- Email: [email protected]
Interference effects in multiple-light scattering with gain
PHYSICA ELSEVIER
Physica A 241 (1997) 82-88
Interference effects in multiple-light scattering with gain Diederik S. Wiersma a'c'*, Ad Lagendijk b a FOM-lnstitute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, Netherlands b Van der Waals-Zeeman Laboratorium, Valckenierstraat 65-67, 1018 XE Amsterdam, Netherlands c European Laboratory for non-Linear Spectroscopy, Largo E. Fermi 2 (Arcetri), 50125 Florence, Italy
Abstract Experimental results are reported on coherent backscattering and speckle from an amplifying random medium. Also, coherent backscattering with gain is calculated completely, using diffusion theory. To conclude, the concept of a random laser is discussed. PACS: 42.25.Bs; 42.55.-f; 78.45.+h Keywords: Multiple-light scattering; Interference; Light amplification; Random laser
1. Introduction Since a couple of years it has been known that interesting interference effects can occur in light which is multiply scattered from the disordered structures, especially if the scattering is very strong [1]. Examples of these phenomena are (weak) localization of light, optical band gaps, and extremely slow transport. In all these cases absorption poses an important problem: due to absorption the light has simply disappeared after a few scattering events. All objects in daily life (even those appearing perfectly white) absorb light to some extent. What would happen if the light is amplified at every scattering event instead of getting absorbed? In that case we have the opposite situation of absorption and the object is so to say 'whiter than white'. The light intensity will increase instead of decreasing on multiple scattering. Light amplification can be obtained in a laser material. We have recently succeeded in realizing a random medium with gain by grinding a titanium sapphire laser crystal. The availability of an amplifying random medium allows one to study interference phenomena in multiple scattering with gain. * Correspondence address: European Laboratory for non-Linear Spectroscopy, Largo E. Fermi 2 (Arcetri), 50125 Florence, Italy. Fax: 3955224072; e-mail: [email protected]. 0378-4371/97/$17.00 Copyright (~) 1997 Published by Elsevier Science B.V. All rights reserved PII S0378-4371(97)00063-0
D.S. Wiersma, A. Lagendijkl Physica .4 241 (1997) 82-88
83
2. Coherent backscattering Coherent backscattering [2] is a general interference effect for waves which are backscattered from a random medium and originates from the time-reversal symmetry, or better, from reciprocity. Because o f reciprocity, any light path from a light source, through a random medium, and back to the light source, can be followed in opposite directions. Light waves which travel along the reciprocal paths will acquire the same phase and interfere constructively in the direction back to the detector. The result is an enhancement of the intensity in backscattering, which has the shape o f a narrow cone. In Fig. 1, backscattering cones are shown as recorded from a powdered Ti:sapphire crystal, which was pumped with 14ns pulses from a frequency doubled N d : Y A G laser. For more details o f the experiment we refer to Ref. [3]. The lower curve is the unamplified cone without pump light. We observed that the overall scattered intensity increases with pump energy. We also observed that the shape o f the cone changes; upon increasing the gain, the top o f the backscattering cone sharpens. This sharpening can be understood if one realizes that the shape o f the cone reflects the path-length distribution in the sample. The wings o f the cone are mostly due to contributions from short paths, and as we approach the top of the cone, contributions from successively longer paths add to the intensity. Because the amplification along a path depends exponentially on its length, the introduction o f gain will sharpen the central region o f the backscattering cone. Diffusion theory provides a very good description o f coherent backscattering from passive random media. To calculate the backscattering cone for an amplifying medium using diffusion theory, we start from a standard stationary diffusion equation with gain [4,5] and perform a Fourier transform over the spatial coordinates x and y, in which
'
'
I
'
'
:~ 3
I
'
'
I
'
'
I
~ overallgain = 71% [] overallgain -- 36 % o unamplified
©
..= ~o 2 N
O
Z
1 ,
,
,
I
0
,
,
,
I
,
,
,
i
2 4 Angle (mrad)
,
,
,
L
6
Fig. 1. Three backscattering cones in the polarization conserving channel at different pump energy. Sample: 30% Ti: Sapphire particles (10 gm diameter) in water, transport mean free path: 40 gm. The overall gain of 0%, 36%, and 71%, corresponds to pump energies: 0, 165, and 190mJ. The intensity is normalized to the diffuse background at zero gain. Solid lines: calculated curves based on diffusion theory. Upon increasing the gain, the overall intensity increases and the top of the cone sharpens.
