r 2, (13 λ s where we write B = s, the ocal length o the optical system perorming the Fourier transorm. This equation inicates the sie o the Fourier transorm through the relation between the spatial coorinate r 2 at the output plane an the spatial requency u o the unction at the input plane. Figure 5 shows two simple optical systems that ulil the Fourier transorm conition. The irst one is the classical 2 optical system that perorms the Fourier transorm between the ront an rear ocal planes o a converging lens. The secon one consists o two converging lenses with ientical ocal length, separate by a istance =. Both systems perorm a Fourier transorm between the input plane P 1 an the output plane P 2 [8]. In both cases, the ray matrix rom P 1 to P 2 takes the orm o equation (10 with B equal to the ocal length. Figure 5 shows the trajectories o some rays, illustrating the Fourier transormation between a point an a plane wave. A thir example involves a quarter-pitch grin lens. Its ray matrix also takes the orm o equation (10 with B = 1/ω. Figure 2(a shows the trajectories o some rays illustrating this example. The above iscussion shows that the ray-matrix metho provies a simple solution or etermining the location o Fourier relate planes an the sie o the Fourier transorm. The speciic shape o the waveront in this plane requires perorming the Fourier transorm o the input transparency (equation (9. However, the simple case o a iraction grating can be very easily analyse in terms o propagation o rays. It is very well known that a iraction grating illuminate with a collimate plane wave generates plane waves at angles

7 Teaching Fourier optics through ray matrices 267 P 1 P 2 (a 1 = 2 = P 1 P 2 (b = Figure 5. Two lens systems that perorm Fourier transorm between planes P 1 an P 2. Diraction grating F F Figure 6. Illustration showing the Fourier transorm o a iraction grating. o constructive intererence given by the law r m sin(r m = mλ/t 0, where T 0 is the perio o the iraction grating an m is an integer number. Figure 6 shows the trajectories in the 2 Fourier transorm system o some rays associate with the plane waves corresponing to m = 0 an m =±1 iracte orers. The result shows how the rays ocus to orm the iraction orers in the Fourier plane Factoriation o a general ray matrix. Application to Fresnel transorm In this section, we present a actoriation o the general ray matrix which provies an easy interpretation in terms o Fourier transorming properties. In [12], a actoriation o the ray matrix was propose relate to the principal planes. Here we consier a ierent actoriation propose in [13] where a general ray matrix is ecompose as ( ( A B 0 B C D = ( 1 0 D/B 1 1/B 0 ( 1 0 A/B 1. (14

8 268 I Moreno et al This actoriation is vali when the input an output planes are not conjugate (B 0. The two matrices in the extremes take the orm o thin lenses (equation (3 while the central matrix takes the orm o the Fourier transorm system (equation (10. Thereore a general system can be viewe as a Fourier transorm system characterie by a ocal length = B, inserte between two lenses with ocal lengths 1 = B/A an 2 = B/D at the input an output planes respectively. This ecomposition provies a useul connection between the amplitue o the waveront at the input an output planes, g 1 (r 1 an g 2 (r 2 respectively. From the wave-optics point o view, the passage through a lens is equivalent to a multiplication o the waveront by a quaratic phase actor given by equation (6. The Fourier transorm operation is given by equation (9 an its scale is ixe by the parameter B through equation (13. Thereore, applying the ecomposition in equation (14, the waveront amplitue at the output g 2 (r 2 is obtaine, except or a constant actor, by multiplying the input waveront g 1 (r 1 by a quaratic phase actor, calculating its Fourier transorm, an multiplying the result by another quaratic phase. The results lea to the ollowing relation, known as the generalie iraction equation or Collins equation [14]: [ g 2 (r 2 = exp j πr2 2 D ] { [ FT exp j πr2 1 A ] } g 1 (r 1. (15 λb λb u=r 2 /λb A case o particular interest is the interpretation o Fresnel iraction. The application o the previous actoriation to the ree space propagation matrix (equation (2 leas to ( ( ( ( = 0 1 1/ 1 1/ 0 1/ 1. (16 This equation shows that the ree space propagation (Fresnel transorm is equivalent to a Fourier transorm system, multiplie at both extremes by iverging lenses o ocal length =. The ecomposition in equation (16 irectly leas to the very well-known equation or the Fresnel iraction approximation [7] g 2 (r 2 = exp [ j πr2 2 λ ] FT { exp [ ] } j πr2 1 g 1 (r 1. (17 λ u=r 2 /λ The exact Fourier transorm system shown in igure 5(b is obtaine by aing compensating converging lenses with ocal lengths = + on either sie o the ree propagation Optical Fourier transorm with a phase istribution (A = 0; D 0 There exist many other optical architectures where a Fourier transorm is obtaine, but multiplie by a quaratic phase istribution. They happen when A = 0butD 0. In this case the actoriation in equation (14 irectly gives an optical Fourier transorm matrix multiplie by a lens with ocal length = B/D, i.e., a quaratic phase actor at the output plane given by equation (6. Oten, in practice, we ignore this quaratic phase shit when we etect the output intensity. As an example we consier a system similar to the 2 shown in igure 5(a, but with the rontal istance between the object an the lens. This system has a ray matrix ( 0 M = 1/ 1. (18 Thereore, the output amplitue is the Fourier transorm o the input, but it is multiplie by a quaratic phase actor which vanishes only when =, i.e., [ ( g 2 (r 2 = exp +j πr2 2 1 ] FT {g 1 (r 1 } λ u=r2 /λ. (19 This result coincies with those obtaine with integral iraction theory [7].

