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An inverse problem in the theory of stochastic electromagnetic beams
Optics Communications 282 (2009) 141–142
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
An inverse problem in the theory of stochastic electromagnetic beams David Kuebel a, Mayukh Lahiri b,*, Emil Wolf b,c a b c
Department of Physics, St. John Fisher College, Rochester, NY 14618, USA Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA Institute of Optics, University of Rochester, Rochester, NY 14627, USA
a r t i c l e
i n f o
Article history: Received 24 June 2008 Received in revised form 9 September 2008 Accepted 10 September 2008
OCIS: 260.0260 260.5430 030.1640
a b s t r a c t The cross-spectral density matrix of an electromagnetic beam has been playing increasingly important role in studies of changes of spectra, of coherence and of polarization as the beam propagates. In this paper we derive solution to an inverse problem, which makes it possible to determine the cross-spectral density matrix of the beam in the source plane z ¼ 0, from the knowledge of the matrix in any cross-section z ¼ z0 > 0 in the half-space into which the beam propagates. We apply the result to the theory of socalled Stokes beams, which were introduced not long ago. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction A central result in the theory of stochastic electromagnetic beams is the propagation law for the so-called cross-spectral density matrix of such beams [1–3]. The elements of the matrix represent correlations between Cartesian components, at a particular frequency, of the electric field at a pair of points in space. The propagation law makes it possible to determine the cross-spectral density matrix of the beam, for any pair of points in the half-space z ¼ z0 > 0 into which the beam propagates, from the knowledge of the cross-spectral density matrix of the beam in the ‘‘source plane” z ¼ 0. Recent developments relating to a possible decomposition of a stochastic electromagnetic beam into a completely polarized and a completely unpolarized beam [4] require the understanding of the inverse propagation problem, namely determination of the cross-spectral density matrix of the beam at all pairs of points in the ‘‘source plane” z ¼ 0 from the knowledge of the cross-spectral density matrix at all pairs of points in any cross-section at z ¼ z0 > 0. In this paper we present solution to this problem and apply it to the ‘‘decomposition” problem mentioned above. Let us first consider a monochromatic beam of frequency x, with its axis along the z direction, propagating into the half-space z > 0. The angular spectrum representation of such a beam ([5], Sections 3.2.2 and 5.6.1) contains only homogeneous waves, whose propagation directions make small angles with z-axis. Since evanescent waves are not present, the space-dependent part
Eðx; y; 0Þ of the electric field Eðx; y; 0Þ expðixtÞ in the plane z ¼ 0 (the source plane) may be expressed in terms of the spacedependent part of the electric field in any transverse plane z ¼ constant > 0 by the formula
Eðq; 0; xÞ ¼ eikz
Z
2
Eðq0 ; z; xÞGðÞ ðq q0 ; z; xÞd q0 :
ð1Þ
In this formula, which is essentially a vector generalization of Eq. (4.9) of Ref. [6] for scalar fields, GðÞ ðq q0 ; z; xÞ is the Green’s function for inverse propagation, specialized for paraxial propagation (cf. [5] – Eq. (5.6)–(17)) and is given by the expression
GðÞ ðq q0 ; z; xÞ ¼
ik exp½ikðq q0 Þ2 =2z: 2p z
ð2Þ
Here k ¼ x=c, c being the speed of light in free space and the integration in Eq. (1) extends over the plane z ¼ constant > 0. Suppose now that the beam is not monochromatic but is stochastic, represented by a statistically stationary ensemble. The cross-spectral density matrices of the beam in the source plane z ¼ 0 and in any transverse plane z ¼ z0 > 0 may be expressed in terms of monochromatic realizations, in the sense of coherence theory in the space-frequency domain [9] by the formulas $ Wðq1 ; q2 ; 0; xÞ ½W ij ðq1 ; q2 ; 0; xÞ ¼ Ei ðq1 ; 0; xÞEj ðq2 ; 0; xÞ ði ¼ x; y; j ¼ x; yÞ ð3aÞ
and $
* Corresponding author. E-mail address: [email protected] (M. Lahiri). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.09.028
Wðq1 ; q2 ; z; xÞ ½W ij ðq1 ; q2 ; z; xÞ ¼ Ei ðq1 ; z; xÞEj ðq2 ; z; xÞ ði ¼ x; y; j ¼ x; yÞ
ð3bÞ
142
D. Kuebel et al. / Optics Communications 282 (2009) 141–142
respectively. On substituting from Eq. (1) into Eq. (3a) it follows that $
Wðq1 ; q2 ; 0; xÞ ¼
Z Z
$
Wðq1 ; q2 ; z; xÞ 2
2
LðÞ ðq1 q01 ; q2 q02 ; z; xÞd q01 d q02 ;
ð4Þ
where
LðÞ ðq1 q01 ; q2 q02 ; z; xÞ ¼ GðÞ ðq1 q01 ; z; xÞGðÞ ðq2 q02 ; z; xÞ: ð5Þ The formula (4) provides solution to the inverse problem; namely it expresses the cross-spectral density matrix of the beam in the plane z ¼ 0 in terms of the cross-spectral density matrix in any cross-sectional plane, z ¼ z0 > 0 say, in the half-space into what the beam propagates [10]. We will now apply the inversion formula (4) to a problem mentioned earlier, which recently became of interest in the theory of so-called Stokes beams [4]. Suppose that the cross-spectral density matrix of a beam in a transverse cross-section z ¼ constant > 0 has the form $
W ðq01 ; q02 ; z;
Ex ðq01 ; z; xÞEx ðq02 ; z; xÞ Ex ðq01 ; z; xÞEy ðq02 ; z; xÞ
xÞ ¼
!
