Spectral Degree of Coherence of a Random Three- Dimensional Electromagnetic Field

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1 University of Miami Scholarly Repository Physics Articles and Papers Physics Spectral Degree of Coherence of a Random Three- Dimensional Electromagnetic Field Olga Korotkova University of Miami, o.korotkova@miami.edu Emil Wolf Recommended Citation Korotkova, Olga and Wolf, Emil, "Spectral Degree of Coherence of a Random Three-Dimensional Electromagnetic Field" (004). Physics Articles and Papers This Article is brought to you for free and open access by the Physics at Scholarly Repository. It has been accepted for inclusion in Physics Articles and Papers by an authorized administrator of Scholarly Repository. For more information, please contact repository.library@miami.edu.

2 38 J. Opt. Soc. Am. A/ Vol. 1, No. 1/ December 004 O. Korotkova and E. Wolf Spectral degree of coherence of a random three-dimensional electromagnetic field Olga Korotkova Department of Physics and Astronomy, University of Rochester, Rochester, New York 1467, and College of Optics/ CREOL, University of Central Florida, Orlando, Florida 3816 Emil Wolf Department of Physics and Astronomy and The Institute of Optics, University of Rochester, Rochester, New York 1467, and College of Optics/CREOL, University of Central Florida, Orlando, Florida 3816 Received April 8, 004; revised manuscript received June 11, 004; accepted July, 004 The complex spectral degree of coherence of a general random, statistically stationary electromagnetic field is introduced in a manner similar to the way it is defined for a beamlike field, namely, by means of Young s interference experiment. Both its modulus and its phase are measurable. We illustrate the definition by applying it to blackbody radiation emerging from a cavity. The results are of particular interest for near-field optics. 004 Optical Society of America OCIS codes: , , , INTRODUCTION A quantitative measure of the degree of coherence of a directional (beamlike) field was introduced, on the basis of the scalar theory, by Zernike in a classic paper 1 (see also Ref. ) that provided the basis for the development of modern coherence theory. The definition of the degree of coherence was later refined and generalized in several ways, 3 10 but the essence of Zernike s approach was retained; namely, that the degree of coherence of an optical field (more precisely, its absolute value) at two points Q 1 and Q was taken to be proportional to the visibility of fringes in Young s interference experiment, with pinholes at the two points. In the present paper we consider a general (i.e., not necessarily beamlike) statistically stationary electromagnetic field and show, with the help of the Rayleigh Luneberg diffraction integrals, 11,1 that for such fields it is also possible to take the visibility of fringes as a measure of coherence. The expression for the degree of coherence of a general three-dimensional statistically stationary electromagnetic field, which we now introduce, is a natural generalization of the definition introduced recently for beamlike electromagnetic fields. 10 We illustrate our main result by determining the degree of coherence of radiation emerging from an opening in a cavity containing a field in thermal equilibrium with the walls of the cavity, i.e., blackbody radiation. Our results may be expected to be useful for near-field optics, because close to radiating sources and scatterers the optical fields are generally not beamlike.. ELECTRIC CROSS-SPECTRAL DENSITY OF A RANDOM ELECTROMAGNETIC FIELD PROPAGATING INTO A HALF-SPACE Let us consider a randomly fluctuating electromagnetic field that propagates from a plane z 0 into the halfspace z 0. We assume that the fluctuations are characterized by a statistical ensemble that is stationary, at least in the wide sense. The second-order statistical properties of the electric field may be characterized by the 3 3 cross-spectral density matrix (see Ref. 11, Sec ) W J r 1, r, W r 1, r, E* r 1, E r, x,y,z; x,y,z, (.1) r 1 and r are the position vectors of two points in the half-space z 0, denotes the frequency, and the asterisk denotes the complex conjugate. The subscripts, label Cartesian components with respect to two mutually orthogonal x, y axes perpendicular to the z direction, and the angule brackets on the right-hand side denote the average taken over a statistical ensemble of spacefrequency realizations (see Ref. 11, Sec. 4.7). Let us consider a typical member E(r, ) of the ensemble. It is known (see references in Ref. 1) that the x, y components on the boundary plane z 0, together with the outgoing condition at infinity in the half-space z 0, determine uniquely the electric field throughout the half-space z 0. The three Cartesian components of the electric field are given by the following formulas, which appear to have been first derived by Luneberg: 13 E x r, 1 E y r, 1 E z r, 1 E x, G z r, d, E y, G z r, d, E x, G x r, E y, G y r, d, (.a) (.b) (.c) /004/138-04$ Optical Society of America

