Enhanced propagation of photon density waves in random amplifying media
Lalruatfela Renthlei, Harshawardhan Wanare, S. A. Ramakrishna
aa r X i v : . [ phy s i c s . op ti c s ] N ov Enhanced propagation of photon density waves in randomamplifying media
Lalruatfela Renthlei, Harshawardhan Wanare, ∗ and S. A. Ramakrishna Department of Physics, Indian Institute of Technology, Kanpur 208016 INDIA
Abstract
We demonstrate enhanced wave-like character of diffuse photon density waves (DPDW) in anamplifying random medium. The amplifying nature makes it contingent to choose the wave solutionthat grows inside the amplifying medium, and has a propagation vector pointing opposite to thegrowth direction. This results in negative refraction of the DPDW at an absorbing-amplifyingrandom medium interface as well as the possibility of supporting “anti”-surface-like modes at theinterface. A slab of amplifying random medium sandwiched between two absorbing random mediasupports waveguide resonances that can be utilized to extend the imaging capabilities of DPDW.
PACS numbers: (170.5270) Medical optics and biotechnology, Photon density waves; (110.0113) Imaging sys-tems, Imaging through turbid media; (290.1990) Scattering ,Diffusion; (110.7050) Imaging systems, Turbidmedia. ∗ Corresponding author: [email protected]
1t is well known that a time-modulated light source placed in a random scattering mediumcreates a modulation in the density of the diffuse photon flux [1]. These density variationspropagate in the source free regions of the random medium according to the diffusion equa-tion ~ ∇ · ( D ~ ∇ φ ) − µ a φ − c ∂φ∂t = 0 , (1)where, D = 1 / µ ′ s + µ a ) is the diffusion coefficient, with µ ′ s being the reduced scatteringcoefficient, µ a is the absorption coefficient, φ is the amplitude of the density variation and c is the speed of light in the medium in which the scatterers are embedded. The waves havebeen popularly known as Diffuse Photon Density Waves (DPDW) in the area of biomedicalimaging where it has been used for a variety of imaging through biological media [2, 3].These waves are known to exhibit interference [4] and diffraction [5], and can be related acrossinterfaces between two distinct scattering media by reflection and transmission coefficients [1,6]. DPDW suffer from strong attenuation due to scattering and typically have penetrationdepths of only a couple of centimeters at ∼
100 MHz modulation frequencies.The diffusion approximation itself breaks down at much higher modulation frequencies [7].We note that other theories for photon migration [8, 9] have been shown to be advantageousin the short time limits (large modulation frequencies). However, the diffusion equation 1describes the DPDW well in the steady state at length scale much larger than the transportmean free path for reasonable modulation frequencies ( <
200 MHz) [1, 2]. Another limitationfor the DPDW intensity arises from the intrinsic absorption in the medium. Amplificationof light has been used to compensate for attenuation of the waves in many contexts, forexample, fiber amplifiers [10], amplifying media in metamaterials/perfect lenses of negativerefractive index [11, 12], extending surface plasmon propagation lengths at the interface ofmetal/amplifying medium [13]. Random amplifying media are well known in the contextof random lasers [14]. The compensation of absorption by laser gain in random media isknown to lead to the sharpening of the coherent backscattering peak and increased weaklocalization [15].In this letter, we examine the behaviour of DPDW in an amplifying random medium.The incorporation of gain in the random medium results in extending the wavelike charac-teristics of the DPDW, wherein the real part of the wavevector can become larger than theimaginary part. For an incident DPDW at an interface of absorbing-amplifying media, wefind that negative refraction of the DPDW can occur. This results from a physical choice of2
