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A variational principle in optics

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A variational principle in optics
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  A variational principle in optics Jacob Rubinstein and Gershon Wolansky Department of Mathematics, Indiana University, Bloomington, Indiana, 47401, and Department of Mathematics,Technion, Haifa 32000, Israel  Received March 16, 2004; revised manuscript received May 14, 2004; accepted May 27, 2004 We derive a new variational principle in optics. We first formulate the principle for paraxial waves and thengeneralize it to arbitrary waves. The new principle, unlike the Fermat principle, concerns both the phase andthe intensity of the wave. In particular, the principle provides a method for finding the ray mapping betweentwo surfaces in space from information on the wave’s intensity there. We show how to apply the new principleto the problem of phase reconstruction from intensity measurements. © 2004 Optical Society of America OCIS codes:  080.0080, 000.3860. 1. INTRODUCTION The Fermat principle is one of the pillars of optics. It liesat the foundations of geometrical optics, where it providesa theoretical and computational tool to find ray trajecto-ries and hence the phase of a wave. The principle,though, only concerns rays and provides no informationon intensity transport. The main goal of this paper is toderive a new variational principle in optics that relatesthe phase and the intensity of a wave. The new principleis formulated in terms of the geometrical-optics approxi-mation of the wave equation.Fermat postulated that a light ray travels between twospecified points so as to minimize the action    n d l , where n  is the refraction index of the medium. It was latershown that this principle is equivalent to the eikonalequation. In our setup, we are not given the terminalpoints of a ray. Instead, we are given two intensity dis-tributions on two planes. Our principle determines boththe end points of each ray and the ray trajectory.One of the promising applications of the new principleis as a means for determining the wave’s phase from in-tensity measurements. We therefore start by recalling inSection 2 the idea of the transport-of-intensity equation(TIE) and curvature sensors. This theory was developedfor paraxial waves. Therefore we first formulate our newprinciple in the paraxial regime and for nonhomogeneousmedia. The theory is developed in Section 3 in full detail.In particular, we explain there the precise meaning of paraxiality in our approximations. This explanationleads us in a natural way to derive (Section 4) the generalform of our principle. Finally, in Section 5 we summarizeand discuss our results. We also briefly discuss there ex-tending the principle to include singular solutions andpractical aspects such as the numerical solution of the variational problem. The numerical questions will be ad-dressed in more detail, together with simulation results,in a sequel. 2. PHASE-RECONSTRUCTION PROBLEM  A central problem in optics is to determine the phase of awave. The problem is particularly hard when the phaseis not very close to being planar or spherical, and there-fore interferometry methods are difficult to apply. Theneed to find the phase arises in a variety of applicationsincluding adaptive optics, astronomy, and ophthalmic op-tics. A widely used general phase sensor is the Hartmann – Shack device. It consists of an array of lenslets that con- vert an incoming beam into spots of light on a detectionscreen. This sensor has a number of drawbacks: Theresolution is limited by the size of lenslets, the location of the spot centroids is hard to determine accurately, andthe transformation from the location of the centroids of the spots to the phase gradient is only approximate.In contrast to phase determination, it is relatively easyto measure the wave’s intensity. It is therefore tempting to seek methods for finding the phase from intensity