3 O. Korotkova and E. Wolf Vol. 1, No. 1/December 004/J. Opt. Soc. Am. A 383 is a two-dimensional position vector of a point in the plane z 0 and G x, G y, G z are the partial derivatives of the outgoing free-space Green s function, so that G r, r G x r, exp ik r r, (.3) r r exp ik r r, (.4) x r r r with similar expressions for G y and G z, and k /c (.5) denotes the wave number associated with the frequency, c being the speed of light in vacuum. For later purposes we need an expression for the boundary value of the z component of the electric field E z ( ) in the plane z 0. It is given by the expression E z, lim E z, z, z 0 1 ik 1 E, 3/ exp ik d. (.6) Formulas (.a) and (.b) are vector analogs of the well-known Rayleigh diffraction integral (Ref. 14; see Sec ) for the outgoing solution of Dirichlet s boundary value problem for the scalar Helmholtz equation in the half-space z 0. Moreover, since the boundary values of the z component E z of the electric field on the plane z 0 are also known [given by Eq. (.6)] and since E z also satisfies the Helmholtz equation throughout the halfspace z 0 and behaves as an outgoing wave in that half-space, it can also be calculated from the Rayleigh diffraction integral. 3. SPECTRAL DEGREE OF COHERENCE OF THE ELECTRIC FIELD We will follow Zernike s precedent of identifying the degree of coherence of the field at two points Q 1 (r 1 ) and Q (r ) as being a measure of the sharpness of interference fringes that would be formed by allowing the fields from the immediate neighborhood of these points to superpose. Thus we have to consider the distribution of the average intensity more precisely, of the averaged electric energy density in the interference pattern formed in Young s interference experiment, with pinholes at the points Q 1 and Q in an opaque screen placed across the field (see Fig. 1). Let da 1 and da be the areas of the two pinholes. As we noted at the end of Section, the Cartesian components of the electric field at a point P(r) in Young s interference pattern are given by Rayleigh s formula, i.e., E r, 1 E, G z r, d A 1 A x,y,z, (3.1) Fig. 1. Illustration of the notation relating to determining the spectral degree of coherence of a three-dimensional electromagnetic field from Young s interference experiment. integration extends over the two small apertures A 1 and A. Assuming that the apertures are sufficiently small, expression (3.1) may be approximated by the formula E r, 1 E 1, G z r, 1 da 1 E, G z r, da x,y,z. (3.) The average electric energy density at the point P(r) in the observation plane is given by the trace of the 3 3 electric correlation matrix (.1), evaluated for r 1 r r; i.e., it is given by the formula S r, Tr W J r, r, S x r, S y r, S z r,, (3.3) S r, E* r, E r, x,y,z, (3.4) with E (r, ) being given by expression (3.). The z derivative of the Green s function [Eq. (.3)] that appear in Eq. (3.) is given by the expression G z r, z ik R 1 exp ikr, (3.5) R R R r. (3.6) Assuming that the distance R is much greater than the wavelength c/, G z may be approximated by the expression G z r, ik R z exp ikr. (3.7) R It follows that the electric field components, given by Eq. (3.), may be expressed in the form E r, i E 1 1, z exp ik E, z R exp ikr da 1 da R x,y,z. (3.8)

4 384 J. Opt. Soc. Am. A/ Vol. 1, No. 1/ December 004 O. Korotkova and E. Wolf On substituting from Eq. (3.8) into Eq. (3.4) we obtain for the spectral density S (r, ) of the electric field component E (r, ) the expression S r, 1 S 1, z da 1 S, z R da R Re W 1,, exp ik R R z R da 1 da x,y,z, (3.9) Re denotes the real part. On substituting from Eq. (3.9) into Eq. (3.3), we obtain for the average electric energy density at a point P(r) in the interference pattern the expression S r, 1 S 1, z da 1 S, z R da R Re Tr W J 1,, exp ik R R z R da 1 da. (3.10) We may rewrite formula (3.10) in a physically more significant form by noting that the first two terms on its right-hand side have a clear meaning. The first term represents the average electric energy density, S 1 r, 1 S 1, z da 1, (3.11) when the field reaches the point of observation P(r) only through the first pinhole, i.e., when da 0. Similarly, the second term represents the average electric spectral density, S r, 1 S, z R da, (3.1) R when the field reaches the point of observation only through the second pinhole. Hence formula (3.10) can be expressed in the form S r, S 1 r, S r, 1,, S 1 r, S r, exp ik R Re 1,, R, (3.13) TrW J 1,, S 1, S, TrW J 1,, TrW J 1, 1, TrW J,, (3.14a) (3.14b) Formula (3.13) expresses a general spectral interference law for random statistically stationary electromagnetic fields. We see that as the path difference R varies, the average electric energy density S(r, ) in the neighborhood of the point of observation P(r) varies sinusoidally between the values S max r, S 1 r, S r, S 1 r, S r, 1,,, S min r, S 1 r, S r,. (3.15a) S 1 r, S r, 1,,. (3.15b) Hence the spectral visibility of the fringes in the neighborhood of the point P(r) in the interference pattern is given by the expression V r, S max r, S min r, S max r, S min r, S 1 r, S r, 1,,. S 1 r, S r, (3.16) If, as is usually the case, S (1) (r, ) S () (r, ), formula (3.16) reduces to V r, 1,, ; (3.17) i.e., the absolute value of the expression (3.14) is equal to the spectral visibility of the interference fringes. Hence, if following Zernike we interpret the absolute value of the degree of coherence of the field at points Q 1 and Q as the visibility of fringes that would be formed by the field reaching the point of observation from the two pinholes, we see that the modulus of expressions (3.14) represents the absolute value of the degree of coherence. 15 The phase of is associated with the location of the maxima in the fringe pattern and may also be measured (see Refs. 16 and 17). Hence we may identify (r 1, r, ), given by formulas (3.14), with the (generally complex) spectral degree of coherence at frequency of any statistically sta-

