A non-perturbative quantization of a paraxial electromagnetic field is
achieved via a generalized dispersion relation imposed on the longitudinal and
the transverse components of the photon wave vector. The theoretical formalism
yields a seamless transition between the paraxial- and the Maxwell-equation
solutions. This obviates the need to introduce either "ad hoc" or
perturbatively-defined field operators. In the limit of narrow beam-like
fields, the theory is in agreement with approximated quantization schemes
provided by other authors.Comment: 4 pages, no figure

Aiello, A.

Woerdman, J. P.

English

arXiv

Woerdman, J.P.

Leiden University Scholary Publications

Exact quantization of a paraxial electromagnetic fieldA. Aiello and J. P. WoerdmanHuygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The NetherlandsReceived 3 March 2005; published 30 December 2005A nonperturbative quantization of a paraxial electromagnetic field is achieved via a generalized dispersionrelation imposed on the longitudinal and the transverse components of the photon wave vector. This theoreticalformalism yields a seamless transition between the paraxial- and the Maxwell-equation solutions. This obviatesthe need to introduce either ad hoc or perturbatively defined field operators. Moreover, our exact formalismremains valid beyond the quasimonochromatic paraxial limit.DOI: 10.1103/PhysRevA.72.060101 PACS numbers: 03.65.Db, 41.85.p, 03.70.kIn this Rapid Communication we seek an answer to thequestion: What is the quantum-mechanical state of a photonin a beam of light whose propagation can be classically de-scribed by a paraxial wave equation? The problem of quan-tum propagation of paraxial fields was considered by severalauthors in the past 1. In these contributions the approachwas either ad hoc or approximated. A notable exception wasthe work of Deutsch and Garrison where the authors devel-oped a perturbative quantization scheme 2. However, al-though their theory formally solves, order by order in a per-turbation expansion, the problem of paraxial quantumpropagation, it suffers from two main limitations. First, itdoes not provide any clear and easily manageable formulafor the paraxial quantum modes. Second, it requires thequasimonochromatic approximation which is unsuitable for,e.g., the description of the very quantum phenomenon ofpropagation of broad-band entangled photons provided byspontaneous parametric down-conversion SPDC. The in-creasing importance, for quantum information in general 3and quantum cryptography 4 in particular, of a proper de-scription of paraxial propagation of SPDC entangled pho-tons, calls for an exact namely, nonperturbative theory ofparaxial quantum fields.The aim of this Rapid Communication is to introduce anonperturbative method to quantize a paraxial electromag-netic field. Our scheme is conceptually simple: we begin byconsidering the quantized transverse electromagnetic vectorpotential calculated at the arbitrary initial time t=0. Thenwe select from the wave-vector space only those field con-figurations which are exact solutions of the paraxial waveequation, and let them evolve in time according to thed’Alembert wave equation. This procedure automatically en-sures the validity of both the canonical commutation rela-tions and the transversality conditions for the fields. Westress that our theory accounts for the axial propagation ofvector fields with any spectral and spatial bandwidth andreproduces the well-known paraxial results in the limit ofnarrow beamlike fields.To begin with, let us consider the vector potential operatorÂr , t= Â+r , t+ Â−r , t in the Coulomb gauge whichcan be written in the plane-wave basis as 5Â+r,t = d3k 1630ck1/2	=12kâkexpik · r − ickt , 1and Â−r , t is the Hermitian conjugate of Â+r , t. Itclearly satisfies the d’ Alembert wave equation Âr , t=0.The two unit polarization vectors k =1,2 are trans-verse k ·k=0 and mutually orthogonal 1k ·2k=0. Moreover, the annihilation and creation operatorssatisfy