- Email: [email protected]
Random gain medium with reflecting boundary
15 July 2000
Optics Communications 181 Ž2000. 379–384 www.elsevier.comrlocateroptcom
Random gain medium with reflecting boundary A.C. Selden ) Department of Physics, UniÕersity of Zimbabwe, P.O. Box MP 167, Mount Pleasant, Harare, Zimbabwe Received 18 November 1999; received in revised form 17 April 2000; accepted 24 May 2000
Abstract The effect of a reflecting boundary on the generation threshold of a random gain medium is analysed for both specular and diffuse reflection, the latter corresponding to feedback scattering from a passive boundary layer. The feedback effect is significant for non-diffusive transfer Žoptically thin layer. but small in the diffusion regime unless 1 y R < 3g , where g is the gain parameter, with a maximum reduction by a factor of two for a perfectly reflecting boundary. A second reflecting boundary further reduces the threshold, the reduction depending on the optical parameters of the medium. q 2000 Elsevier Science B.V. All rights reserved.
'
PACS: 42.20; 42.55 Keywords: Random gain media; Diffuse reflectance; Scattering
1. Introduction Experiments on random gain media in the form of crystalline powders and colloidal solutions have been carried out for a wide range of scattering particle densities and concentrations such that the gain parameter g s a l has varied from g < 1 Ždiffusion regime. to g ) 1 Žweak scattering., where a is the gain coefficient and l the Žtransport. mean free path w1–3x. Feedback of emitted photons by diffuse or specular reflection at the gain boundaries can have a significant effect on the threshold fluence, as demonstrated by Lawandy and co-workers w4,5x, yet no theoretical analysis of its dependence on the optical parameters of the medium appears to have been made, although Monte Carlo simulations have been ) Tel.: q263-4-303-211; fax: q263-4-333-407; e-mail: [email protected]
used to evaluate specific cases. In this paper we employ a two-stream model of the random gain medium w6x, which has the advantage of providing simple analytic solutions for the full range of transport parameters, thereby making the dependence of the generation threshold on specified conditions particularly transparent. A similar approach has previously been applied in studying fluctuations in the albedo of a random gain medium embedded in an optical waveguide w7x. Here we consider feedback scattering provided by diffusion of photons in a passive boundary layer adjacent to the random gain medium and evaluate its effect on the threshold parameters of the system w4x. The additional effect of specular reflection e.g. by a dichroic mirror at the input or Fresnel reflection at the interface, is evaluated by the same method w5x. Non-uniform gain is easily included in the analysis. Although the twostream model is approximate, it can be useful for
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 7 8 9 - 6
380
A.C. Selden r Optics Communications 181 (2000) 379–384
predicting the properties of specific gain media and in preliminary analysis of the experimental data w6x. 2. Theory Radiative transfer in a plane layer can be analysed using a two-stream model derived from the radiative transfer equation, which is analytically simple yet sufficiently accurate for practical use. The two-stream equations w6x dF1rdt s g 1 F1 y g 2 F2
Ž 1a . Ž 1b .
dF2rdt s g 2 F1 y g 1 F2
relate the dependence of the forward and backward hemi-spherical flux densities F1 and F2 on optical depth t Žin mean free paths. via the transfer coefficients g 1 and g 2 defined below. Setting F2 s RF1 and eliminating F1 and F2 from Eqs. Ž1a. and Ž1b. we obtain the albedo equation dRrdt s 2 Ž g 2 y g1 . R q g 2 Ž 1 y R .
2
Ž 2.
where R is defined as the ratio of the hemi-spherical fluxes of the two streams of radiation. This definition
of the albedo provides an extremely simple means of handling reflecting boundary conditions: for specular reflection the exact boundary condition is F2rF1 s R s R s , where R s is the specular reflectance. For diffuse reflection we may write R s R d , where R d is the diffuse reflectance averaged over the angular flux distribution. Thus the effect of a reflecting boundary on a random gain medium may be readily evaluated in the two-stream approximation using existing solutions of the albedo equation, which can also be used for calculating the diffuse reflectance of the passive scattering medium w7,8x.
3. Reflecting boundary With the appropriate value R b F 1 for the boundary reflectance, the critical length LXc for the gain medium may be found by applying the boundary condition R s R b to the albedo equation, equivalent to subtracting a segment of length L R with albedo R s R b from the critical length L c for a plane layer without a reflecting boundary: thus LXc s L c y L R
Fig. 1. Relative reduction in critical length L R rL c vs. gain parameter g for selected values of boundary reflectance R b . The inset shows the albedo curves for active and passive random media and the effect of a reflecting boundary on the threshold optical depth.