Ey ðq01 ; z; xÞEx ðq02 ; z; xÞ Ey ðq01 ; z; xÞEy ðq02 ; z; xÞ
$
Wðq01 ; q02 ; z; xÞ ¼ aðq01 ; q02 ; z; xÞ
1 0 0 1
ð8Þ
i.e., for each pair of points in the cross-section z ¼ constant > 0, it is proportional to the unit matrix. As noted in [4], such a matrix represents a beam which is completely unpolarized at each point (i.e. when q01 ; z q02 ; z ¼ ðq0 ; zÞ ). On substituting from Eq. (8) into the inversion formula (4) one finds that the cross-spectral density matrix of the beam in the plane z ¼ 0 is also of the form (8), with
aðq1 ; q2 ; 0; xÞ ¼
Z Z
aðq01 ; q02 ; z; xÞ 2
2
LðÞ ðq1 q01 ; q2 q02 ; z; xÞd q01 d q02 :
ð9Þ
The two results which we have just derived, imply that each of the cross-spectral density matrices (6) and (8) necessarily imply that they represent beams which are generated by fields in the plane z ¼ 0 whose cross-spectral density matrices are of the same form. This result provides a rigorous justification of the main conclusion stated in [4], concerning necessary and sufficient conditions for a light beam to be a Stokes beam. Acknowledgements
ð6Þ i.e. each element of the matrix is a product of two functions, one of which depends on the position of a point q01 ; z and the other on the 0 position of the point q2 ; z . As shown in Ref. [4] and as can be verified by simple calculation, such a matrix represents a beam which at every point (i.e. when q01 ; z q02 ; z ¼ ðq0 ; zÞ ) in the half-space z P 0 is completely polarized. On substituting from Eq. (6) into the inversion formula (4) it follows at once that the cross-spectral density matrix of the beam in the plane z ¼ 0 is also of the factorized form (6), and, consequently the field in that plane is also completely polarized at each point, with
Ei ðq; 0; xÞ ¼ eikz
Z
2
Ei ðq0 ; z; xÞGðÞ ðq q0 ; z; xÞd q0 ;
ði ¼ x; yÞ; ð7Þ
as implied by Eqs. (4) and (6). Next suppose that the cross-spectral density matrix in a transverse cross-section z ¼ constant > 0 has the form
The research was supported by the US Air Force Office of Scientific Research under Grant No. F49260-03-1-0138, and by Air Force Research Laboratory (AFRL) under contract 9451-04-C-0296. References [1] E. Wolf, Phys. Lett. A 312 (2003) 263. [2] E. Wolf, Opt. Lett. 28 (2003) 1078. [3] E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge University Press, Cambridge, 2007. [4] E. Wolf, Opt. Lett. 33 (2008) 642. [5] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. [6] J.R. Shewell, E. Wolf, J. Opt. Soc. Am. 58 (1968) 1596. [7] J. Tervo, T. Setala, A.T. Friberg, J. Opt. Soc. Am. A 21 (2004) 2205. [8] M. Alonso, E. Wolf, Opt. Commun. 281 (2008) 2393. [9] For a discussion of the coherence theory for scalar fields in the space-frequency domain, see [5], Sec 4.7. The theory was generalized to stochastic electromagnetic fields in [7] and [8]. [10] One can, of course, generalize formula (4) to provide solution to the inverse problem from a plane z ¼ z0 > 0 to a plane z ¼ z1 , located between the source plane and the plane z ¼ z0 . One then needs to replace z by z z1 in Eqs. (2), (4) and (5).