IG. 1. The highlighted regions indicate all possible values that the propagation vector ( k and k ) can take in the complex plane, (a,b) and (c,d) show these for absorbing and amplifying randommedia, respectively. (e) Schematic of the incident, reflected and the transmitted wavevectors atthe interface of absorbing-amplifying random media ( z = 0 plane). the wavevector that is mandated by the requirement of a growing density wave inside theamplifying random medium in spite of the DPDW having a positive phase velocity in boththe media. Although the finite modulation frequency of the DPDW renders the waves ex-ponentially decaying, a slab of amplifying random medium can support resonant waveguidemodes.Assuming plane wave solutions of the form, φ ∼ φ exp[ i ( ~k · ~r − ωt )], for the diffusion3quation results in the following dispersion relation k = − cµ a + iωcD . (2)In order to obtain the wave-vector k , we are faced with the classic problem of choosing thesign of the square root [18] from the dispersion relation in Eqn. 2, which yields k = ± (cid:20) − cµ a + iωcD (cid:21) / . (3)Note that the coefficient ( µ a ) can either be positive (for absorption) or negative (for ampli-fication). The validity of the diffusion equation to describe photon migration is contingenton the magnitude | µ a | ≪ µ ′ s , whereby scattering dominates over absorption/amplification.The level of amplification of the diffuse flux due to µ a < dissipative random mediumand choosing a wave-vector that causes decay of the wave with increasing distance of prop-agation [Im( k ) > k in Eqn.(3). We note that k lies in the second quadrant of the complex plane and hence k in the absorbing medium will lie in the first quadrant (see Fig. 1). This gives rise to theusual DPDW that exponentially decays with distance due to an imaginary part of k that islarger than the real part of k , thereby limiting the wavelike character.We now turn our attention to an amplifying random medium. In this case, k lies in thefirst quadrant of the complex plane and k could lie either in the first quadrant or in thethird quadrant. Due to amplification, the amplitude of the diffuse photon flux will increaseexponentially with increasing distance of propagation under conditions of linear unsaturatedgain. Hence, we require the wavevector to have a negative imaginary part Im( k ) < k will lie in the third quadrant as shown in Fig. 1 ( k x = 0 forour discussion here). Note, that regardless of the choice of the sign of k , the Re( k ) isalways greater than the Im( k ) for the random amplifying medium. This implies that theoscillatory aspect of the DPDW is enhanced over the exponential decay arising intrinsicallyfrom multiple scattering, which also contributes to amplification owing to the resulting longerpath through the gain medium. A discussion of the relevant physical length and frequencyscales is imperative at this juncture: the amplifying medium provides unbiased amplification4 −6 −1 (k x / k ) | R | −1 −0.5 0 0.5 110 −10 z × Re (k ) / 2 π | φ | ω = 2 π ×
10 Hz ω = 2 π ×
500 kHz ω = 2 π ×
10 MHz ω = 2 π ×
100 MHz medium 1(absorbing) medium 2(absorbing) x / k ) | R | −1 −0.5 0 0.5 110 z × Re(k ) / 2 π | φ | ω = 2 π ×
10 Hz ω = 2 π ×
500 kHz ω = 2 π ×
10 MHz ω = 2 π ×
100 MHz medium 2(amplifying)medium 1(absorbing)
FIG. 2. (a) The reflection of a DPDW at an absorbing-absorbing random medium interfaceresults in a Brewster-like condition, with µ a = 0 . − , D = 0 . µ a = 0 .
025 cm − ,and D = 0 .
033 cm. (b) The reflection of a DPDW at an absorbing-amplifying random mediuminterface is shown for µ a = − . − , and µ a , D and D are same as in (a). The insetsindicate the DPDW amplitude across the interface in the two media at ω = 2 π ×