A.C. Selden r Optics Communications 181 (2000) 379–384
381
w9x. The dependence of L R rL c on g s a l for selected values of the boundary reflectance is shown in Fig. 1, where we have used the radiative transfer coefficients g 1 s Ž3 y 7g .r4 and g 2 s Ž3 q g .r4 w6x, which give the diffusion result L crl f pr 3g for the plane layer when g < 1 w10x. The feedback effect is small in the diffusion regime unless 1 y R b < 3g , the ‘turnover’ becoming more acute as R b 1. The ratio L R rL c approaches a limiting value r F 0.5 as g increases, depending on the boundary reflectance. In the limit R b 1 we find L R s L cr2 for all g ) 0. Thus a perfectly reflecting boundary halves the critical length for all scattering regimes, as would be expected from symmetry considerations w11x, whereas a high reflectance boundary Ž1 y R < 1. nearly does so.
passive regions will be related: for example, the available gain and the absorption length depend on dye concentration and the mean free path on scattering particle density. Hence the critical length LXc may be found as a continuous function of the gain parameter g s a l . In contrast to the case of a fixed boundary reflectance, the ratio LXcrL c slowly increases with increasing mean free path l as a consequence of diminishing feedback by diffuse reflection. This is shown in Fig. 2, where we have used the diffuse reflectance of a passive scattering medium of infinite depth w7x with transfer coefficients gX1 s Ž3 q 7gy .r4 and gX2 s Ž3 y gy .r4 w6x, where gys lrl a and l a is the absorption length in the passive medium. The effect becomes more pronounced as the gainrloss ratio a l a decreases.
4. Diffuse reflection
5. Specular reflection
When considering a random gain medium e.g. a colloidal dye solution, excited to a finite depth beyond which it acts as a passive reflector, we may find the effective critical length Žand gain threshold. for the active medium with the diffuse reflectance of the passive medium Ž g F 0. as the boundary condition. In this case, the parameters of the active and
The effect of a second reflecting boundary, e.g. a dichroic mirror at the input, will be to further reduce the critical length w5x. Thus LYc s L c y L R1 y L R2 , where R 1 and R 2 are the reflectances at the respective gain boundaries. In analysing a specific system we may combine results of the type shown in Fig. 1 and Fig. 2 with a calculation of the diffuse re-
'
´ '
´
X
Fig. 2. Diffuse reflectance R d and reduced critical length ratio L crL c vs. gain parameter g for random gain medium with diffusely reflecting boundary layer.
A.C. Selden r Optics Communications 181 (2000) 379–384
382
Table 1 Reduced critical length vs. specular reflectance R s for a colloidal dye medium with parameters a s 4 mmy1 l s 5 mm and l a s 5 mm X
6. Plane layer with reflecting boundaries
L c rL c s 0.58
R d s 0.93
Rs
L Rs rL c
L c rL c
L c rL c
0.98 0.95 0.9 0.8 0.7 0.6 0.5
0.48 0.45 0.39 0.29 0.21 0.15 0.1
0.1 0.13 0.19 0.29 0.37 0.43 0.48
0.17 0.22 0.33 0.5 0.64 0.74 0.83
Y
there is no significant reduction in critical length for R s - 0.5 in this case.
Y
X
flectance R d of the passive boundary layer to find LYc . For example, the dependence of the critical length on specular reflectance R s is given in Table 1 for a gain medium with parameters a s 4 mmy1 , l s 5 mm and l a s 5 mm, appropriate to a colloidal dye solution w4x. The diffuse reflectance of the passive medium R d s 0.93, LXcrL c s 0.58 and LYcrLXc s 0.17 for specular reflectance R s s 0.98, corresponding to a more than five-fold reduction in threshold compared with that for diffuse feedback scattering alone w5x. It is also clear from the data that
If the gain medium has two boundaries of equal reflectance, e.g. Fresnel reflection at the dielectric surfaces, the reduced critical length LYc s L c y 2L R . The critical length approaches a limiting value as g increases and can be reduced by a large factor, as for a conventional laser. However, the effect of the reflecting boundaries progressively diminishes in the diffusion regime, where multiple scattering is dominant. The relation between boundary reflectance R b and threshold gain parameter gth s a th l for a plane layer with reflecting boundaries is shown in Fig. 3 for selected values of optical depth t 0 s L 0rl . Clearly gth is very sensitive to the boundary reflectance for optically thin layers Ž t 0 - 3., but relatively insensitive for optically thick layers Ž t 0 ) 10., where the threshold is already low. For example, the threshold gain coefficient for a plane layer of width L 0 s 0.8 mm with reflecting boundaries and transport length l s 40 mm Ž t 0 s 20. falls in the range 0 F a th F 0.2 mmy1 for 1 G R b G 0 w10x, while a th ; 10 mmy1 for a 30 mm layer of colloidal dye
Fig. 3. Relation between boundary reflectance R b and threshold gain parameter gth for plane layers with reflecting boundaries and specified optical depth t 0 s L 0 rl .
