Journal of Optics J. Opt. 9 (07) 03560 (5pp) https://doi.org/0.088/040-8986/aa5d5 Effects of phase conjugation on electromagnetic optical fields propagating in free space Bhaskar Kanseri Experimental Quantum Interferometry and Polarization (EQUIP), Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delh06, India E-mail: bkanseri@physics.iitd.ac.in Received 6 October 06, revised 8 November 06 ccepted for publication 9 December 06 Published 6 February 07 bstract By using the property of phase conjugation, we demonstrate that the inverse of van Cittert Zernike theorem holds for electromagnetic (EM) fields propagating in free space. This essentially implies that spatially incoherent partially polarized field distributions can be generated from spatially coherent partially polarized optical fields. We further utilize phase conjugation with a polarization rotator to swap the spatial coherence properties of orthogonal polarization components of EM fields on propagation, at least in free space. This study suggests that the method of phase conjugation could be potentially useful in arbitrarily manipulating spatial coherence properties of vector optical fields in the field plane. Keywords: polarization, electromagnetic fields, coherence. Introduction The statistical properties of light fields, such as coherence and polarization features, may vary appreciably on propagation, even in free space. The van Cittert Zernike theorem [, ], propounded during the 930s, advocates an increase in the spatial coherence properties of light on propagation in free space. This important theorem has found several potential applications in optics, such as in determination of angular diameter of stars using Michelson stellar interferometer [3, 4], in reconstruction of the brightness distribution of distant sources (called aperture synthesis) [5], in optical and ultrasonic imaging [6, 7], in adaptive optics [8] etc. The van Cittert-Zernike theorem states that in the far zone, for scalar fields, the developed coherence in consequence of propagation of a field distribution in free space is proportional to the Fourier transform of the intensity distribution at the source plane. Mathematically, the complex degree of spectral coherence at field points r and r as shown in figure can be expressed as [3] ( r, r, ) exp[ ik( r r )] S( ) exp( i kn. ) d, S( ) d ( ) in the space frequency domain, where n is the difference of unit vectors specifying the two directions, S ( ) is the spectral density at the source, k is the wavevector, and ω is the angular frequency of light. The integration is done over area of the incoherent source. recent experimental observation of van Cittert Zernike theorem can be found in [9]. For scalar fields, Devaney et al [0] have demonstrated the existence of an inverse of the van Cittert Zernike theorem. In this study, using the property of phase conjugation, it has been shown that highly incoherent field distribution can be generated from a highly coherent one for scalar fields propagating in free space. This study concluded that the increase in spatial coherence due to propagation of field distribution in free space can be nullified once the property of phase conjugation is utilized. The developments in the research that have taken place in the area of electromagnetic (EM) optical fields over the last couple of decades emphasized the involvement of polarization in the existing scalar coherence theories []. Consequently, a set of new parameters namely two-point (generalized) Stokes parameters were proposed [ 4] and experimentally determined [5] for several kinds of sources governing the spatial coherence properties of different polarization components of EM fields [6 8]. The coherence properties of these vector 040-8978/7/03560+05$33.00 07 IOP Publishing Ltd Printed in the UK
and U ( ) (, ) exp( i t). Shewell and Wolf (968) [0, 6] have demonstrated a reciprocity theorem valid for free optical fields, according to which, if the two scalar field distributions in a plane 0 0 are complex conjugates of each other, i.e., if U (, ) [ U (, )], ( 3) ( ) 0 ( ) 0 Figure. Spatially incoherent source of area and the notations. is the position vector at the source plane and n and n are the unit vectors in the directions corresponding to points r and r. fields are shown to be quantified in terms of the EM degree of coherence [9]. Further, a natural extension of scalar optical fields to the EM fields has been demonstrated to follow the EM spectral interference law and it has been shown that the EM degree of coherence provides a measure of the modulation contrast for all the usual Stokes parameters [0, ]. The generalization of the van Cittert Zernike theorem has already been investigated for EM fields by several authors in the past [ 5]. Using one such extension, as discussed in [5], the modulation