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
An inverse problem in the theory of stochastic electromagnetic beams David Kuebel a, Mayukh Lahiri b,*, Emil Wolf b,c a b c
Department of Physics, St. John Fisher College, Rochester, NY 14618, USA Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA Institute of Optics, University of Rochester, Rochester, NY 14627, USA
a r t i c l e
i n f o
Article history: Received 24 June 2008 Received in revised form 9 September 2008 Accepted 10 September 2008
OCIS: 260.0260 260.5430 030.1640
a b s t r a c t The cross-spectral density matrix of an electromagnetic beam has been playing increasingly important role in studies of changes of spectra, of coherence and of polarization as the beam propagates. In this paper we derive solution to an inverse problem, which makes it possible to determine the cross-spectral density matrix of the beam in the source plane z ¼ 0, from the knowledge of the matrix in any cross-section z ¼ z0 > 0 in the half-space into which the beam propagates. We apply the result to the theory of socalled Stokes beams, which were introduced not long ago. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction A central result in the theory of stochastic electromagnetic beams is the propagation law for the so-called cross-spectral density matrix of such beams [1–3]. The elements of the matrix represent correlations between Cartesian components, at a particular frequency, of the electric field at a pair of points in space. The propagation law makes it possible to determine the cross-spectral density matrix of the beam, for any pair of points in the half-space z ¼ z0 > 0 into which the beam propagates, from the knowledge of the cross-spectral density matrix of the beam in the ‘‘source plane” z ¼ 0. Recent developments relating to a possible decomposition of a stochastic electromagnetic beam into a completely polarized and a completely unpolarized beam [4] require the understanding of the inverse propagation problem, namely determination of the cross-spectral density matrix of the beam at all pairs of points in the ‘‘source plane” z ¼ 0 from the knowledge of the cross-spectral density matrix at all pairs of points in any cross-section at z ¼ z0 > 0. In this paper we present solution to this problem and apply it to the ‘‘decomposition” problem mentioned above. Let us first consider a monochromatic beam of frequency x, with its axis along the z direction, propagating into the half-space z > 0. The angular spectrum representation of such a beam ([5], Sections 3.2.2 and 5.6.1) contains only homogeneous waves, whose propagation directions make small angles with z-axis. Since evanescent waves are not present, the space-dependent part
Eðx; y; 0Þ of the electric field Eðx; y; 0Þ expðixtÞ in the plane z ¼ 0 (the source plane) may be expressed in terms of the spacedependent part of the electric field in any transverse plane z ¼ constant > 0 by the formula
Eðq; 0; xÞ ¼ eikz
Z
2
Eðq0 ; z; xÞGðÞ ðq q0 ; z; xÞd q0 :
ð1Þ
In this formula, which is essentially a vector generalization of Eq. (4.9) of Ref. [6] for scalar fields, GðÞ ðq q0 ; z; xÞ is the Green’s function for inverse propagation, specialized for paraxial propagation (cf. [5] – Eq. (5.6)–(17)) and is given by the expression
GðÞ ðq q0 ; z; xÞ ¼
ik exp½ikðq q0 Þ2 =2z: 2p z
ð2Þ
Here k ¼ x=c, c being the speed of light in free space and the integration in Eq. (1) extends over the plane z ¼ constant > 0. Suppose now that the beam is not monochromatic but is stochastic, represented by a statistically stationary ensemble. The cross-spectral density matrices of the beam in the source plane z ¼ 0 and in any transverse plane z ¼ z0 > 0 may be expressed in terms of monochromatic realizations, in the sense of coherence theory in the space-frequency domain [9] by the formulas $ Wðq1 ; q2 ; 0; xÞ ½W ij ðq1 ; q2 ; 0; xÞ ¼ Ei ðq1 ; 0; xÞEj ðq2 ; 0; xÞ ði ¼ x; y; j ¼ x; yÞ ð3aÞ
and $
* Corresponding author. E-mail address: [email protected] (M. Lahiri). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.09.028
Wðq1 ; q2 ; z; xÞ ½W ij ðq1 ; q2 ; z; xÞ ¼ Ei ðq1 ; z; xÞEj ðq2 ; z; xÞ ði ¼ x; y; j ¼ x; yÞ
ð3bÞ
142
D. Kuebel et al. / Optics Communications 282 (2009) 141–142
respectively. On substituting from Eq. (1) into Eq. (3a) it follows that $
Wðq1 ; q2 ; 0; xÞ ¼
Z Z
$
Wðq1 ; q2 ; z; xÞ 2
2
LðÞ ðq1 q01 ; q2 q02 ; z; xÞd q01 d q02 ;
ð4Þ
where
LðÞ ðq1 q01 ; q2 q02 ; z; xÞ ¼ GðÞ ðq1 q01 ; z; xÞGðÞ ðq2 q02 ; z; xÞ: ð5Þ The formula (4) provides solution to the inverse problem; namely it expresses the cross-spectral density matrix of the beam in the plane z ¼ 0 in terms of the cross-spectral density matrix in any cross-sectional plane, z ¼ z0 > 0 say, in the half-space into what the beam propagates [10]. We will now apply the inversion formula (4) to a problem mentioned earlier, which recently became of interest in the theory of so-called Stokes beams [4]. Suppose that the cross-spectral density matrix of a beam in a transverse cross-section z ¼ constant > 0 has the form $
W ðq01 ; q02 ; z;
Ex ðq01 ; z; xÞEx ðq02 ; z; xÞ Ex ðq01 ; z; xÞEy ðq02 ; z; xÞ
xÞ ¼
!