A.C. Selden r Optics Communications 181 (2000) 379–384
383
Fig. 4. Threshold of random gain medium with reflecting boundaries: comparison of uniform and linear gain profiles for L 0 rl s 9.75 and l s 10 mm.
solution for which l s 10 mm Ž t 0 s 3. between boundaries with reflectance R b s 0.5 w4,5x. 7. Non-uniform gain In general the active medium will have a gain profile determined by the transport of pump radiation and the local level of excitation in the medium. The form of the gain profile will depend on the scattering regime and the energy and duration of the pump pulse: e.g. it will be asymptotically linear in the weakly pumped diffusion regime for single-sided pumping, but symmetric and almost uniform when pumped from both sides w10x. Assuming the gain profile to be determined, we may solve the albedo equation numerically with the gain parameter g as a Žknown. variable. An example is shown in Fig. 4, where the thresholds for uniform Žconstant gain. and linear gain profiles are compared for a plane layer with reflecting boundaries, indicating a slightly lower mean gain for the latter w10x. 8. Conclusions Application of the albedo equation to the analysis of a random gain medium with reflecting boundary
reveals the essential physics involved without recourse to numerical solution. The analysis shows that the critical length LXc , and thus the generation threshold, is uniquely determined by the gain parameter g and the boundary reflectance R b . A perfectly reflecting boundary halves the critical length, as would be expected from symmetry considerations. This can be further reduced by introducing a second reflecting boundary, e.g. by placing a dichroic mirror in the path of the pump beam close to the surface of the gain medium w5x. The reduction in threshold caused by a reflecting boundary depends on the scattering regime: for diffuse scattering Ž g < 1. the effect is small unless 1 y R < 3g , the gain medium itself providing the major part of the feedback, while for an optically thin layer between reflecting boundaries the gain threshold is a sensitive function of boundary reflectance w12x. The albedo of the passive medium is critically dependent on residual absorption, but can provide effective feedback in colloidal dye systems, resulting in a significant lowering of the gain threshold w4x. Comparison of the thresholds for a plane layer with uniform gain and one with a linear gain profile shows a slight advantage in favour of the latter, in agreement with the remarks made in w10x. The applicability of the two-stream analysis to ran-
'
384
A.C. Selden r Optics Communications 181 (2000) 379–384
dom gain media is discussed in detail in a forthcoming paper.
References w1x N.M. Lawandy, R.M. Balachandran, A.S.L. Gomes, E. Sauvain, Nature 368 Ž1994. 436–438. w2x W.L. Sha, C.H. Liu, R.R. Alfano, Opt. Lett. 19 Ž1994. 1922–1924. w3x G. Beckering, S.J. Zilker, D. Haarer, Opt. Lett. 22 Ž1997. 1427–1429. w4x R.M. Balachandran, N.M. Lawandy, J.A. Moon, Opt. Lett. 22 Ž1997. 319–321.
w5x P.C. de Oliveira, J.A. McGreevy, N.M. Lawandy, Opt. Lett. 22 Ž1997. 895–897. w6x R.M. Goody, Y.L. Yung, Atmospheric radiation, 2nd edn., OUP, 1989. w7x C.W.J. Beenakker, J.C.J. Paaschens, P.W. Brouwer, Phys. Rev. Lett. 76 Ž1996. 1368–1371. w8x A.C. Selden, Opt. Commun. 10 Ž1974. 1–3. w9x V.S. Letokhov, JETP 26 Ž1968. 835–840. w10x D.S. Wiersma, A. Lagendijk, Phys. Rev. E 54 Ž1996. 4256– 4265. w11x An identical conclusion follows for the critical thickness of the homogeneous slab, where the flux density is symmetric about the mid-plane: K.M. Case, P.F. Zweifel, Linear Transport Theory, Addison Wesley, 1967. w12x Similar conclusions apply to diffusive transfer in passive scattering media: see A. Lagendijk, R. Vreeker, P. de Vries., Phys. Lett. A 136 Ž1989. 81–88.