mea-surements. Indeed, Teague 1 proposed such a phase sen-sor. His method was further developed by Roddier 2 andothers. To explain the idea behind such sensors (some-times called curvature sensors), we consider a complex- valued wave  u  in the Fresnel regime where the waveequation in a homogeneous medium is approximated by  i   u    z   ku   12 k  u . (1)Here  z  is the main direction of propagation,  u  is the wavefunction,  k  is the wave number, and    and    denote, re-spectively, the gradient and Laplacian operators in theplane orthogonal to  z . Writing   u    A  exp( ik   ), we ob-tain for the real and imaginary parts of Eq. (1)     A    z  12  A    A 2      , (2)        z   1   1 k 2  A   A   12       2 .(3)The first equation can be written more conveniently as anequation for the intensity  I     A 2 : 2164 J. Opt. Soc. Am. A/Vol. 21, No. 11/November 2004 J. Rubinstein and G. Wolansky1084-7529/2004/112164-09$15.00 © 2004 Optical Society of America      I     z     •    I       . (4)Equation (4) is called the TIE. Teague 1 pointed out thatEq. (4) can be thought of as an elliptic partial differentialequation for the phase     in terms of the intensity  I  . Thushe considered Eq. (4) over some domain  D  in a plane  z   z 0  and solved it under prescribed boundary conditions(the Dirichlet problem). We note that, strictly speaking,the phase is  k   , but we shall refer here to     alone also asthe phase.The difficulty with Teague’s method is that the valuesof the phase at the boundary     D  are not easy to measure.Therefore a number of people suggested alternative algo-rithms related to Eq. (4) that attempt to resolve this is-sue.Roddier 2 proposed to use homogeneous Neumannboundary conditions at the boundary     D  instead of theDirichlet conditions but did not justify this proposal.Gureyev and Nugent 3 analyzed more carefully the bound-ary behavior of the wave  u . They pointed out that, inpractice, the domain  D  is determined by the regime wherethe intensity is positive (essentially the image of the ap-erture). Then they argued that since  I   vanishes at     D ,Eq. (4) is singular and has a unique solution without anyboundary condition. Lee and Rubinstein 4 showed that amore delicate analysis of the boundary behavior of   I   isneeded. Indeed, Eq. (4) has a unique (up to an additiveconstant) solution without any boundary condition only if   I   vanishes at     D  at a suitable rate. They also devisednumerical methods to solve such equations.Notice that the TIE is only one half of the Fresnel equa-tion. Clearly, a proper solution must satisfy the otherhalf [Eq. (3)], too. This raises the following question:Suppose we measure the intensity  I   at  two  planes  z   Z 1 ,  z    Z 2 ; can we use this information to determinethe phase by considering jointly Eqs. (4) and (3)? In fact,a measurement of the intensity at two planes is also re-quired for the TIE, since we need to find not only the in-tensity  I   but also its derivative  I   z . Computing this de-rivative requires measuring the intensity at two nearbyplanes. In the question we posed above, however, thetwo planes can be arbitrarily located. Apartial answer to our question was given by van Damand Lane. 5 They realized that if the wave, confined toboth observation planes, depends only on one variableand if the rays do not intersect, one can order the initialand terminal points of the rays on the two respectivescreens such that all successive pairs of rays hold be-tween them the same amount of total intensity. Once theray end points are known, one can determine the phaseslopes and from them the phase itself. Van Dam andLane also tried to extend this approach to the generaltwo-dimensional case. They proposed to sample the in-tensity, as in the one-dimensional case, in many orienta-tions and to apply the Radon transform to deduce thephase slopes from the obtained integrals. No justifica-tion, however, was given for this method, and it is notclear why it should give a good approximation to thephase slopes for