100 M Hz .
5f both the background diffuse flux as well as the flux associated with the DPDW, as thesetwo fluxes cannot be distinguished by the stimulated emission process. The modulationfrequency (typically 10 to 100 MHz) responsible for the DPDW is much smaller comparedto the bandwidth of the gain medium ( ∼ THz for typical dye molecules). The gain can beconsidered uniform as the diffusion length scales are much shorter than the wavelength ofthe DPDW.Now let us understand the solutions with our requirement for Im( k ) < ~J = − D ~ ∇ φ , points in the same direction as the realpart of the wavevector. The complex nature of the the ~J arises due to the complex phasornotation for the PDW used in the paper. The real and imaginary parts should simply beviewed as representing the in-phase and in-quadrature components of the response in relationto the oscillatory source term. We have a wave with positive phase velocity where both thewavevector and the flux vector point in the same direction. Both these consequences aredictated by the need for having the DPDW amplitude as well as the background diffuse fluxthat exponentially increase with distance from a source in a gain medium. This solutionshould be compared with solutions pertaining to the electromagnetic waves in the negativerefractive index media [19]. A negative choice of the wavevector (real part) is required inthat context also. However, the Poynting vector representing the energy flux associatedwith the electromagnetic wave is oppositely oriented to the wavevector, which is an intrinsiccharacter of a wave with negative phase velocity. Note that these differences principallyarise from the governing equations: the Maxwell’s equations for the electromagnetic wavesand the diffusion equation for the DPDW.Consider the refraction of the DPDW at an interface between an absorbing randommedium and an amplifying random medium when the DPDW is incident from the absorbingside. Without loss of generality, we choose the z = 0 plane to be the interface and invariancealong the y − direction, so that the incident wavevector lies in the x − z plane such that k x + k z = k = ( − cµ a + iω ) /cD (see Fig. 1e). Continuity of the interface in the x − directionimplies equality of k x in both the media. The normal component k z is given by k z = ± (cid:20) − c ( µ a + Dk x ) + i ωcD (cid:21) / , (4)with the choice of sign in the two media governed by the previous discussion. The evidence6 −4 (k x / k ) | R s l ab | ω = 2 π ×
10 Hz ω = 2 π × ω = 2 π ×
10 MHz (k x / k ) | T s l ab | ω = 2 π ×
10 Hz ω = 2 π × ω = 2 π ×
10 MHz (z / d) | φ | k x / k = 0.136k x / k = 0.386k x / k = 0.474 FIG. 3. For a slab of amplifying random medium with µ a = − . − and D = 0 .
33 cmsandwiched between two identical absorbing random media with µ a = 0 . − and D = 0 .
166 cm,the reflection and transmission are shown in (a) and (b), respectively. (c) The amplitude of theDPDW within the slab at the transmission peaks indicated in (b) for ω = 2 π M Hz .
7f negative refraction at the interface lies in the decrease of the phase of the DPDW withincreasing distance from the interface into the amplifying medium. Ensuring the conditionsof continuity of the amplitude of the DPDW and the normal component of the flux yields φ I + φ R = φ T → R = T ( ~J I + ~J R ) · ˆ z = ~J T · ˆ z → D k z (1 − R ) = D k z T (5)where the subscripts I,R and T refer to the incident, reflected and the transmitted DPDWat the interface, respectively. The reflection and transmission coefficients are given by R = φ R φ I = D k z − D k z D k z + D k z , T = 2 D k z D k z + D k z . (6)Ripoll and Nieto-Vesperinas [6] have shown the existence of a Brewster-like condition,where the reflection coefficient ( R ) goes to zero, for a DPDW with ω → z . This impedance matching occurs at k x at which D k z = D k z ,see Fig. 2(a), which in turn is determined by the various parameters of the medium, where k x = µ a D − µ a D D − D + iωD + D . (7)For ω ≪ µ a /D the imaginary part is negligible and a proper choice of µ a and D allows oneto tailor its occurrence and this can be used to realize a directional filter [6].It appears intriguing that a pair of rather isotropic random media should choose a specific k x for an efficient coupling of the DPDW. The conventional Brewster angle observed forelectromagnetic waves reflected across interfaces is a polarization dependent feature thatarises from the requirement of orthogonality of the induced material polarization to thereflected direction. What would then underly the Brewster-like condition of the scalardensity waves? The above Eqn.