contrasts at a far zone for the generalized Stokes parameters have been shown to be fully determined by the respective usual Stokes parameters at the source plane, i.e. l ( r, r, ) exp[ ik( r r)] s l (, ) exp( i kn. ) d, ( ) where sl(, ) Sl(, ) S (, ) d 0 are the normalized usual (one-point) Stokes parameters of the source plane (see figure ). Thus it shows that the generalized (twopoint) Stokes parameters, or the spatial coherence properties of different polarization components at the far zone, are governed by the usual Stokes parameters, i.e. polarization properties of the incoherent EM source. Clearly the EM degree of coherence changes on propagation following the van Cittert Zernike theorem for EM fields. In this paper, we study the outcome of phase conjugation theoretically, and demonstrate that it leads to the inverse of van Cittert Zernike theorem for vector fields on propagation in free space. To our knowledge, effects of phase conjugation on propagation of EM fields have not been demonstrated previously. We further show that as a byproduct, it offers a means to arbitrarily swap the spatial coherence properties of orthogonal polarization components of EM fields propagating in free space, by using a polarization rotator.. Electromagnetic fields and phase conjugation Let us consider that two spatially incoherent, free quasimonochromatic fields propagating in the ζ direction (see figure ) with frequency ω are given by U ( ) (, ) exp( i t) U (, L) [ U (, L)], ( 4) ( ) ( ) 0 0 throughout the half-space, for all values of L and, where 0 L 0. Thus fields at planes L and -L, which are mirror images of each other with respect to the 0 plane, would be complex conjugates of each other. One can now consider the generalization of the reciprocity theorem [6] for freely propagating electromagnetic fields by implementing the scalar result of equations (3) and (4) for each polarization component of a vector field. Thus, considering fields U ( ) (, ) and U ( ) (, ) to be partially polarized, if different polarization components, i.e., i x, y of the EM field in some plane 0 0 are complex conjugates (shown by asterisk) of each other, i. e., if for all i ) 0 U (, ) [ U (, )], for i x, y ( 5) ( ) ( ) U (, L) [ U (, L)], ( 6) throughout the half-space. This infers that at planes L and -L, the EM field components would be complex conjugates of each other. Clearly equations (5) and (6) hold only for the plane, which is phase conjugate in both x and y axes. Since the elements of cross-spectral density matrix are defined as (angle brackets denote the ensemble average) [] W ( r, r, ) U ( r, ) U ( r, ), ( 7) ij 0 i 0 j 0 by considering the ensemble of free EM fields following the generalized reciprocity theorem (equations (5) and (6)), with each element of the ensemble propagating into the half space 0, one can show in a straightforward manner that if W ( ) ( ) ij 0 ij 0 ( r, r, ) [ W ( r, r, )], ( 8) ij 0 ij ) 0 W ( r, r, L) [ W ( r, r, L)]. ( 9) The generalized (two-point) Stokes parameters which are represented in terms of the elements of the cross-spectral density matrix are expressed as [3] S ( r, r, ) W ( r, r, ) W ( r, r, ), ( 0a) 0 xx yy S ( r, r, ) W ( r, r, ) W ( r, r, ), ( 0b) xx yy S ( r, r, ) W ( r, r, ) W ( r, r, ), ( 0c) xy yx S ( r, r, ) i[ W ( r, r, ) W ( r, r, )]. ( 0d) 3 yx xy
Figure. Geometrical illustration of the behavior of phase conjugation on the propagation of EM field. Optical fields at the source plane and at the field plane are spatially incoherent partially polarized, whereas at the phase conjugate plane, the field is spatially coherent partially polarized. We note that the generalized (two-point) Stokes parameters can be transformed into usual (single-point) Stokes parameters once the position coordinates are equalized, i.e. r r. In a straightforward manner, one can define the normalized generalized Stokes parameters corresponding to different polarization components given by [0, ] Sl ( r, r, ) l ( r, r, ) [ S ( r, ) S ( r, )] 0 0, l 0,,,3; ( ) where S 0 ( r, i ) for ( i, ) is the spectral density (first usual Stokes parameter) at point r i. These parameters govern the spatial coherence properties for different polarization components of the EM field. One can determine the EM degree of coherence from the normalized generalized Stokes parameters using the following relation [] 3 ( r, r, ) ( r, r, ). ( ) l0 Using equations (8) () one can show that if 0 0 ( r, r, ) [ ( r, r, )], ( 3) 0 0 ( r, r, L) [ ( r, r, L)], ( 4) Now if we consider L 0, it follows immediately that if ( r, r, L) [ ( r, r, L)], ( 5) ( r, r, L) [ ( r, r, 0 )], ( 6) which infers that the modulation contrasts or the absolute value of the normalized generalized (two-point) Stokes parameter for different polarization components of the EM field at distance L would be the same as that of the source