the canonical commutation rules âk , â† k=3k−k. Since the fields considered in this paper aremainly beams which propagate close to the z direction, wefind it convenient to introduce a finite quantization length Lalong the z axis with discrete wave vector Cartesian z com-ponents kz→	n= 2 /Ln n=0,1 , . . . . The choice n0 im-plies that only the parts of the field which propagate in thepositive z direction are included in Eq. 1. The integral withrespect to dkz in Eq. 1 is then replaced by a sum6,7dkxdkydkz→ 2 /L	nd2kT, where kT= kx ,ky andd2kTdkxdky. Moreover, we scale the annihilation and cre-ation operators by defining âk=L / 2âkT ,n, in sucha way thatâkT,n, â† kT,n = nn2kT − kT , 2where the Kronecker symbol replaces the delta function ac-cording to kz−kz→ L /2nn.A paraxial field is usually expressed as an envelope fieldmodulating a carrier plane wave with wave vector k0 andangular frequency 0=ck0. Without lack of generality weassume k0=k0ẑ and we choose k00 so that the carrier planewave propagates in the positive z direction. By using thetrivial identity1 =L202/Ldk0expik0z − j0texpik0z − i0t, 3where L is an arbitrary length, we can rewrite the vectorpotential operator asPHYSICAL REVIEW A 72, 060101R 2005RAPID COMMUNICATIONS1050-2947/2005/726/0601014/$23.00 ©2005 The American Physical Society060101-1Â+r,t = Â+r,t  1=L202/Ldk0 expik0z − i0t̂r,t , 4where we have introduced the envelope field̂r,t = 	n d2kT 820ckL1/2	=1,2kâkT,n expik − k0ẑ · r − ick − 0t , 5where k=kT+ ẑ	n. Now, the key idea is to find a subspace ofthe three-dimensional wave vector k space where the initialtime envelope field ̂r̂r , t=0 satisfies the paraxialwave equation 82̂rx2+2̂ry2+ 2ik0̂rz= 0. 6If we substitute from Eq. 5 into Eq. 6 we obtain	n/k0 = 1 − kT2/2k02 . 7This generalized dispersion relation plays a key role throughthis paper. It defines a two-dimensional domain in the kspace where both d’Alembert wave equation and paraxialwave equation are satisfied. For sake of clarity we writeq=kT, q= q and define the dimensionless parameter=q / 2k02. If we denote with  0, /2 the angle be-tween the wave vector k and the axis z, then q /k · ẑ=tan and from Eq. 7, it follows that 2=−cot +2+cot2 .This relation is exact; however, we can gain someinsight if we consider it in the limit 1 where2−3 /6+O5. This equations shows that 2=q /k0 is approximately equal to the divergence angle of aGaussian beam9. Moreover, by comparing  with Eq. 2.8by Deutsch and Garrison2, one recognizes 21 as theirperturbative expansion parameter. However, in our case theonly constraint is 1, as follows from Eq. 7 and thecondition 	n0.From Eq. 7 it readily follows that the exponential factorin Eq. 5 can be writtenexpik − k0ẑ · r − ick − 0t= expiq · x − i2k0zexp− i0t1 + 4 − 1 , 8where xx ,y. Clearly Eq. 7 affects also the value of thepolarization unit vectors in Eq. 5. To see this, we firstwrite the total wave vector k in terms of q and k0 ask= q̂q+ ẑk01−2, where q̂q / q, then we arbitrarilychoose we always have this freedom 2q ,= ẑ q̂. Theremaining unit vector 1 is then uniquely fixed by the cyclicrelation 12k to the value1q, = q̂1 − 2 − ẑ2/1 + 41/2. 9It is now possible to write explicitly the envelope field ̂rrestricted to the k subspace defined by the dispersion relationEq. 7. From the definition of 	n and Eq. 7, it follows thatwe must select from the sum over n in Eq. 5 only thoseterms corresponding to n=nk0L /21−2IP,where “IP” stands for integer part. This objective can beachieved by replacing in Eq. 5 	n→	nn,n, wheren,n=1 for n=n and n,n=0 otherwise. Finally, wecan write from Eq. 5̂r = 	n d2q 82001 + 41/2L1/2 	=1,21q,âq,nn,nexpiq · x − i2k0z , 10where q , are given by Eq. 9 and previous formulas.We have substituted everyplace in Eq. 10 	n by Eq. 7; thisoperation is permitted by the presence of n,n within thesum. The only exception to these substitutions is representedby the operators âq ,n which, for the moment, are leftunchanged.At this point, we note that since the restriction to theparaxial k subspace has already been achieved via Eq. 10,it is possible to make a step backward from the discretemomentum 	n to the continuous paraxial frequency  :c	n= 2c /Ln→. The reason for this choice for the name of, will become clear later see discussion after Eq. 23. Theother required replacements are 	n→ L /2cd,âq ,n=2c /Lâq , andn,n =2cL„ − ck01 − 2… , 11in such a way thatâq,, â† q, =  − 2q − q . 