D.S. Wiersma, A. LaoendijkIPhysica A 241 (1997) 82 88
84
the sample is translationally invariant. The resulting equation is
~(2(~1-q~)
F(q±,z,,z2)+f~cgF(q±,zl,z2)+6(zl - z 2 ) = 0 .
(1)
Applying the boundary condition of a slab of thickness L, [F(q±, zl, z 2 ) = 0 for z2 = -z0 and z2 = L +z0, and for every zl, with z0 ~ 0.71(], we find
F(q±,zl,z2) =F(q±,Zs,Za) =
3 cos[fl(L - zs)] - 3 cos[fl(L + 2z0 2 ( 2 fl sin[fl(L + 2z0)]
Izol)] ,
(2)
where Zs = zl + z2, zd -- z, -- z2, and fl =- V/fadp - q~_, with •amp the amplification length in the medium. The amplification length is defined as the (rms) average distance at which the intensity is amplified in the medium by a factor e +1 , while the gain length (g is defined as the actual traveled distance to obtain this amplification. From Eq. (2) we can see that F(q±,zbz2) becomes unstable (i.e. diverges for small angles) for L + 2zo ~>Lcr: ~gCamp,where Lcr is the critical thickness. From the diffusion propagator F(q±,zl,z2) one can calculate the total bistatic coefficient (or normalized intensity), which is the sum of an interference term 7c(0s) and a diffuse background term 7~(0s). The interference term describes the backscattering cone and is given by [6] L 2L--zd
F(q±, ZI, 7-2 ) COS(Zd~)e-uz~ dzs d&l,
7c(0s) = ~ 0
(3)
zd
whereas for the diffuse background we have [6] L 2L--z,l
1
f
7t(0s)=~/
0
F(q± =O, Zl,Z2)e-(1/2)KeZs(~-'+l)e-(I/2)~zd(~7'-l)dzsdzd.
(4)
zd
Here r/=k0(1-/~s), u = ~1 x e ( l + # s--1 ), and #s = C O S ( 0 s ) with 0s the angle of the outgoing wave vector k2 relative to the z-axis. Furthermore, Ke is defined by ~:~ = {-1 _ ( - 1 , where E~ is the scattering mean free path and Eg is the gain length in the medium. The incoming wave vector is taken along the z-axis. The above integrals can be performed explicitly using the intensity propagator from Eq. (2). The result is 3e -~L
1 7~(0~) = 2(3flsin[fl(L + 2zo)] (f12 + q2 + u2)2 _ (2flq)2 × [2(fl 2 + r/2 + u 2) cos(2flzo) cos(D/) - 4fir/sin(2flzo) sin(D/)
+ 2-~(-fl 2 + r/2 - u 2) sin(fl(L + 2z0)) sinh(uL) U
D.S. Wiersma, A. LaoendijklPhysica A 241 (1997) 82-88
85
+ 2(/~2 --/12 -- U2 ) cos(L?1) - 2(fl2 + 712 + u 2) cos(//(L + 2Zo)) cosh(uL)
+4~usin(flL)sinh(uL) + 2 ( - f i 2 + 712+ u2)cos(flL)cosh(uL)]
/
(5)
and 3
?~(Os) = 2d3flsin[fl( L + 2zo)]
Zl(1 + e -2uL) + Z2(1 - e -2uL) + Z3e -L(v+u) U[(U 2 + fl2)2 + V2(2f12 _ 2U2 +/)2)] ,
(6)
where Zl = / / ( / ) 2 _ u2 _ f12) c o s [ f l ( L q_ 2Zo)]
+ u(u 2 - 1)2 - f12) c o s ( / ~ L )
V2 _.}..f12 __ 3U 2
+ 2uvfl sin[/~(L + 2zo)] + uvfl
Z2 = v ( u 2 - v 2 - f12) c o s [ f l ( L q- 2 z o ) ]
u £ 7- -fi2
(7)
sin(ilL),
+ 2uZflsin(~L)
- f l ( u 2 + v2 + f12) sin[fl(L + 2zo)] + u2v
V2 __ U2 -~
3[/2
U 2 _]_ f12
cos(/ L)
(8)
and
Z3 = 2u(v 2 -
u 2 '}-/~2) q_
2U(U2 _ V2 +/~2) COS(2ZOfl)_ 4UVfl sin(2zo/3),
(9)
with v = g1 Ke(#s--1 - 1), and u, 71, /~, and #s as defined before. (Note that q± = 0 in the expression for 7e.) For small angles, this solution agrees with the results of Ref. [7], where the central region around the cusp of the cone was calculated. For passive media, usually the limit of L ~ oc is taken to obtain a much simpler expression. For amplifying media this is impossible due to the divergence of the intensity for L + 2z0 > Lcr. The solid lines in Fig. 1 are the calculated intensities from the sum of Eqs. (5) and (6). We see that the theory and experimental results are in very good agreement, considering the fact that the only free parameter for the upper two theoretical curves is the gain length dg.