5 O. Korotkova and E. Wolf Vol. 1, No. 1/December 004/J. Opt. Soc. Am. A 385 tionary electromagnetic field that propagates into the half-space z 0. Formulas (3.14) are a natural generalization of the expression for the spectral degree of coherence introduced not long ago for beamlike fields. 10,18 Our main formula (3.14) may be illustrated by applying it to a well-known three-dimensional random field, namely, blackbody radiation. The 3 3 electric crossspectral density matrix of blackbody radiation was derived in Ref. 19. When Eq. (3.14b) is applied to blackbody radiation emerging from the cavity, one finds after straightforward calculation that in this case, 1,, sin k 1. (3.18) k 1 ACKNOWLEDGMENTS The research was supported by the U.S. Air Force Office of Scientific Research under grant F , by the Engineering Research Program of the Office of Basic Energy Sciences at the U.S. Department of Energy under grant DE-FG0-ER4599, and by the Air Force Research Laboratory under contract FA C-096. REFERENCES 1. F. Zernike, The concept of degree of coherence and its application to optical problems, Physica 5, (1938).. F. Zernike, Diffraction and optical image formation, Proc. Phys. Soc. London 61, (1948). 3. E. Wolf, A macroscopic theory of interference and diffraction of light from finite sources II. Fields with a spectral range of arbitrary width, Proc. R. Soc. London Ser. A 30, (1955). 4. B. Karczewski, Coherence theory of the electromagnetic field, Nuovo Cimento 30, (1963). 5. W. H. Carter and E. Wolf, Far-zone behavior of electromagnetic fields generated by fluctuating current distributions, Phys. Rev. A 36, (1987). 6. L. Mandel and E. Wolf, Spectral coherence and concept of cross-spectral purity, J. Opt. Soc. Am. 66, (1976). 7. E. Wolf and W. H. Carter, Angular distribution of radiant intensity from sources of different degrees of spatial coherence, Opt. Commun. 13, (1975). 8. E. Wolf and W. H. Carter, A radiometric generalization of the van Cittert Zernike theorem for fields generated by sources of arbitrary state of coherence, Opt. Commun. 16, (1976). see also Ref M. J. Bastiaans, A frequency-domain treatment of partial coherence, Opt. Acta 4, (1977). 10. E. Wolf, Unified theory of coherence and polarization of statistical electromagnetic beams, Phys. Rev. Lett. 31, (003). 11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995). 1. B. Karczewski and E. Wolf, Comparison of three theories of electromagnetic diffraction at an aperture. Part I: Coherence matrices, J. Opt. Soc. Am. 56, (1966). 13. R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, Calif., 1964), pp M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK, 1999). 15. Expression (3.14) was noted previously (Ref. 5, Eq. 6.16) as the degree of coherence in a particular case, namely, in the far field generated by three-dimensional fluctuating charge current distribution in free space. 16. D. F. V. James and E. Wolf, Determination of the degree of coherence of light from spectroscopic measurements, Opt. Commun. 145, 1 4 (1998). 17. V. N. Kumar and D. N. Rao, Two-beam interference experiments in the frequency-domain to measure the complex degree of spectral coherence, J. Mod. Opt. 48, (001). 18. An analogous definition of the degree of coherence of a beamlike field in the space time domain was obtained many years ago by Karczewski in a little-known paper C. L. Mehta and E. Wolf, Coherence properties of blackbody radiation. III. Cross-spectral tensors, Phys. Rev. 161, (1967).