arbitrary intensity distributions on thedetection screens.We shall use the new variational principle to solve theproblem we posed. We also present a preliminary analy-sis of a number of numerical schemes for actually comput-ing the phase. To incorporate Eq. (3) in the analysis, wefurther express the phase     in the form      z        x ,  z  , (5)where  x  denotes a point in the plane  R 2 and     is the per-turbation of the phase about the planar term  z . Substi-tuting Eq. (5) into Eq. (3), we obtain for     (  x ,  z )        z  12      2  12 k 2  A   A . (6)In the small-wavelength approximation we neglect theterm on the right-hand side and replace Eq. (6) with        z  12      2   0. (7)Our discussion on the phase reconstruction was limitedto homogeneous media. The variational principle weshall derive, however, is applicable to arbitrary media.When the refraction index  n  is not constant, we need toinclude the term  12 ( n 2   1) in the right-hand side of Eq.(6). Thus the optical problem we consider consists of thefollowing equations and boundary conditions:    I     z     •    I         0, (8)        z  12      2  12  n 2   1  ª  P   x ,  z  . (9)Here  x    R 2 ,  z      Z 1 ,  Z 2  ,  I  (  z    Z 1 ,  x )    I  1 (  x ), and  I  (  z    Z 2 ,  x )    I  2 (  x ), where  I  1  and  I  2  are two given in-tensity distributions. Equations (8) and (9) together withthe side conditions will be denoted collectively as problem( Op ). In Section 3 we show that problem ( Op ) can besolved by certain optimization problems. 3. VARIATIONAL PROBLEM I: THEPARAXIAL LIMIT Consider two planes  P 1 :  z    Z 1  and  P 2 :  z    Z 2 . Let  I  1  and  I  2  be two nonnegative functions given on  P 1  and  P 2 . Optically, the functions  I  1  and  I  2  are the measuredintensities; mathematically, however, we can considerthem arbitrary density functions. We assume that theintensities are normalized to 1, and that they have finitesecond moments:   I  1   x  d  x      I  2   x  d  x    1,   x 2  I  i   x  d  x     ,  i    1,2. (10)We recall from geometrical optics that if a point  x   P 1  is mapped by a ray into a point  y    P 2 , if the re-fraction index near  P 1  and  P 2  is the same, and if the rayis approximately orthogonal to the planes, then the inten-sities are related by 6 J. Rubinstein and G. Wolansky Vol. 21, No. 11/November 2004/J. Opt. Soc. Am. A 2165   I  1   x     I  2 ( T    x  )  J   T   . (11)Here  T  (  x ) is the ray mapping from  P 1  to  P 2 , and  J  ( T  ) isthe Jacobian of this mapping. We shall say that a map-ping   T   satisfying the relation (11) transports  I  1  to  I  2 .We use the formal notation T  #  I  1    I  2 . (12)More generally, a mapping   T   (not necessarily continuous)transports  I  1  to  I  2  if and only if      ( T    x  )  I  1   x  d  x          x   I  2   x  d  x ,        C 0   R 2  ,(13)where  C 0 (  R 2 ) is the space of all continuous functions inthe plane with compact support. Our first variationalprinciple, denoted by problem ( Mp ), is the following:Find a map  T  ¯   such that  T  ¯  #  I  1    I  2  and  M    I  1 ,  I  2 ,  T  ¯     ª   Q (  x ,  T  ¯     x  )  I  1   x  d  x     Q (  x ,  T    x  )  I  1   x  d  x ,   T  #  I  1    I  2 ,(14)where the action  Q  is given by Q   x ,  y   ª  Q   x ,  y ,  Z 1 ,  Z 2    min    Z 1  Z 2  12  d  x d  z  2   P   x   z  ,  z   d  z  (15)and where the minimization is among all orbits  x (  z ) suchthat  x (  Z 1 )    x ,  x (  Z 2 )    y .In the homogeneous case (  P    0) the action reduces to Q   x ,  y      x    y  2 2   Z 2    Z 1  .In this case our variational principle ( Mp ) becomes thequadratic Monge problem ( Mpq ):Find a map  T  ¯   such that  T  ¯  #  I  1    I  2  and    T  ¯     x     x  2  I  1   x  d  x       T    x     x  2  I  1   x  d  x ,  T  #  I  1    I  2 . (16)We shall show that the optimal mapping   T  ¯   is the raymapping of the optical problem. For this purpose, we re-late problem ( Op ) to problem ( Mp ) through several addi-tional equivalent optimization problems. We start by in-troducing a new problem, denoted by ( Wp ), and provethat its solution is the pair (  I  ,    ) that solves ( Op ). Theorem 1 . Let         (  x ,  z )    0 and  v    v (  x ,  z )   R 2 be solutions of the following optimization problem:inf    , v W    I  1 ,  I  2 ;  P     inf    , v   Z 1  Z 2    12    v  2   P    d  x d  z ,(17)subject to the constraints     z     •      v     0,  Z 1    z    Z 2 ,     x ,  Z i     I  i   x  ,  i    1,2. (18)Then      I  ,  v       , (19)where  I   and     solve ( Op ).  