(7) indicates that it is the matching of the absorption length l abs = p l s l a / l abs at the interface of these media is augmented by the k x component of theDPDW. Here, l s = µ ′ − s , l a = µ − a and the absorption length l abs is the (rms) averagedistance between begin and end points of paths of length l a in the random medium. Thus,the reflection minimum leading to strong coupling occurs such that the wave propagationdictated by the periodicity of the k − x is matched with the difference of the absorption lengthsin the two media. 8n the case of the absorbing-amplifying media interface, the above situation simply doesnot exist. The sign of k z changes, and the expressions of R and T Eqn.(6) transformappropriately, leaving behind a denominator that is positive definite because of the reversalof the sign of µ a of the amplifying medium [20]. For ω ≪ | µ a | /D one obtains R ∼ k x < | µ a | /D . At k x = | µ a | /D , the real part of k z becomeszero and beyond this point the reflection coefficient continues to increase as the input angleis increased due to the dominance of the imaginary part of the wavevector (Fig. 1d). Noteagain that the pertinent length scale is the amplification length ( l amp defined similarly tothe l abs above) which determines the condition on the different regimes of k x . In Fig. 2(b)we present reflection and transmission coefficients at an interface of an absorbing-amplifyingmedium for various modulation frequencies.One can also arrange to obtain an “anti”-surface mode at an absorbing-amplifying in-terface having a minima at the interface and the field growing with distance on both sidesof the interface (see inset in Fig.2b). These fields do not correspond to a true surface modeas the DPDW diverges at infinity, and hence cannot exist in isolation without the excitinginput density wave. Such a mode has been envisaged in the context of impedance matchedperfect lenses made of a negative refractive index material [21]. This physical realization canoccur only in conjunction with an amplifying medium and an absorbing-absorbing interfacecannot support such a mode.We present the response of a slab of amplifying random medium sandwiched betweenidentical absorbing random media, with the reflection and transmission given as R slab = 2 i ∆ [( D k z ) − ( D k z ) ] sin k z d, (8) T slab = − D D k z k z e − ik z d , (9)where, d is the thickness of the slab and ∆ = ( D k z − D k z ) e ik z d − ( D k z + D k z ) e − ik z d .The above expressions for a finite slab are invariant with respect to the sign of k z [22]. Thereflection and transmission as a function of k x show large peaks at the resonant conditionsfor an amplifying slab as in Fig. 3. Such a configuration leads to localized waveguide modeswithin the amplifying slab. The density variation with distance inside the slab is shown inFig. 3(c). The integral half-wavelength-like conditions are apparent. We have chosen theparameters in Fig.3 such that | k | > | k | and hence for k x > | k | the field in the slab isevanescent. Due to this, the transmission does not contain resonant features for k x ≥ | k | .9he ability of a layer of amplifying random medium to guide DPDW can be exploited inimaging situations, wherein coupling to the random medium and extraction of the DPDWcan be greatly enhanced.In conclusion, we have demonstrated that incorporating amplification (laser gain) ina random medium can substantially enhance the wavelike characteristics of the DPDW.This aspect may be used for increasing the depth over which imaging can be carried outin a random medium with DPDW. The physical requirement of a growing density wavein an amplifying random medium results in a backward flux towards the interface withthe absorptive random medium. This gives rise to negative refraction across the interfaceand conditions can result to produce “anti”-surface like states at such interfaces. A slabof amplifying random medium is shown to waveguide DPDW and we suggest that suchwaveguides can be used for improving the imaging capabilities of the DPDW, whose non-invasive aspects continue to hold great promise in diagnostic imaging.We thank S. Guenneau for initial discussions on possibility of negative refraction indiffusive systems. [1] M.A. O’Leary, A.G. Yodh and B. Chance, Phys. Rev. Lett. , 2658 (1992).[2] A.G. Yodh and B. Chance, Phys. Today, , 34 (1995).[3] T. Durduran, R Choe, W. B. Baker and A. G. Yodh, Rep. Prog. Phys. , 076701 (2010).[4] J. M. Schmitt, A. Knttel, and J. R. Knutson, J. Opt. Soc. Am. A, , 1832-1843 (1992).[5] D.A. Boas, M.A. O’Leary, B. Chance and A.G. Yodh, Phys. Rev. E, , R2999 (1993)[6] J.Ripoll and M. Nieto-Vesperinas, Opt. Lett., , 796-798 (1999)[7] Q. C. Su, G.H. Rutherford, W. Harshawardhan, R Grobe Laser Phys. , 94-101 (2001).[8] M.Porr‘a, J. Masoliver and G. H.Weiss, Phys. Rev. E , 1381 (1999).[10] E. Desurvire, D. Bayart, B. Desthieux, and S. 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