12Equation 4 can be then written asÂ+r,t = LL02/Ldk0 expik0z − i0t  d d2q 1630k01 + 41/21/2 	=1,2q,âq,„ − ck01 − 2…expiq · x − i2k0zexp− i0t1 + 4 − 1 ,13where the irrelevant term L /L can be absorbed in the defi-nition of Â+r , t. This is not yet our final expression sincewe can perform explicitly the integration with respect to which is still present in the expression for the annihilationoperator: dâq,„ − ck01 − 2…  1 + 2âq,0 ,14whereA. AIELLO AND J. P. WOERDMAN PHYSICAL REVIEW A 72, 060101R 2005RAPID COMMUNICATIONS060101-2âq,0, â† q,0 = 2q − q0 − 0 .15The use of Eq. 14 has been suggested to us by Visser andNienhuis who first introduced this technique 11. Finally, wecan write, after redefining ck0,Â+r,t = d exp− it − z/c 1630c1/2 d2q 	=1,2Eq,,z,tâq,expiq · x − iq2c2 z ,16where we have defined the slowly varying polarization vec-torsEq,,z,t  q, 1 + 21 + 41/2exp− it1 + 4 − 1 , 17and the unit vectors q , are given by Eq. 9 and pre-vious formulas.Equation 16 is the first main result of this work. It rep-resents a vector potential field operator which is a bona fidetransverse field obeying the d’Alembert wave equation forany time t0, whose corresponding envelope field satisfiesthe paraxial wave equation 6 at t=0. We stress that thisexpression is exact; no approximations were made.Now, we are ready to address the problem of building thequantum mechanical state describing a photon in a paraxialbeam. To this end, we first note that until now the creationoperators â†q , have passed untouched through all ouroperations. In fact, they still satisfy the canonical commuta-tion relations Eq. 12. However, their form is not the mostsuitable one to deal with paraxial fields; therefore we intro-duce the Fourier-transformed operators7,10âx, =12 d2qâq,expiq · x , 18such thatâx,, â† x, =  − 2x − x . 19If we substitute Eq. 18 in Eq. 16 we obtain, after somealgebra,Â+r,t = 	=12  d e−it−z/c40c/1/2Âx,z,,t ,20where r= x ,z and we have introduced the exact slowlyvarying photon annihilation vector operatorsÂx,z,,t   d2xFx,z,x,,tâx, .21Moreover, in Eq. 21 we have defined the Maxwell-paraxialMP slowly varying modesFx,z,x,,t =122  d2qEq,,z,texpiq · x − x − iq2c2z . 22Equation 22 displays the second main result of this paper. Itdescribes the field in the plane z at time t due to a pointsource with paraxial frequency  located at x in thetransverse plane z=0. As its shape clearly suggests,Fx ,z ,x , , t is the quantum analog of the classicalHuygens-Fresnel diffracted field. This may be seen moreclearly by writing Eq. 22 in the narrow-beam limit which isachieved by restricting the transverse momentum integral tothe domain C= q :q /c. It is easy to see that within thisdomain =qc /21, and Eq , ,z , teq, wheree1q= q̂+O and e2q= ẑ q̂, are the zeroth order po-larization unit vectors. In this limit Eq. 22 reduces to thewell-known paraxial Green’s function Px ,z ,x ,8Px,z,x, = Cd2q22eqeiq·x−x−q2c/2z,23here generalized to vector fields. From Eq. 23 it is clearthat in the narrow-beam limit the paraxial frequency reducesto the actual frequency  and Âx ,z , , t becomes timeindependent. Now, it is straightforward to show that at afixed time t, in each transverse plane z, the MP functionsFx ,z ,x , , t are quasiorthogonal d2xFx,z,x,,t · Fx,z,x,,t=  d2q22 1 + 21 + 41/2expiq · x − x ,24that is, the right side of this equation approaches2x−x in the narrow-beam limit →0, as expectedfrom the orthogonality of the classical paraxial Green’s func-tions for free space propagation. This result was alreadyfound by Visser and Nienhuis11 who suggested to interpretthe Fourier-transformed creation