Proof  . Recall that any vector field in  R 2 is the orthogo-nal sum of a gradient      and a vector field  w  such that   •  (   (  x ,  z ) w )   0. This decomposition holds for any  z .Setting   v          w , we obtain       x ,  z   v   x ,  z   2 d  x          x ,  z        x ,  z   2 d  x         x ,  z   w   x ,  z   2 d  x .Clearly, for any candidate    , the choice  w    0 reduces theenergy  W   without affecting constraint (18). Thereforethe optimal choice for  v  must be of the form  v        forsome potential    .To further characterize    , we equate to zero the first variation of the energy in Eq. (17), taking into accountconstraints (18). We therefore write       I       ,             , where  I   and  v        solve constraint (18) and     isa small positive number. Substituting      and     into con-straints (18), we obtain        z     •              •    I         O     . (20)The first variation of the energy is   W          12        2   I      •           P  d  x d  z   O    2  . (21)Integrating the second term in the integrand by partswith respect to the  x  variable, we obtain for any  z     Z 1 ,  Z 2  :   I      •     d  x          •    I      d  x .Integrating now on  R 2     Z 1 ,  Z 2  , using Eq. (20), andthen performing another integration by parts, we get   I      •     d  x d  z              z     •            d  x d  z   O                    z        2  d  x d  z   O     ,where we used the fact that constraints (18) imply   (  x ,  Z 1 )      (  x ,  Z 2 )    0. Substituting this equationinto Eq. (21) and equating the first variation to zero, weobtain that     solves Eq. (9), and then constraint (18) im-ply that  I   solves Eq. (8), as required.We proceed to show that the infimum of the functional W   equals  M   defined in expression (14). Let (  I  ,    ) be the 2166 J. Opt. Soc. Am. A/Vol. 21, No. 11/November 2004 J. Rubinstein and G. Wolansky  solution of problem ( Wp ). We use the phase    , i.e., thesolution to the Hamilton – Jacobi equation (9), to definethe following flow:d  x ¯  d  z      (  x ¯    z  ,  z ) ,  x ¯    Z 1     x . (22)The flow (22) induces a mapping  T   Z 1  z   x   ª  x ¯    z  . (23) Proposition 2 . The mapping (23) transports  I  1  to  I  (  x ,  z ).  Proof  . We define  I  ˜    x ,  z     J   T   Z 1  z   I  ( T   Z 1  z   x  ,  z ) . (24) A standard result in the theory of ordinary differentialequations states that the Jacobian  j  of the mapping      in-duced by the flow generated by an equation of the formd  x  /d t    f  (  x ) satisfies the identity d  j  /d t    j   •  f  . Apply-ing this identity to the flow (22), we findd  J   T   z  d  z   J   T   z      T   z ,  z  . (25)Therefore    I  ˜     z   J   T   Z 1  z    I          I   •          I     z   T   Z 1  z ,  z    0,where the last equality follows from the assumption that  I   satisfies Eq. (8). Since  I  ˜   does not depend on  z , we canwrite  I  ˜  (  x ,  z )    I  ˜  (  x ,  Z 1 )    I  1 . Replacing   I  ˜  (  x ,  z ) in Eq.(24) with  I  1  we obtain  I  1   x     J   T   Z 1  z   I  ( T   Z 1  z   x  ,  z ) , (26)which, on recalling Eq. (11), proves our assertion.We are now ready to state the main result of this sec-tion. Theorem 3 . The mapping   T     T   Z 1  z   Z 2 , where  T   Z 1  z isdefined in Eq. (23), is the optimal mapping   T  ¯   , i.e., T  ¯     T   Z 1  Z 2 . (27)In addition,inf    , v W    I  1 ,  I  2 ;  P       Q (  x ,  T  ¯    x  )  I  1   x  d  x .  