Optics Communications 181 Ž2000. 379–384 www.elsevier.comrlocateroptcom
Random gain medium with reflecting boundary A.C. Selden ) Department of Physics, UniÕersity of Zimbabwe, P.O. Box MP 167, Mount Pleasant, Harare, Zimbabwe Received 18 November 1999; received in revised form 17 April 2000; accepted 24 May 2000
Abstract The effect of a reflecting boundary on the generation threshold of a random gain medium is analysed for both specular and diffuse reflection, the latter corresponding to feedback scattering from a passive boundary layer. The feedback effect is significant for non-diffusive transfer Žoptically thin layer. but small in the diffusion regime unless 1 y R < 3g , where g is the gain parameter, with a maximum reduction by a factor of two for a perfectly reflecting boundary. A second reflecting boundary further reduces the threshold, the reduction depending on the optical parameters of the medium. q 2000 Elsevier Science B.V. All rights reserved.
'
PACS: 42.20; 42.55 Keywords: Random gain media; Diffuse reflectance; Scattering
1. Introduction Experiments on random gain media in the form of crystalline powders and colloidal solutions have been carried out for a wide range of scattering particle densities and concentrations such that the gain parameter g s a l has varied from g < 1 Ždiffusion regime. to g ) 1 Žweak scattering., where a is the gain coefficient and l the Žtransport. mean free path w1–3x. Feedback of emitted photons by diffuse or specular reflection at the gain boundaries can have a significant effect on the threshold fluence, as demonstrated by Lawandy and co-workers w4,5x, yet no theoretical analysis of its dependence on the optical parameters of the medium appears to have been made, although Monte Carlo simulations have been ) Tel.: q263-4-303-211; fax: q263-4-333-407; e-mail: [email protected]
used to evaluate specific cases. In this paper we employ a two-stream model of the random gain medium w6x, which has the advantage of providing simple analytic solutions for the full range of transport parameters, thereby making the dependence of the generation threshold on specified conditions particularly transparent. A similar approach has previously been applied in studying fluctuations in the albedo of a random gain medium embedded in an optical waveguide w7x. Here we consider feedback scattering provided by diffusion of photons in a passive boundary layer adjacent to the random gain medium and evaluate its effect on the threshold parameters of the system w4x. The additional effect of specular reflection e.g. by a dichroic mirror at the input or Fresnel reflection at the interface, is evaluated by the same method w5x. Non-uniform gain is easily included in the analysis. Although the twostream model is approximate, it can be useful for
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 7 8 9 - 6
380
A.C. Selden r Optics Communications 181 (2000) 379–384
predicting the properties of specific gain media and in preliminary analysis of the experimental data w6x. 2. Theory Radiative transfer in a plane layer can be analysed using a two-stream model derived from the radiative transfer equation, which is analytically simple yet sufficiently accurate for practical use. The two-stream equations w6x dF1rdt s g 1 F1 y g 2 F2
Ž 1a . Ž 1b .
dF2rdt s g 2 F1 y g 1 F2
relate the dependence of the forward and backward hemi-spherical flux densities F1 and F2 on optical depth t Žin mean free paths. via the transfer coefficients g 1 and g 2 defined below. Setting F2 s RF1 and eliminating F1 and F2 from Eqs. Ž1a. and Ž1b. we obtain the albedo equation dRrdt s 2 Ž g 2 y g1 . R q g 2 Ž 1 y R .
2
Ž 2.
where R is defined as the ratio of the hemi-spherical fluxes of the two streams of radiation. This definition
of the albedo provides an extremely simple means of handling reflecting boundary conditions: for specular reflection the exact boundary condition is F2rF1 s R s R s , where R s is the specular reflectance. For diffuse reflection we may write R s R d , where R d is the diffuse reflectance averaged over the angular flux distribution. Thus the effect of a reflecting boundary on a random gain medium may be readily evaluated in the two-stream approximation using existing solutions of the albedo equation, which can also be used for calculating the diffuse reflectance of the passive scattering medium w7,8x.
3. Reflecting boundary With the appropriate value R b F 1 for the boundary reflectance, the critical length LXc for the gain medium may be found by applying the boundary condition R s R b to the albedo equation, equivalent to subtracting a segment of length L R with albedo R s R b from the critical length L c for a plane layer without a reflecting boundary: thus LXc s L c y L R
Fig. 1. Relative reduction in critical length L R rL c vs. gain parameter g for selected values of boundary reflectance R b . The inset shows the albedo curves for active and passive random media and the effect of a reflecting boundary on the threshold optical depth.