plane, as shown in figure. Using equation () the EM degree of coherence, defined in terms of the absolute values, would thus remain the same both at the source plane P and at the field plane P 3, whereas the phases of the EM degree of coherence will be opposite in sign (equation (6)). In other words, the spatial coherence properties of different polarization components of the EM field would increase with the propagation of the incoherent, partially polarized beam from the source plane P towards the phase conjugate plane P.t plane P, the coherence properties of different polarization components of the field distribution would be the Fourier transform of the source polarization properties (at P ) owing to the van Cittert Zernike theorem [5]. Now, due to the phase conjugation properties of plane P, the spatial coherence of different polarization components of the EM field would start decreasing on propagation towards plane P 3, making the field at the P 3 plane spatially incoherent and partially polarized, similar to the field at source plane P (see figure ). Thus one can see the the inverse of van Cittert Zernike theorem exists for the EM fields provided the fields are phase conjugate at an intermediate plane. Optical phase conjugation at any plane can be achieved by several standard methods proposed in [7 9].Degenerate four wave mixing (backward) is quite a popular technique to produce optical phase conjugation involving two counter-propagating strong plane fields and a signal field with the same frequency but different propagation directions illuminating a third-order nonlinear medium simultaneously [30]. This generates a new field which is the phase conjugate version of the signal field. Similarly, non-degenerate four wave mixing has also been shown to generate optical phase conjugation [3]. In addition, several studies demonstrated that photo-refractive effects using materials such as BaTiO 3,LiNbO 3,BSO,BGO etc can be utilized to produce optical phase conjugation [3, 33]. Other methods to realize optical phase conjugation include various types of stimulated scatterings [9], among which backward stimulated Brillouin scattering was the first one to achieve optical phase conjugation [34]. Implementation of one of these methods could offer a future possibility to experimentally demonstrate the effects of phase conjugation on EM fields propagating in free space. It is worthwhile to mention that if plane P is phase conjugate for one polarization only (either x or y) [9], one can see that equation (5) would be valid for that polarization 3
component only. In that situation, on propagation in free space, only those field components facing phase conjugation would have a decrease in spatial coherence. On the other hand, for the orthogonal field components spatial coherence would increase owing to the fact that such fields would not follow the generalized reciprocity theorem given in ( ) ( ). equation (6), i.e., U (, L) [ U (, L)] Use of such polarization selective phase conjugate plane might manifest more complex features in the EM field propagating in free space, which, however, are not considered under the framework of the study presented in this paper. 3. n application: swapping of spatial coherence One application of this interesting behavior could be to swap the spatial coherence properties of different orthogonal planes of a polarized, partially coherent light beam by using a polarization rotator at the phase conjugate plane. To visualize this, let us consider that a half wave plate with its fast axis making 45 with the polarization of the input field is placed at the phase conjugate plane, which rotates orthogonal polarization components of the field by 90. In this situation one can see if ( ) ( ) j 0 U ( r, ) [ U ( r, )], for ( i, j) ( x, y) ( 7) j ) 0 U ( r, L) [ U ( r, L)]. ( 8) Using equation (7), the cross-spectral density matrix in this situation yields, if W ij 0 ji ) 0 ( r, r, ) [ W ( r, r, )], ( 9) ij 0 ji ) 0 W ( r, r, L) [ W ( r, r, L)]. ( 0) This finally in terms of the normalized generalized Stokes parameters gives for even parameters if ( ) ( ) ( r, r, L) [ ( r, r, L)], for l 0, ( ) ( ) ( ) and for odd parameters if ( r, r, L) [ ( r, r, 0 )]. ( ) ( r, r, L) [ ( r, r, L)], for l, 3 ( 3) ( r, r, L) [ ( r, r, 0 )]. ( 4) From equations () and (4) we infer that at field plane P 3 the normalized two-point Stokes parameters associated with the intensity pattern and with 45 linear polarization remain unchanged, whereas those associated with linear and circular polarizations become opposite in sign. These observations together imply that the spatial coherence properties of orthogonal polarization components at planes P and P 3 would be swapped due to the effect of a polarization rotator placed in the phase conjugate plane. lso from equation (), we note that the