March 25, 1999 TERRESTRIAL REDSHIFTS FROM A DIFFUSE LIGHT SOURCE Ad Lagendijk a FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands (Received ) Abstract Enhanced (elastic) backscattering of light from a disordered medium shows a redshift. This phenomenon belongs to a class of redshifts recently introduced by Wolf. The magnitude of the redshifts is calculated on the basis of a simple diffusion theory. 1
Recently Wolf pointed out in a series of papers 1 3 that an important difference can exist between the near-field spectrum, S Q (ω), and the observed far-field spectrum, S V (ω), of an extended light source on the basis of source correlations. In the case of highly coherent laser beams this effect might not be so surprising but the occurrence for almost incoherent sources is quite unexpected. Possible implications for astrophysics were pointed out. It is the intention of this work to demonstrate that in the relatively new field of localization of light diffuse secondary light sources are being studied which exhibit precisely those correlations required to show the Wolf effect. As a result of intensive research the field correlations of these secondary sources are now known to a very high accuracy. I will discuss the occurrence and magnitude of the redshift in these media and I will use this example to comment on some aspects of Wolf s work. Weak localization is the phenomenon of enhanced backscattering from a strongly scattering disordered medium. 4,5 The origin of this enhancement comes from constructive interference between time-reversed paths. The enhancement has an angular extension around the direction of backscattering of the order of λ/(2πλ mf ), λ being the wavelength and λ mf the mean free path of the light. This feature is enough to appreciate the existence of a redshift in the scattered light: as the angular extent of the backscattering is wavelength dependent, at a fixed angle the red part of the spectrum is scattered more efficiently than the blue part. This shift is purely due to interference as the scattering properties of the individual scatterers are assumed to be independent of wavelength. Let us look at the scattered light intensity from an plane wave incident on a semiinfinite slab (z >0) containing isotropic point scatterers, 6 I(r 1 ) = Ψ inc (r 1 ) Ψ inc (r 1) + G(r 1, r 2 ) G (r 1, r 3 ) R(r 2, r 3, r 4, r 5 Ψ inc (r 4 ) Ψ inc(r 5 ) dr 2 dr 3 dr 4 dr 5, (1) in which the triangular brackets denote averaging over the realizations of the disorder. Ψ inc is the incoming field amplitude and G(r 1, r 2 ) is the Green s function for the amplitude. Eq. (1) has a very simple interpretation: the incoming fields are being transported through the medium by the vertex function R and are being transported outside the medium by the Green s functions. To make this interpretation even clearer the integrand in Eq. (1) should 2
be integrated over the internal coordinates r 4 and r 5. The result can be written as I(r 1 ) = Ψ inc (r 1 ) Ψ inc (r 1) + G(r 1, r 2 ) G (r 1, r 3 ) W Q (r 2, r 3 )dr 2 dr 3, (2) and the newly introduced function W Q can be understood as the source-correlation function. The remaining integrations can be then be viewed as the propagation of the radiation produced by these sources. This representation makes it now possible to map these equations on the formalism used by Wolf. If all the secondary sources in the medium would radiate incoherently the correlation function W Q would be proportional to a delta function, W Q (r 1, r 2 ) δ(r 1 r 2 ). Some workers, including Wolf, subtract this incoherent contribution from the function W Q, and the remaining part is referred to as the cross spectral-density. In the weakly scattering regime, λ mf λ/2π, the vertex R has to different contributions: the incoherent sum (usually partitioned in a first-order and a multiple-order contribution to allow for application of a diffusion model on the latter) and the coherent component originating from the constructive interference of time-reversed paths. Borrowing terminology from diagrammatic perturbation theories the three different constituents are usually labelled single scattering (S), ladder (L) and most-crossed (C). In the language of Wolf the term C would be considered to give rise to source correlations and would make up the mutual correlation function. The incoherent contributions, S + L, were not considered by Wolf. We have derived rigorous integral equations for L and C, 6 but as was shown by us the diffusion approach of Akkermans et al. 7 is more than sufficient to describe L and C accurately. In this approach S, L, andc are given by S(r 1, r 2, r 3, r 4 ) = (4πa/λ mf )δ(r 1 r 2 )δ(r 1 r 3 )δ(r 1 r 4 ), (3) L(r 1, r 2, r 3, r 4 ) = F (r 1, r 2 )δ(r 1 r 3 )δ(r 2 r 4 ), (4) C(r 1, r 2, r 3, r 4 ) = F (r 1, r 2 δ(r 1 r 3 )δ(r 1 r 4 ), (5) in which F (r 1, r 2 ) is simply the Green s function for the time-independent diffusion equation with appropriate boundary conditions. The albedo a describes the effect of absorption: a = 1 corresponds to no absorption. For an infinite lossless medium F (r 1, r 2 ) would be equal to 3a 2 π 1 λ 3 mf r 1 r 2 1. The particular forms of the delta-functions in S and L demonstrate that these contributions are incoherent. This becomes even more evident when in Eq. (1) we put R equal to S + L + C and perform the integrations: the contribution of S + L is a sum of squares. 3