3. Laser speckle When a coherent laser light is scattered by a random collection of particles or even a rough surface, the angular dependence of the scattered intensity shows strong fluctuations known as laser speckles [8]. These fluctuations are of the order of the average scattered intensity, and at some angles the scattered intensity is truly zero. This effect is due to interference between (multiply) scattered waves. A speckle pattern is very sensitive to the relative positions of the particles. The pattern changes completely if the particles move over a distance much smaller than the wavelength because the change of phase of a scattered wave is due to the cumulative effect of the motion of all the particles from which this wave is scattered. This effect
86
D.S. Wiersma, A. Laoendijk/Physica A 241 (1997) 82-88
4
.9
Z 1 0 0
20
40 60 Angle (mrad)
0
20
40 60 Angle (mrad)
Fig. 2. Normalized transmitted intensity in a solid angle of 125 I-trad, from a powdered Ti:sapphire sample, versus rotation angle of the sample. Sample thickness 100/am, average particle diameter 10 I.tm. Probe light: wavelength 780 nm, pulse duration 14 ns, beam diameter 200 lam. The dotted line is a reference measurement without pump light. In the right plot the pump intensity is higher than in the left plot (2.4 x 1011 J/mZs resp. 1.6 x 1011 j/m2 s).
is used to study the motion of small particles in a technique often referred to as 'diffusing-wave spectroscopy' [9]. One can describe a random sample as a waveguide that couples incident waves from incoming modes ~ and a' to outgoing modes /~ and /3'. One mode is a solid angle containing one coherence area (i.e. one speckle spot). It is interesting to see if the introduction of gain can alter the mode structure. For instance, at very large gain, one mode (or speckle spot) might win the competition with others, like in a regular laser. In Fig. 2, a speckle pattem is shown from a pumped Ti:sapphire powder. The transmitted intensity in a solid angle of 125 Ixrad is plotted versus the rotation angle of the sample. The dotted line is a reference measurement without the pump light, which was recorded by blocking the pump beam at every other probe shot and averaging even and odd probe shots separately. In the left figure the pump intensity was 1.6 × 1011 J/m 2 s. The overall gain is reasonably low (1.18). We see that the introduction of gain does not change the observed pattern to a large extend. It merely amplifies the pattern by a certain factor. This behavior is found for various measurements at modest pump intensities. If we increase the pump intensity to 2.4 x 1011 J/m 2 s, the observed pattern changes completely (right plot). A strong increase of one specific speckle spot is not observed however. Because the pump intensity lies close to the damage threshold of the sample (about a factor 2-3 higher) it is very difficult to distinguish between effects from an increased gain and heating effects that induce small movements of the particles. An alternative would be a single shot experiment in which the whole speckle pattem is recorded at once e.g. with a CCD camera. If the laser pulse is sufficiently short, the movement of the particles during the pulse will then be very small even at high pump intensities.