Ey ðq01 ; z; xÞEx ðq02 ; z; xÞ Ey ðq01 ; z; xÞEy ðq02 ; z; xÞ
$
Wðq01 ; q02 ; z; xÞ ¼ aðq01 ; q02 ; z; xÞ
1 0 0 1
ð8Þ
i.e., for each pair of points in the cross-section z ¼ constant > 0, it is proportional to the unit matrix. As noted in [4], such a matrix represents a beam which is completely unpolarized at each point (i.e. when q01 ; z q02 ; z ¼ ðq0 ; zÞ ). On substituting from Eq. (8) into the inversion formula (4) one finds that the cross-spectral density matrix of the beam in the plane z ¼ 0 is also of the form (8), with
aðq1 ; q2 ; 0; xÞ ¼
Z Z
aðq01 ; q02 ; z; xÞ 2
2
LðÞ ðq1 q01 ; q2 q02 ; z; xÞd q01 d q02 :
ð9Þ
The two results which we have just derived, imply that each of the cross-spectral density matrices (6) and (8) necessarily imply that they represent beams which are generated by fields in the plane z ¼ 0 whose cross-spectral density matrices are of the same form. This result provides a rigorous justification of the main conclusion stated in [4], concerning necessary and sufficient conditions for a light beam to be a Stokes beam. Acknowledgements
ð6Þ i.e. each element of the matrix is a product of two functions, one of which depends on the position of a point q01 ; z and the other on the 0 position of the point q2 ; z . As shown in Ref. [4] and as can be verified by simple calculation, such a matrix represents a beam which at every point (i.e. when q01 ; z q02 ; z ¼ ðq0 ; zÞ ) in the half-space z P 0 is completely polarized. On substituting from Eq. (6) into the inversion formula (4) it follows at once that the cross-spectral density matrix of the beam in the plane z ¼ 0 is also of the factorized form (6), and, consequently the field in that plane is also completely polarized at each point, with
Ei ðq; 0; xÞ ¼ eikz
Z
2
Ei ðq0 ; z; xÞGðÞ ðq q0 ; z; xÞd q0 ;
ði ¼ x; yÞ; ð7Þ
as implied by Eqs. (4) and (6). Next suppose that the cross-spectral density matrix in a transverse cross-section z ¼ constant > 0 has the form
The research was supported by the US Air Force Office of Scientific Research under Grant No. F49260-03-1-0138, and by Air Force Research Laboratory (AFRL) under contract 9451-04-C-0296. References [1] E. Wolf, Phys. Lett. A 312 (2003) 263. [2] E. Wolf, Opt. Lett. 28 (2003) 1078. [3] E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge University Press, Cambridge, 2007. [4] E. Wolf, Opt. Lett. 33 (2008) 642. [5] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. [6] J.R. Shewell, E. Wolf, J. Opt. Soc. Am. 58 (1968) 1596. [7] J. Tervo, T. Setala, A.T. Friberg, J. Opt. Soc. Am. A 21 (2004) 2205. [8] M. Alonso, E. Wolf, Opt. Commun. 281 (2008) 2393. [9] For a discussion of the coherence theory for scalar fields in the space-frequency domain, see [5], Sec 4.7. The theory was generalized to stochastic electromagnetic fields in [7] and [8]. [10] One can, of course, generalize formula (4) to provide solution to the inverse problem from a plane z ¼ z0 > 0 to a plane z ¼ z1 , located between the source plane and the plane z ¼ z0 . One then needs to replace z by z z1 in Eqs. (2), (4) and (5).