operator â†x , as the op-erator which creates at t=0 a photon with polarization  inthe paraxial mode expiz /cPx ,z ,x ,. More gener-ally, the interpretation of Fx ,z ,x , , t as single-photonwave function can be put on a rigorous basis by introducingthe “transverse-position” states x , , â†x ,0.Then it readily follows that 0Â+x ,z , tx , ,Fx ,z ,x , , texp−it−z /c.Now, the definition of x , , makes it possible to asso-EXACT QUANTIZATION OF A PARAXIAL… PHYSICAL REVIEW A 72, 060101R 2005RAPID COMMUNICATIONS060101-3ciate to any single-photon state , its correspondingMaxwell-paraxial wave function x ,x , , .Then, for example, the MP wave function associatedto the plane wave state q , , â†q ,0, is simplygiven by the Fourier relation x , , q , ,= expiq ·x /2−. More generally, for a givencomplete set of orthogonal transverse functions nmx , as,e.g., the Hermite- or the Laguerre-Gaussian beams 9, it ispossible to build the corresponding MP single-photon stateasn,m,, = d2xnmx,x,, , 25where x , , n ,m , ,=nmx ,−.This equation is our third and final main result: it repre-sents the exact quantum-mechanical state of each of the pho-tons in the classical paraxial beam nmx ,. It should benoticed that while Eq. 25 is always well defined in ourperfectly general framework, its interpretation in terms ofparaxial beams is limited by the classical condition̃nmq ,0 for q /c, where ̃nmq , is the two-dimensional Fourier transform of nmx ,.In conclusion, a nonperturbative quantization scheme forelectromagnetic paraxial fields has been introduced. It relieson the fact that it is possible to select some initial field con-figurations which are exact solutions of the paraxial waveequation. These configurations are then evolved at later timeswith the d’Alembert wave equation. In this way we wereable to find explicit and manageable expressions for the ex-act field in both momentum Eq. 16 and in position Eqs.20–22 representation. Moreover we gave an unambigu-ous definition for Maxwell-paraxial quantum states Eq.25. The usefulness of the present theory, first introducedby us in 12, has been very recently confirmed by Calvo etal. 13, who used our formalism to illustrate some propertiesof photon angular momentum.It is a pleasure to acknowledge Jorrit Visser and GerardNienhuis for fruitful discussions. We also acknowledge sup-port from the EU under the IST-ATESIT contract. Thisproject was also supported by FOM.1 R. Graham and H. Haken, Z. Phys. 213, 420 1968; H. Yuenand J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 1978;T. A. B. Kennedy and E. M. Wright, Phys. Rev. A 38, 2121988; J. C. Garrison, H. Nathel, and R. Y. Chiao, J. Opt. Soc.Am. B 5, 1528 1988; M. I. Kolobov, Rev. Mod. Phys. 71,1539 1999.2 I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 24981991.3 M. A. Nielsen and I. L. Chuang, Quantum Computation andQuantum Information, reprinted 1st ed. Cambridge UniversityPress, Cambridge, UK, 2002.4 N. Gisin, G. Ribody, W. Tittel, and H. Zbinden, Rev. Mod.Phys. 74, 145 2002.5 R. Loudon, The Quantum Theory of Light, 3rd ed. OxfordUniversity Press, Oxford, UK, 2000.6 T. D. Lee, Particle Physics and Introduction to Field Theory,revised and updated 1st ed. Harwood Academic, Chur, Swit-zerland, 1988.7 K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd,Phys. Rev. A 42, 4102 1990.8 L. Mandel and E. Wolf, Optical Coherence and Quantum Op-tics 1st ed. Cambridge University Press, Cambridge, 1995.9 A. E. Siegman, Lasers University Science Books, Mill Valley,CA, 1996.10 A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C.Teich, Phys. Rev. Lett. 87, 123602 2001.11 G. Nienhuis private communication; J. Visser, Ph.D. thesis,Leiden University, 2005, unpublished.12 A. Aiello and J. P. Woerdman, e-print quant-ph/0502164.13 G. F. Calvo, A. Picón, and E. Bagan, e-print quant-ph/0509040.A. AIELLO AND J. P. WOERDMAN PHYSICAL REVIEW A 72, 060101R 2005RAPID COMMUNICATIONS060101-4

Exact Paraxial Quantization

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