D.S. Wiersma, A. LagendijklPhysica A 241 (1997) 82-88
87
4. Random lasers Disordered materials with gain are also interesting from a completely different point of view. Instead of performing an experiment with a probe pulse, one can study the spontaneously emitted output of a laser material in which disorder is introduced. This spontaneous emission can have (e.g. spectral) properties which are very similar to laser light. The effects depend on the amount of scattering. We can distinguish three regimes.
4.1. Weak scatterin O and gain If the scattering is very weak, that is Y is of the order of the sample size, the role of the scatterers is trivial. Spontaneous emission which is amplified considerably can have a very narrow spectrum. This effect also occurs for normal (homogeneous) laser materials and is called amplified spontaneous emission or ASE. The introduction of some scatterers will diffuse the (already present) ASE and will not influence its spectral properties. This is the explanation for the effects observed in Ref. [10], where weakly scattering dye solutions were studied.
4.2. Modest scatterin 9 and gain If the scattering is stronger, i.e. f is much smaller than the sample size but still larger than the wavelength, the presence of scatterers influences the spectral and temporal properties of the output. Due to modest scattering, the residence time of the light in the sample is largely increased and gain narrowing will therefore be much stronger. Modest scattering with gain can also lead to a pulsed output when the gain is switched on [11,12], similar to laser spiking in regular lasers.
4.3. Strong scattering and gain If we further increase the scattering strength we reach the situation where d becomes equal to or smaller than the wavelength. This is the regime where Anderson localization of light is expected to occur [1]. Due to very strong scattering, recurrent scattering events arise [13]. These are scattering events in which light returns to a scatterer from which it was scattered before, thereby forming closed loop paths. If the amplification along such loop paths would be strong enough, they might be able to serve as random ring cavities for the light.
Acknowledgements We wish to acknowledge valuable discussions with Meint van Albada. The work in this paper was part of the research program of the 'Stichting voor Fundamenteel Onderzoek der Materie' (Foundation for Fundamental Research on Matter) and was made
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D.S. Wiersma, A. Layendijk/Physica A 241 (1997) 82-88
possible by financial support from the ' N e d e r l a n d s e Organisatie v o o r W e t e n s c h a p p e l i j k O n d e r z o e k ' (Netherlands Organization for the A d v a n c e m e n t o f Research).
References [1] For recent reviews, in: Analogies in Optics and Micro Electronics, eds. W. van Haeringen and D. Lenstra (Kluwer, Dordrecht, 1990); P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic Press, San Diego, 1995). [2] Y. Kuga and A. Ishimaru, J. Opt. Soc. Amer. A 8 (1984) 831; M.P. van Albada and A. Lagendijk, Phys. Rev. Lett. 55 (1985) 2692; P.E. Wolf and G. Maret, Phys. Rev. Lett. 55 (1985) 2696. [3] D.S. Wiersma, M.P. van Albada and A. Lagendijk, Phys. Rev. Lett. 75 (1995) 1739. For all details on experiments and theory of coherent backscattering with gain, see: D.S. Wiersma, Light in Strongly Scattering and Amplifying Random Media, PhD Thesis, University of Amsterdam (1995). [4] B. Davison and J.B. Sykes, Neutron Transport Theory (Oxford University Press, Oxford, 1958). [5] V.S. Letokhov, Sov. Phys. JETP 26 (1968) 835. [6] M.B. van der Mark, M.P. van Albada and A. Lagendijk, Phys. Rev. B 37 (1988) 3575. [7] A.Yu. Zyuzin, Europhys. Lett. 26 (1994) 517. [8] J.C. Dainty ed. Laser Speckle and Related Phenomena, Topics in Applied Physics 9 (Springer, Berlin, 1984). [9] D.J. Pine, D.A. Weitz, P.M. Chaikin and E. Herbolzheimer, Phys. Rev. Lett. 60 (1988) 1134. [10] N.M. Lawandy, R.M. Balachandran, A.S.L. Gomes and E. Sauvain, Nature 368 (1994) 436. [11] C. Gouedard, D. Husson, C. Sauteret, F. Auzel and A. Migus, J. Opt. Soc. Amer. B 10 (1993) 2358. [12] D.S. Wiersma and A. Lagendijk, Phys. Rev. E (1996). [13] D.S. Wiersma, M.P. van Albada, B.A. van Tiggelen and A. Lagendijk, Phys. Rev. Lett. 74 (1995) 4193.