9 Teaching Fourier optics through ray matrices Fractional Fourier transorm systems (A = D The ractional Fourier transorm (FRFT is the generaliation to ractional orers o the Fourier transorm operation [15 19]. In the last ecae, it has attracte a lot o interest in the optical community since it can be easily obtaine by means o optical systems, either with classical lenses [16] or with more versatile programmable iractive lenses [17]. Optical systems that prouce FRFT can also be treate using the ray-matrix ormalism [18]. Here we present a simple erivation o the FRFT systems base on the Sylvester theorem [6]. Let us consier an optical elemental system escribe by a general matrix M 0 with parameters ABCD. The ray matrix o the m-repetition o this elemental system can be calculate by means o the Sylvester theorem [6], which states that M m 0 = 1 ( A sin(mθ sin((m 1θ B sin(mθ, (20 sin(θ C sin(mθ D sin(mθ sin((m 1θ where m is an integer an the angle θ is given by the relation cos(θ = 1 (A + D. (21 2 I the m-repetition o this optical elemental system prouces a Fourier transorm, the elemental system can be regare as proucing a ractional Fourier transorm o orer p = 1/m. Thereore, an optical system that prouces a FRFT o orer p = 1/m must have a ray matrix M 0 such that ( A M m 0 = C m ( B 0 B = D 1/B, (22 0 where B is the parameter characteriing the ocal length o the Fourier transorm system. By comparing equations (20 an (22, the FRFT conition is satisie i sin((m 1θ A = D =, (23 sin(mθ where now cos(θ = A = D. These two last relations lea to the conition cos(mθ = 0, which has the non-trivial solution θ = p π 2 = π 2m, (24 where p = 1/m is the ractional orer o the FRFT. Thereore, the ray matrix o an optical system perorming a FRFT can be written in the ollowing general orm: ( cos(θ b sin(θ M FRFT = 1 b sin(θ cos(θ, (25 where the parameter b acts as a scaling actor. As examples to implement the FRFT we consier the two systems propose by Lohmann in [16], which are sketche in igure 7. The irst system consists o a ree propagation o istance, a converging lens o ocal length, an a secon ree propagation o istance. Its ray matrix is M 0 = ( 1 1 ( 2 1. (26 The secon system consists o a lens o ocal length, a ree propagation o istance an a secon lens o the same ocal length. In this case the ray matrix is ( 1 M 0 = ( 1 2. (27 1