Proof  . Let (  I  ,    ) be a solution to the problem ( Wp ).Integrating the Hamilton – Jacobi equation (9) along anarbitrary orbit  x      (  z ), we obtaindd  z   (     z  ,  z )       • d   d  z         z   12  d   d  z       2  12  d   d  z  2   P (     z  ,  z ) . (28)We first use identity (28) for the special case       x ¯  . In-tegrating Eq. (28) from  Z 1  to  Z 2 , we write   ( T   Z 1  Z 2   x  ,  Z 2 )       x ,  Z 1      Z 1  Z 2  12  d  x ¯  d  z  2   P (  x ¯    z  ,  z )  d  z    Q (  x ,  T    x  ) . (29)Thanks to expression (29) and to the conclusion we de-rived above that  T   transports  I  1  to  I  2 , we can write       x ,  Z 2   I  2   x  d  x        x ,  Z 1   I  1   x  d  x     Q (  x ,  T    x  )  I  1   x  . (30)We now show that the left-hand side of expression (30)is nothing but inf    , v  W  (  I  1 ,  I  2 ;  P ) in a disguised form.We thus calculate  E     ,  I  1 ,  I  2   ª       x ,  Z 2   I  2   x  d  x        x ,  Z 1   I  1   x  d  x     Z 1  Z 2      z (     x ,  z   I    x ,  z  ) d  x d  z     Z 1  Z 2        I     z      I         z   d  x d  z .Using Eqs. (8) and (9) and then integrating by parts, wefind that the last expression equals     Z 1  Z 2         •    I        12  I       2   IP  d  x d  z     Z 1  Z 2    12  I       2   IP  d  x d  z    W    I  1 ,  I  2 ;  P  .We therefore obtain from expression (30) W    I  1 ,  I  2 ;  P       Q (  x ,  T    x  )  I  1   x  d  x     Q (  x ,  T  ¯    x  )  I  1   x  d  x . (31)To complete the proof, we shall establish now the re- verse inequality in expression (31). For this purpose weprove the following: Proposition 4 . Let  T   be any mapping satisfying  T  #  I  1    I  2  and let     be any function satisfying theHamilton – Jacobi equation (9). Then  E    ,  I  1 ,  I  2       Q (  x ,  T    x  )  I  1   x  d  x . (32)In particular,  E     ,  I  1 ,  I  2     max          x ,  Z 2   I  2   x  d  x         x ,  Z 1   I  1   x  d  x  , (33)where the maximum is taken over all functions        (  x ,  z ), which satisfy     z  12    x    2   P   x ,  z  . Assuming this proposition, we can substitute the func-tion     that solves Eq. (9) for     in the left-hand side of ex- J. Rubinstein and G. Wolansky Vol. 21, No. 11/November 2004/J. Opt. Soc. Am. A 2167  pression (32), substitute the optimal mapping   T  ¯   for  T   inthe right-hand side, and conclude that the two inequali-ties in expression (31) must, in fact, be equalities; this es-tablishes Theorem 3.To prove Proposition 4, we return to the integration for-mula (28). This formula holds for any solution of theHamilton – Jacobi equation, so, in particular, it holds for    .For the orbit we choose the curve that connects  x  with T  (  x ) and minimizes the action  Q   x ,  T  (  x )  . We thus ob-tain the inequality   ( T    x  ,  Z 2 )       x ,  Z 1     Q (  x ,  T    x  ) .Multiplying the last inequality by  I  1 (  x ) and integrating with respect to  x , we get     ( T    x  ,  Z 2 )  I  1   x  d  x          x ,  Z 1   I  1   x  d  x     Q (  x ,  T    x  )  I  1   x  d  x .Since  T   transports  I  1  into  I  2 , the first term on the left-hand side of the last inequality equals     (  x ,  Z 2 )  I  2 (  x )d  x .This completes the proof of the proposition and the theo-rem.Theorem 1 says that the minimizing pair (  I  ,    ) for thefunctional  W   is a solution to the problem ( Op ). Theorem3 states that the flow induced by     generates the optimalmapping   T  ¯   . Therefore by solving the variational prob-lem we obtain complete information on the ray mapping and hence the phase     of problem ( Op ). Notice that thefunctional  M  (  I  1 ,  I  2 ;  T  ¯   ) defines a metric measuring thedistance between the intensities  I  1  and  I  2 .In the special case of the quadratic Monge problem, cor-responding to the optical setup of a homogeneous me-dium, the action  Q  is minimized by the straight line (ray)connecting   x  and  T  (  x ). Therefore the optimal map  T  ¯   inthis case is given explicitly by T   Z 1  Z 2   x     x      x     x ,  Z 1   