Physica A 241 (1997) 82-88
Interference effects in multiple-light scattering with gain Diederik S. Wiersma a'c'*, Ad Lagendijk b a FOM-lnstitute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, Netherlands b Van der Waals-Zeeman Laboratorium, Valckenierstraat 65-67, 1018 XE Amsterdam, Netherlands c European Laboratory for non-Linear Spectroscopy, Largo E. Fermi 2 (Arcetri), 50125 Florence, Italy
Abstract Experimental results are reported on coherent backscattering and speckle from an amplifying random medium. Also, coherent backscattering with gain is calculated completely, using diffusion theory. To conclude, the concept of a random laser is discussed. PACS: 42.25.Bs; 42.55.-f; 78.45.+h Keywords: Multiple-light scattering; Interference; Light amplification; Random laser
1. Introduction Since a couple of years it has been known that interesting interference effects can occur in light which is multiply scattered from the disordered structures, especially if the scattering is very strong [1]. Examples of these phenomena are (weak) localization of light, optical band gaps, and extremely slow transport. In all these cases absorption poses an important problem: due to absorption the light has simply disappeared after a few scattering events. All objects in daily life (even those appearing perfectly white) absorb light to some extent. What would happen if the light is amplified at every scattering event instead of getting absorbed? In that case we have the opposite situation of absorption and the object is so to say 'whiter than white'. The light intensity will increase instead of decreasing on multiple scattering. Light amplification can be obtained in a laser material. We have recently succeeded in realizing a random medium with gain by grinding a titanium sapphire laser crystal. The availability of an amplifying random medium allows one to study interference phenomena in multiple scattering with gain. * Correspondence address: European Laboratory for non-Linear Spectroscopy, Largo E. Fermi 2 (Arcetri), 50125 Florence, Italy. Fax: 3955224072; e-mail: [email protected]. 0378-4371/97/$17.00 Copyright (~) 1997 Published by Elsevier Science B.V. All rights reserved PII S0378-4371(97)00063-0
D.S. Wiersma, A. Lagendijkl Physica .4 241 (1997) 82-88
83
2. Coherent backscattering Coherent backscattering [2] is a general interference effect for waves which are backscattered from a random medium and originates from the time-reversal symmetry, or better, from reciprocity. Because o f reciprocity, any light path from a light source, through a random medium, and back to the light source, can be followed in opposite directions. Light waves which travel along the reciprocal paths will acquire the same phase and interfere constructively in the direction back to the detector. The result is an enhancement of the intensity in backscattering, which has the shape o f a narrow cone. In Fig. 1, backscattering cones are shown as recorded from a powdered Ti:sapphire crystal, which was pumped with 14ns pulses from a frequency doubled N d : Y A G laser. For more details o f the experiment we refer to Ref. [3]. The lower curve is the unamplified cone without pump light. We observed that the overall scattered intensity increases with pump energy. We also observed that the shape o f the cone changes; upon increasing the gain, the top o f the backscattering cone sharpens. This sharpening can be understood if one realizes that the shape o f the cone reflects the path-length distribution in the sample. The wings o f the cone are mostly due to contributions from short paths, and as we approach the top of the cone, contributions from successively longer paths add to the intensity. Because the amplification along a path depends exponentially on its length, the introduction o f gain will sharpen the central region o f the backscattering cone. Diffusion theory provides a very good description o f coherent backscattering from passive random media. To calculate the backscattering cone for an amplifying medium using diffusion theory, we start from a standard stationary diffusion equation with gain [4,5] and perform a Fourier transform over the spatial coordinates x and y, in which
'
'
I
'
'
:~ 3
I
'
'
I
'
'
I
~ overallgain = 71% [] overallgain -- 36 % o unamplified
©
..= ~o 2 N
O
Z
1 ,
,
,
I
0
,
,
,
I
,
,
,
i
2 4 Angle (mrad)
,
,
,
L
6
Fig. 1. Three backscattering cones in the polarization conserving channel at different pump energy. Sample: 30% Ti: Sapphire particles (10 gm diameter) in water, transport mean free path: 40 gm. The overall gain of 0%, 36%, and 71%, corresponds to pump energies: 0, 165, and 190mJ. The intensity is normalized to the diffuse background at zero gain. Solid lines: calculated curves based on diffusion theory. Upon increasing the gain, the overall intensity increases and the top of the cone sharpens.