10 270 I Moreno et al P 1 P 2 P 1 P 2 (a (b Figure 7. Lohmann lens systems that perorm FRFT between planes P 1 an P 2. (a Propagation lens propagation, (b lens propagation lens Orer o the FRFT (p Fraction / Figure 8. Evolution o the FRFT orer p as a unction o the raction / in the Lohmann lens systems. In both cases the ray matrix ulils A = D = 1 /. Thereore both systems perorm a ractional Fourier transorm o orer p = 2θ/π, where cos(θ = 1 /. Consequently, i a FRFT o orer p is esire, the relation between an is given by = 1 cos ( pπ 2. (28 Again, this equation coincies with the ormula erive using iraction integrals [19]. In both cases, the perect Fourier transorm systems shown in igure 5 are recovere when =, which correspons to a FRFT orer p = 1. Figure 8 shows the evolution o the orer p as a unction o the quotient / in the range p [0, 2], which covers rom the image plane to the Fourier transorm plane. A inal example o a FRFT system is a grin lens. Early proposals o FRFT systems were base on the propagation in grin meia [15]. These properties can be erive very easily using the ray-matrix approach. The ray matrix o a grin lens (equation (4 ollows the FRFT conition through A = D = cos(ωl. In this case the angle θ in equation (25 is equal to ωl. Consequently, a grin lens o length L prouces a FRFT o orer p = 2ωL/π. When L = π/2ω, i.e., the quarter-pitch grin lens, the Fourier transorm is recovere. All these results coincie with those presente in [15, 16], an are obtaine irectly rom a very simple analysis o the ray matrix o the optical system.

11 Teaching Fourier optics through ray matrices Conclusions We have presente an analysis o lens systems that perorm optical Fourier transorms base on the ray-matrix ormalism, as a useul tool or teaching Fourier optics. With this ormalism we avoi the use o more complicate iraction integrals in the resolution o problems such as the location o Fourier relate planes, the sie o the Fourier transorm or the orer o a ractional Fourier transorm. We have presente a ull sel-containe erivation o these Fourier optics items base on the ray-matrix ormalism, incluing several examples with reractive lenses or grae inex lenses. We have extene this ormalism to analyse optical systems that perorm ractional Fourier transorms. Acknowlegments This work receive support rom Ministerio e Ciencia y Tecnología rom Spain (projects BFM C02 an FIS , an Generalitat Valenciana (project GRUPOS03/117 Reerences [1] 1998 Harnessing Light. Optical Science an Engineering or the 21st century National Research Council, National Acaemic Press [2] Yuel M J 1991 Basic eucation in optics or physicists Proc. SPIE [3] Saleh B E A an Teich M K 1991 Funamentals o Photonics (New York: Wiley [4] Gerrar A an Burch J M 1975 Introuction to Matrix Methos in Optics (New York: Dover [5] Siegman A E 1986 Lasers (Mill Valley, CA: University Science Books [6] Yariv A 1989 Quantum Electronics 3r en (New York: Wiley [7] Gooman J 1996 Introuction to Fourier Optics 2n en (New York: McGraw-Hill [8] Jutamulia J an Asakura T 2002 Optical Fourier-transorm theory base on geometrical optics Opt. Eng [9] Davis J A an Lilly R A 1993 Ray-matrix approach or iractive optics Appl. Opt [10] Naarathy M an Shamir J 1982 First-orer optics a canonical operator representation: lossless systems J. Opt. Soc. Am [11] Bracewell R N 1986 The Fourier Transorm an its Applications (New York: McGraw-Hill [12] Arsenault H H an Macukow B 1983 Factoriation o the transer matrix or symmetrical optical systems J. Opt. Soc. Am [13] Shamir J an Cohen N 1995 Root an power transormations in optics J. Opt. Soc. Am. A [14] Collins S A 1970 Lens-system iraction integral written in terms o matrix optics J. Opt. Soc. Am [15] Menlovic D an Oaktas H M 1993 Fractional Fourier transorms an their optical implementation J. Opt. Soc. Am. A [16] Lohmann A 1993 Image rotation Wigner rotation, an the ractional Fourier transorm J. Opt. Soc. Am. A [17] Moreno I, Davis J A an Crabtree K 2003 Fractional Fourier transorm optical system with programmable iractive lenses Appl. Opt [18] Bernaro L M 1996 ABCD matrix ormalism o ractional Fourier optics Opt. Eng [19] Dorsch R G an Lohmann A 1995 Fractional Fourier transorm use or a lens-esign problem Appl. Opt