S and L give rise to backscattering which is almost angular independent (on angular scales of λ/(2πλ mf )) and the contribution of S is about 15% that of L in the diffusion limit. The contribution C gives rise to the by now well-known enhanced backscattering. The actual integrations can be done analytically both for the semi-infinite slab and a finite slab. In all cases the function F (r 1, r 2 ) is obtained from the result for the infinite medium by applying the reflection principle of Green s function for totally absorbing boundaries. 8 A convenient measure for the intensity of the scattered beam at direction θ s (θ s = 0 corresponding to backscattering) given an incoming beam at incoming angle θ i, is the socalled bistatic coefficient 9 γ. I just quote the result concerning the bistatic coefficient for the semi-infinite slab, broken down into the contributions for single scattering (S), incoherent multiple scattering (L) and enhanced backscattering (C): aµ s γ S =, (6) µ i + µ s [ ] γ L = ( 3a2 µ s 1 µ 2 ) {1+µi c 0 (1 µ i c 0 )exp( 2c 0 τ 0 )} µ2 i, (7) i c2 2c 0 (1 + µ s ) µ i + µ s γ C = 3a2 [c + ν{1 exp( 2cτ 0 )}] 2µ i cν[(ν + c) 2 + µ 2, (8) ] in which µ i,s cos θ i,s,ν (1/µ i +1/µ s )/2, µ 2π(µ i µ s )λ mf /λ, c 0 [3(1 a)] 1/2,and c [3(1 a) +{2π(λ mf /λ)sinθ i } 2 ] 1/2. To mimic in the diffusion approach the rigorous solution of the ladder contribution as much as possible the trapping plane is located slightly outside the real boundary, at z = τ 0 λ mf, τ 0 0.71. These equations describe a strongly angular dependent coherent backscattering riding on an almost angular independent incoherent background. The contribution of the coherent backscattering [Eq. (8)] at exactly 180 o backscattering equals the contribution of L [Eq. (7)] and its angular halfwidth is about 0.67 λ/(2πλ mf ). This enhanced backscattering shows beautifully the Wolf redshift. At a particular angle the scattered light from a beam characterized by a spectral function S Q (ω) will be redshifted as larger wavelengths are scattered more just because of interference in the secondary source. Wolf has indicated that the conditions for the absence of redshifts induced by interference can be formulated in terms of the (length-)scaling behavior of the cross spectral density. The source correlation functions induced by enhanced backscattering have associated with them two independent length scales, wavelength and mean free path, and for that reason do not obey a simple scaling law. 4
Let us now discuss the actual magnitude of the redshift. Results will be presented for two types of spectral functions: a gaussian and a lorentzian shape, both with varying linewidths. As can be anticipated the shifts will be rather small unless the spectral width of the incoming source is very large. The role of the incoherent background is prominent and cannot be overlooked. As a spectator observes both the coherent backscattering and incoherent background a redshift of the coherent contribution will be diluted in the total backscattering through the presence of the incoherent background. It is evident that for the enhanced backscattering the redshift will be largest far in the wings of the enhanced backscattering. However in that case the scattered coherent intensity is at the same time very small compared to the incoherent background and the total shift of background plus enhancement will be very small. The largest redshift of the total intensity will be obtained for angles in the neighborhood of the halfwidth of the angular cone, and a numerical calculation shows that this maximum occurs at around 0.4 λ/(2πλ mf ). In Table I several typical results are presented for the redshifts occurring for this angle. The measurements of these shifts in an actual experiment are rather difficult as they are a small part of the linewidth. It might be interesting to speculate about the possibilty to induce much larger shifts and a promising situation to consider is evidently the occurrence of strong localization. In the present scattering model, consisting of isotropic