D.S. Wiersma, A. LaoendijkIPhysica A 241 (1997) 82 88
84
the sample is translationally invariant. The resulting equation is
~(2(~1-q~)
F(q±,z,,z2)+f~cgF(q±,zl,z2)+6(zl - z 2 ) = 0 .
(1)
Applying the boundary condition of a slab of thickness L, [F(q±, zl, z 2 ) = 0 for z2 = -z0 and z2 = L +z0, and for every zl, with z0 ~ 0.71(], we find
F(q±,zl,z2) =F(q±,Zs,Za) =
3 cos[fl(L - zs)] - 3 cos[fl(L + 2z0 2 ( 2 fl sin[fl(L + 2z0)]
Izol)] ,
(2)
where Zs = zl + z2, zd -- z, -- z2, and fl =- V/fadp - q~_, with •amp the amplification length in the medium. The amplification length is defined as the (rms) average distance at which the intensity is amplified in the medium by a factor e +1 , while the gain length (g is defined as the actual traveled distance to obtain this amplification. From Eq. (2) we can see that F(q±,zbz2) becomes unstable (i.e. diverges for small angles) for L + 2zo ~>Lcr: ~gCamp,where Lcr is the critical thickness. From the diffusion propagator F(q±,zl,z2) one can calculate the total bistatic coefficient (or normalized intensity), which is the sum of an interference term 7c(0s) and a diffuse background term 7~(0s). The interference term describes the backscattering cone and is given by [6] L 2L--zd
F(q±, ZI, 7-2 ) COS(Zd~)e-uz~ dzs d&l,
7c(0s) = ~ 0
(3)
zd
whereas for the diffuse background we have [6] L 2L--z,l
1
f
7t(0s)=~/
0
F(q± =O, Zl,Z2)e-(1/2)KeZs(~-'+l)e-(I/2)~zd(~7'-l)dzsdzd.
(4)
zd
Here r/=k0(1-/~s), u = ~1 x e ( l + # s--1 ), and #s = C O S ( 0 s ) with 0s the angle of the outgoing wave vector k2 relative to the z-axis. Furthermore, Ke is defined by ~:~ = {-1 _ ( - 1 , where E~ is the scattering mean free path and Eg is the gain length in the medium. The incoming wave vector is taken along the z-axis. The above integrals can be performed explicitly using the intensity propagator from Eq. (2). The result is 3e -~L
1 7~(0~) = 2(3flsin[fl(L + 2zo)] (f12 + q2 + u2)2 _ (2flq)2 × [2(fl 2 + r/2 + u 2) cos(2flzo) cos(D/) - 4fir/sin(2flzo) sin(D/)
+ 2-~(-fl 2 + r/2 - u 2) sin(fl(L + 2z0)) sinh(uL) U
D.S. Wiersma, A. LaoendijklPhysica A 241 (1997) 82-88
85
+ 2(/~2 --/12 -- U2 ) cos(L?1) - 2(fl2 + 712 + u 2) cos(//(L + 2Zo)) cosh(uL)
+4~usin(flL)sinh(uL) + 2 ( - f i 2 + 712+ u2)cos(flL)cosh(uL)]
/
(5)
and 3
?~(Os) = 2d3flsin[fl( L + 2zo)]
Zl(1 + e -2uL) + Z2(1 - e -2uL) + Z3e -L(v+u) U[(U 2 + fl2)2 + V2(2f12 _ 2U2 +/)2)] ,
(6)
where Zl = / / ( / ) 2 _ u2 _ f12) c o s [ f l ( L q_ 2Zo)]
+ u(u 2 - 1)2 - f12) c o s ( / ~ L )
V2 _.}..f12 __ 3U 2
+ 2uvfl sin[/~(L + 2zo)] + uvfl
Z2 = v ( u 2 - v 2 - f12) c o s [ f l ( L q- 2 z o ) ]
u £ 7- -fi2
(7)
sin(ilL),
+ 2uZflsin(~L)
- f l ( u 2 + v2 + f12) sin[fl(L + 2zo)] + u2v
V2 __ U2 -~
3[/2
U 2 _]_ f12
cos(/ L)
(8)
and
Z3 = 2u(v 2 -
u 2 '}-/~2) q_
2U(U2 _ V2 +/~2) COS(2ZOfl)_ 4UVfl sin(2zo/3),
(9)
with v = g1 Ke(#s--1 - 1), and u, 71, /~, and #s as defined before. (Note that q± = 0 in the expression for 7e.) For small angles, this solution agrees with the results of Ref. [7], where the central region around the cusp of the cone was calculated. For passive media, usually the limit of L ~ oc is taken to obtain a much simpler expression. For amplifying media this is impossible due to the divergence of the intensity for L + 2z0 > Lcr. The solid lines in Fig. 1 are the calculated intensities from the sum of Eqs. (5) and (6). We see that the theory and experimental results are in very good agreement, considering the fact that the only free parameter for the upper two theoretical curves is the gain length dg.