EM degree of coherence of the light field does not experience any change, owing to the fact that the transformation of polarization rotation is a unitary, reversible transformation [35]. These results could be generalized for swapping spatial coherence properties of any set of polarization axes, and at any set of equidistant mirror-image planes with respect to plane P, once the polarization rotator could be used to make an appropriate angle between its fast axis and input polarization. It is worth mentioning here that the use of a half wave plate without a phase conjugate plane would also rotate the polarization components, however, owing to the propagation of the EM field between the two planes P and P 3 in free space, the spatial coherence properties of the rotated field components (in plane P 3 ) would be inevitably different from their original counterparts (in plane P ). Introduction of an intermediate phase conjugate plane reverses the change in spatial coherence due to propagation of EM field and thus ensures that the spatial coherence properties of original and rotated polarization components of the EM field at the two planes in consideration are identical. 4. Conclusions In conclusion, we have considered the effects of phase conjugation on EM optical fields on propagation in free space. This study demonstrates that the inverse of recently proposed van Cittert Zernike theorem for EM fields holds if the property of phase conjugation is utilized. Thus spatial coherence of different polarization components of the EM field may decrease on propagation, at least in free space. This study further reveals that one may swap the spatial coherence properties of orthogonal polarization components of polarized field distributions, by making simple reversible polarization transformation at the phase conjugate plane. The analysis also indicates that changes in spatial coherence properties of EM fields due to propagation in free space can be revert back by using the property of phase conjugation. We believe that in view of various experimental developments in producing optical phase conjugation took place during last several decades, effects of phase conjugation on EM fields propagating in free space could be demonstrated and studied experimentally in the near future. cknowledgments The author acknowledges the new faculty research grants (seed grant and project MI035) received from Indian Institute of Technology Delhi, India. This research was also supported by young scientist research grant YSS/05/ 000743 from Science and Engineering Research Board (SERB) India. 4
References [] Van Cittert P H 934 Physica 0 [] Zernike F 938 Physica 5 785 [3] Mandel L and Wolf E 995 Optical Coherence and Quantum Optics (New York: Cambridge University Press) [4] Marathay 983 Elements of Optical Coherence Theory (New York: Wiley) [5] Thompson R, Moran J M and Swenson G W 004 Interferometry and Synthesis in Radio stronomy (Weinheim: Wiley) [6] Barakat R 000 J. Mod. Opt. 47 607 [7] Liu D L and Waag R C 995 IEEE Trans. Ultras. Ferroelec. Freq. Control 4 590 [8] Hardy J W 998 daptive Optics for stronomical Telescopes (Oxford: Oxford University Press) [9] Kanseri B and Kandpal H C 00 Opt. Lett. 35 70 [0] Devaney J, Friberg T, Kumar T N and Wolf E 997 Opt. Lett. 67 [] Wolf E 007 Introduction to the Theory of Coherence and Polarization of Light (New York: Cambridge University Press) [] Ellis J and Dogariu 004 Opt. Lett. 9 536 [3] Korotkova O and Wolf E 005 Opt. Lett. 30 98 [4] Tervo J, Setala T, Rouff, Refregier P and Friberg T 009 Opt. Lett. 34 3074 [5] Kanseri B 03 Optical Coherence and Polarization: n Experimental Outlook (Saarbrucken: Lambert cademic) [6] Kanseri B and Kandpal H C 008 Opt. Lett. 33 40 [7] Kanseri B, Rath S and Kandpal H C 009 Opt. Lett. 34 79 [8] Kanseri B and Kandpal H C 00 Opt. Commun. 83 4558 [9] Tervo J, Setala T and Friberg T 003 Opt. Express 37 [0] Setala T, Tervo J and Friberg T 006 Opt. Lett. 3 08 [] Setala T, Tervo J and Friberg T 006 Opt. Lett. 3 669 [] Jacobson D 967 IEEE Trans. ntennas Propag 5 4 [3] Gori F, Santarsiero M, Borghi R and Piquero G 000 Opt. Lett. 5 9 [4] Rodriguez-Herrera O G and Tyo J S 0 J. Opt. Soc. m. 9 939 [5] Tervo J, Setala T, Turunen J and Friberg T 03 Opt. Lett. 38 30 [6] Shewell J R and Wolf E 968 J. Opt. Soc. m. 58 596 [7] Zel dovich B Y, Pilipetsky N F and Shkunov V V 985 Principles of Phase Conjugation (Berlin: Springer) [8] Bouchal Z and Perina J 998 J. Mod. Opt. 45 45 [9] He G S 00 Progr. Quant. Electron 6 3 [30] Hellwarth R W 977 J. Opt. Soc. m 67 [3] Martin G and Hellwarth R W 979 ppl. Phys. Lett. 34 37 [3] Feinberg J 980 Opt. Lett. 5 330 [33] Cronin-Golomb M 99 J. Crystal Growth 09 345 [34] Zeldóvich B Ya, Popovichev V I, Ragulśkii V V and Faizullov F S 97 JETP Letters 5 09 [35] Gori F, Santarsiero M and Borghi R 007 Opt. Lett. 3 588 5