scatterers with a wavelength-independent scattering length, the situation is quite clear. The mobility edge, separating localized modes from extended ones, will be situated at some wavelength and longer wavelengths will be much stronger reflected than the shorter wavelengths to be found at the other side of the mobility edge. The reflected wave would be greatly redshifted and the transmitted wave would be blue-shifted. Our model needs however a serious correction, as the scattering efficiency at long wavelengths (Rayleigh limit) is proportional to 1/λ 4 in contrast to the wavelength independence of s-wave potential scattering like the bouncing of low-energy electrons off impurities. This modification implies that the presence of localization presupposes the existence of at least two mobility edges. Light falling within frequency band will strongly reflected and outside will be much more strongly transmitted. This could lead to redshifts and blue-shifts. Recently we introduced the phenomenon of transverse localization of light. 10 In this case a beam is propagating in a medium with anisotropic disorder. In the extreme case of 5
exclusively transverse disorder (defined as perpendicular to the direction of propagation) the beam profile will not extend diffusively but will remain within a certain profile defined by the localization length. This interference effect will also give rise to Wolf-type redshifts. In addition it has been shown recently that if the disorder is anisotropic and consequently not strictly transverse the propagation of light will be still strongly reflected by interference contributions. 11 To appreciate fully the extent and importance of the ideas of Wolf in the field of strong localization it is clear that the theory should address the question of the influence of interference on the incoherent background. In a conservative system energy and intensity is conserved. The enhanced backscattering must occur at the cost of the incoherent background. Surprisingly as far as I know nobody has proven this in the case of weak localization of a semi-infinite slab. In an infinite medium Ward identities can be used to clarify this point. From this discussion it is clear that one cannot just suggest a source correlation function and neglect its influence on the incoherent correlation function as was for instance done by Wolf. Anticipating that the enhancement will occur at the expense of the incoherent background it is evident that in the present case we can neglect it as the angular extent of the incoherent background is π and of the enhancement λ/(2πλ mf ). There are several ways in which light might change its spectrum even in a linear system. The interacting medium acts just as a filter. More or less trivial wavelength dependent single-particle properties, like Rayleigh scattering and wavelength-dependent absorption, can easily cause such shifts. Wolf has shown that collective interference effects could also act as a filter. The fascinating aspect is that there are no single-particle wavelengthdependent aspects involved. I have shown here that in diffuse light sources due to weak or strong-localization effects these shifts will be present. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). 6
a The author is also at the Natuurkundig Laboratorium der Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands References 1 E. Wolf, Phys. Rev. Lett. 56, 1370 (1986). 2 E. Wolf, Nature 326, 363 (1987). 3 E. Wolf, Opt. Commun. 62, 12 (1987). 4 M.P. van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985). 5 P.E Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985). 6 M.B. van der Mark, M.P. van Albada and A. Lagendijk Phys. Rev. B. 37 3575 (1988). 7 E. Akkermans, P.E. Wolf, and R. Maynard, Phys. Rev. Lett. 56, 1471 (1986). 8 P.M. Morse and H. Feshbach, Methods of theoretical physics (McGraw-Hill, New York, (1953). 9 A. Ishimaru, Wave propagation and scattering in random media (Academic, New York, 1978). 10 H. De Raedt, A. Lagendijk and P. de Vries, Phys. Rev. Lett. 62, 47 (1989). 11 S. Feng (private communication). 7
TABEL I redshift linewidth gaussian lorentzian coherent total coherent total 0.001 1.29 10 7 3.59 10 8 8.49 10 8 1.89 10 8 0.01 1.29 10 5 3.59 10 6 8.97 10 6 2.49 10 6 0.1 1.29 10 3 3.59 10 4 8.96 10 4 2.49 10 4 0.2 5.13 10 3 1.43 10 3 3.57 10 3 9.94 10 4 Caption to Table Table I: Redshifts ( λ/λ) of enhanced backscattering in random media for various spectral linewidths ( fwhh/λ, fwhh refers to full width at half height). The shifts of the coherent (enhanced) scattering and of the total intensity, incoherent background plus coherent backscattering, are presented. The angle of observation is 0.408 λ/(2πλ mf ) radians from backscattering. 8