3. Laser speckle When a coherent laser light is scattered by a random collection of particles or even a rough surface, the angular dependence of the scattered intensity shows strong fluctuations known as laser speckles [8]. These fluctuations are of the order of the average scattered intensity, and at some angles the scattered intensity is truly zero. This effect is due to interference between (multiply) scattered waves. A speckle pattern is very sensitive to the relative positions of the particles. The pattern changes completely if the particles move over a distance much smaller than the wavelength because the change of phase of a scattered wave is due to the cumulative effect of the motion of all the particles from which this wave is scattered. This effect
86
D.S. Wiersma, A. Laoendijk/Physica A 241 (1997) 82-88
4
.9
Z 1 0 0
20
40 60 Angle (mrad)
0
20
40 60 Angle (mrad)
Fig. 2. Normalized transmitted intensity in a solid angle of 125 I-trad, from a powdered Ti:sapphire sample, versus rotation angle of the sample. Sample thickness 100/am, average particle diameter 10 I.tm. Probe light: wavelength 780 nm, pulse duration 14 ns, beam diameter 200 lam. The dotted line is a reference measurement without pump light. In the right plot the pump intensity is higher than in the left plot (2.4 x 1011 J/mZs resp. 1.6 x 1011 j/m2 s).
is used to study the motion of small particles in a technique often referred to as 'diffusing-wave spectroscopy' [9]. One can describe a random sample as a waveguide that couples incident waves from incoming modes ~ and a' to outgoing modes /~ and /3'. One mode is a solid angle containing one coherence area (i.e. one speckle spot). It is interesting to see if the introduction of gain can alter the mode structure. For instance, at very large gain, one mode (or speckle spot) might win the competition with others, like in a regular laser. In Fig. 2, a speckle pattem is shown from a pumped Ti:sapphire powder. The transmitted intensity in a solid angle of 125 Ixrad is plotted versus the rotation angle of the sample. The dotted line is a reference measurement without the pump light, which was recorded by blocking the pump beam at every other probe shot and averaging even and odd probe shots separately. In the left figure the pump intensity was 1.6 × 1011 J/m 2 s. The overall gain is reasonably low (1.18). We see that the introduction of gain does not change the observed pattern to a large extend. It merely amplifies the pattern by a certain factor. This behavior is found for various measurements at modest pump intensities. If we increase the pump intensity to 2.4 x 1011 J/m 2 s, the observed pattern changes completely (right plot). A strong increase of one specific speckle spot is not observed however. Because the pump intensity lies close to the damage threshold of the sample (about a factor 2-3 higher) it is very difficult to distinguish between effects from an increased gain and heating effects that induce small movements of the particles. An alternative would be a single shot experiment in which the whole speckle pattem is recorded at once e.g. with a CCD camera. If the laser pulse is sufficiently short, the movement of the particles during the pulse will then be very small even at high pump intensities.