Z 2    Z 1  . (34)Given two intensity distributions, and assuming thatthey are related by the paraxial Fresnel equations, the variational problem ( Mp ) provides us with a theoreticaland practical tool to find a phase map that connects theseintensities. The optimization formulation involves theaction Q . It would be interesting to see how this action isrelated to the classical Fermat action. This analysis re-quires us to first understand the asymptotic regime inwhich the parabolic wave equation holds for uniform ornonuniform media.To study this regime, we introduce a small positive pa-rameter   . We assume that the refraction index is of theform n   x ,  z     1      P   x ,   1/2  z  , (35)where we assume without loss of generality that the back-ground refraction index is 1. We then seek solutions of the eikonal equation (      /     z ) 2        2   n 2 , where     isthe wave’s phase, of the form     x ,  z     z     1/2     x ,   1/2  z  . (36)Substituting      into the eikonal equation, we find that toleading order     satisfies Eq. (9).The scaling above means that the variation in the re-fraction index is weak and slowly varying in the  z  direc-tion. It also means that we deal with approximatelyparaxial waves. Consider now the Fermat variationalprinciplemin    x y n   x  d l , (37)where the minimization is over all orbits connecting   x and  y  and d l  is a length element of the orbit. The paraxialapproximation amounts to d l     1    12 (d  x  /d  z ) 2  d  z . Thescaling for     in Eq. (36) implies that the initial conditionfor the ray  x (  z ) must satisfy d  x  /d  z (  z    Z 1 )    O (  1/2 ).Substituting the expansion for d l  and the form (35) for  n into the Fermat action [expression (37)], we obtainmin    x y n   x  d l    min    Z 1  Z 2  1      P   1   12   d  x d  z   2  d  z   o        Z 2    Z 1     Q   x ,  y     o    . (38)Therefore the action  Q  is indeed an approximation of theFermat action.The mathematical analysis is valid for the optical prob-lem ( Op ) regardless of its origin. It is particularly inter-esting to note that Eqs. (8) and (9) form the semiclassicallimit of the Schro¨dinger equation. The function  P  thenhas the interpretation of the potential of the physical sys-tem, and the  z  coordinate represents time. Therefore the variational principle ( Mp ) means that if we are given theabsolute value of the wave function everywhere in spaceat two different times  Z 1  and  Z 2  we can find the phase of the wave function at all times  t    (  Z 1 ,  Z 2 ).Some of the results presented in this section, and, inparticular, the connection between problems ( Mp ) and( Wp ) were also derived (by different arguments) for thespecial case of the quadratic Monge problem in Refs. 7 – 9.Our proofs are formal in the sense that we tacitly assumethat all the functions are sufficiently smooth. Acompleterigorous analysis of existence, uniqueness, and regularityof the solutions to problems ( Mp ) and ( Wp ) is delicate andlies beyond the scope of this paper. We refer the readerto Refs. 7 and 8. For the sake of completeness, though,we list a number of basic results that can be obtained forour action  Q  by the tools of these references:1. There exists a minimizer  I     I  (  x ,  z ) of ( Wp ) thatsatisfies the end conditions  I  (  x ,  Z 1 )    I  1 (  x ),  I  (  x ,  Z 2 )   I  2 (  x ). This minimizer may be nonunique.2. If   P  is continuously differentiable, then there existsa maximizer     of   E  that is a Lipschitz function and satis-fies the equation (      /     z )    12     x    2   P  almost every-where.3. A lot more is known in the special but importantcase of homogeneous media where  P    0. For example,if the intensities  I  1  and  I  2  are continuous (or even just  L 1 ) functions, then the minimizer of the Monge problemis unique. Furthermore, a wealth of regularity resultsare known in this case. 7 2168 J. Opt. Soc. Am. A/Vol. 21, No. 11/November 2004 J. Rubinstein and G. Wolansky
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