D.S. Wiersma, A. LagendijklPhysica A 241 (1997) 82-88
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4. Random lasers Disordered materials with gain are also interesting from a completely different point of view. Instead of performing an experiment with a probe pulse, one can study the spontaneously emitted output of a laser material in which disorder is introduced. This spontaneous emission can have (e.g. spectral) properties which are very similar to laser light. The effects depend on the amount of scattering. We can distinguish three regimes.
4.1. Weak scatterin O and gain If the scattering is very weak, that is Y is of the order of the sample size, the role of the scatterers is trivial. Spontaneous emission which is amplified considerably can have a very narrow spectrum. This effect also occurs for normal (homogeneous) laser materials and is called amplified spontaneous emission or ASE. The introduction of some scatterers will diffuse the (already present) ASE and will not influence its spectral properties. This is the explanation for the effects observed in Ref. [10], where weakly scattering dye solutions were studied.
4.2. Modest scatterin 9 and gain If the scattering is stronger, i.e. f is much smaller than the sample size but still larger than the wavelength, the presence of scatterers influences the spectral and temporal properties of the output. Due to modest scattering, the residence time of the light in the sample is largely increased and gain narrowing will therefore be much stronger. Modest scattering with gain can also lead to a pulsed output when the gain is switched on [11,12], similar to laser spiking in regular lasers.
4.3. Strong scattering and gain If we further increase the scattering strength we reach the situation where d becomes equal to or smaller than the wavelength. This is the regime where Anderson localization of light is expected to occur [1]. Due to very strong scattering, recurrent scattering events arise [13]. These are scattering events in which light returns to a scatterer from which it was scattered before, thereby forming closed loop paths. If the amplification along such loop paths would be strong enough, they might be able to serve as random ring cavities for the light.
Acknowledgements We wish to acknowledge valuable discussions with Meint van Albada. The work in this paper was part of the research program of the 'Stichting voor Fundamenteel Onderzoek der Materie' (Foundation for Fundamental Research on Matter) and was made
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possible by financial support from the ' N e d e r l a n d s e Organisatie v o o r W e t e n s c h a p p e l i j k O n d e r z o e k ' (Netherlands Organization for the A d v a n c e m e n t o f Research).
References [1] For recent reviews, in: Analogies in Optics and Micro Electronics, eds. W. van Haeringen and D. Lenstra (Kluwer, Dordrecht, 1990); P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic Press, San Diego, 1995). [2] Y. Kuga and A. Ishimaru, J. Opt. Soc. Amer. A 8 (1984) 831; M.P. van Albada and A. Lagendijk, Phys. Rev. Lett. 55 (1985) 2692; P.E. Wolf and G. Maret, Phys. Rev. Lett. 55 (1985) 2696. [3] D.S. Wiersma, M.P. van Albada and A. Lagendijk, Phys. Rev. Lett. 75 (1995) 1739. For all details on experiments and theory of coherent backscattering with gain, see: D.S. Wiersma, Light in Strongly Scattering and Amplifying Random Media, PhD Thesis, University of Amsterdam (1995). [4] B. Davison and J.B. Sykes, Neutron Transport Theory (Oxford University Press, Oxford, 1958). [5] V.S. Letokhov, Sov. Phys. JETP 26 (1968) 835. [6] M.B. van der Mark, M.P. van Albada and A. Lagendijk, Phys. Rev. B 37 (1988) 3575. [7] A.Yu. Zyuzin, Europhys. Lett. 26 (1994) 517. [8] J.C. Dainty ed. Laser Speckle and Related Phenomena, Topics in Applied Physics 9 (Springer, Berlin, 1984). [9] D.J. Pine, D.A. Weitz, P.M. Chaikin and E. Herbolzheimer, Phys. Rev. Lett. 60 (1988) 1134. [10] N.M. Lawandy, R.M. Balachandran, A.S.L. Gomes and E. Sauvain, Nature 368 (1994) 436. [11] C. Gouedard, D. Husson, C. Sauteret, F. Auzel and A. Migus, J. Opt. Soc. Amer. B 10 (1993) 2358. [12] D.S. Wiersma and A. Lagendijk, Phys. Rev. E (1996). [13] D.S. Wiersma, M.P. van Albada, B.A. van Tiggelen and A. Lagendijk, Phys. Rev. Lett. 74 (1995) 4193.