A.C. Selden r Optics Communications 181 (2000) 379–384
381
w9x. The dependence of L R rL c on g s a l for selected values of the boundary reflectance is shown in Fig. 1, where we have used the radiative transfer coefficients g 1 s Ž3 y 7g .r4 and g 2 s Ž3 q g .r4 w6x, which give the diffusion result L crl f pr 3g for the plane layer when g < 1 w10x. The feedback effect is small in the diffusion regime unless 1 y R b < 3g , the ‘turnover’ becoming more acute as R b 1. The ratio L R rL c approaches a limiting value r F 0.5 as g increases, depending on the boundary reflectance. In the limit R b 1 we find L R s L cr2 for all g ) 0. Thus a perfectly reflecting boundary halves the critical length for all scattering regimes, as would be expected from symmetry considerations w11x, whereas a high reflectance boundary Ž1 y R < 1. nearly does so.
passive regions will be related: for example, the available gain and the absorption length depend on dye concentration and the mean free path on scattering particle density. Hence the critical length LXc may be found as a continuous function of the gain parameter g s a l . In contrast to the case of a fixed boundary reflectance, the ratio LXcrL c slowly increases with increasing mean free path l as a consequence of diminishing feedback by diffuse reflection. This is shown in Fig. 2, where we have used the diffuse reflectance of a passive scattering medium of infinite depth w7x with transfer coefficients gX1 s Ž3 q 7gy .r4 and gX2 s Ž3 y gy .r4 w6x, where gys lrl a and l a is the absorption length in the passive medium. The effect becomes more pronounced as the gainrloss ratio a l a decreases.
4. Diffuse reflection
5. Specular reflection
When considering a random gain medium e.g. a colloidal dye solution, excited to a finite depth beyond which it acts as a passive reflector, we may find the effective critical length Žand gain threshold. for the active medium with the diffuse reflectance of the passive medium Ž g F 0. as the boundary condition. In this case, the parameters of the active and
The effect of a second reflecting boundary, e.g. a dichroic mirror at the input, will be to further reduce the critical length w5x. Thus LYc s L c y L R1 y L R2 , where R 1 and R 2 are the reflectances at the respective gain boundaries. In analysing a specific system we may combine results of the type shown in Fig. 1 and Fig. 2 with a calculation of the diffuse re-
'
´ '
´
X
Fig. 2. Diffuse reflectance R d and reduced critical length ratio L crL c vs. gain parameter g for random gain medium with diffusely reflecting boundary layer.
A.C. Selden r Optics Communications 181 (2000) 379–384
382
Table 1 Reduced critical length vs. specular reflectance R s for a colloidal dye medium with parameters a s 4 mmy1 l s 5 mm and l a s 5 mm X
6. Plane layer with reflecting boundaries
L c rL c s 0.58
R d s 0.93
Rs
L Rs rL c
L c rL c
L c rL c
0.98 0.95 0.9 0.8 0.7 0.6 0.5
0.48 0.45 0.39 0.29 0.21 0.15 0.1
0.1 0.13 0.19 0.29 0.37 0.43 0.48
0.17 0.22 0.33 0.5 0.64 0.74 0.83
Y
there is no significant reduction in critical length for R s - 0.5 in this case.
Y
X
flectance R d of the passive boundary layer to find LYc . For example, the dependence of the critical length on specular reflectance R s is given in Table 1 for a gain medium with parameters a s 4 mmy1 , l s 5 mm and l a s 5 mm, appropriate to a colloidal dye solution w4x. The diffuse reflectance of the passive medium R d s 0.93, LXcrL c s 0.58 and LYcrLXc s 0.17 for specular reflectance R s s 0.98, corresponding to a more than five-fold reduction in threshold compared with that for diffuse feedback scattering alone w5x. It is also clear from the data that
If the gain medium has two boundaries of equal reflectance, e.g. Fresnel reflection at the dielectric surfaces, the reduced critical length LYc s L c y 2L R . The critical length approaches a limiting value as g increases and can be reduced by a large factor, as for a conventional laser. However, the effect of the reflecting boundaries progressively diminishes in the diffusion regime, where multiple scattering is dominant. The relation between boundary reflectance R b and threshold gain parameter gth s a th l for a plane layer with reflecting boundaries is shown in Fig. 3 for selected values of optical depth t 0 s L 0rl . Clearly gth is very sensitive to the boundary reflectance for optically thin layers Ž t 0 - 3., but relatively insensitive for optically thick layers Ž t 0 ) 10., where the threshold is already low. For example, the threshold gain coefficient for a plane layer of width L 0 s 0.8 mm with reflecting boundaries and transport length l s 40 mm Ž t 0 s 20. falls in the range 0 F a th F 0.2 mmy1 for 1 G R b G 0 w10x, while a th ; 10 mmy1 for a 30 mm layer of colloidal dye
Fig. 3. Relation between boundary reflectance R b and threshold gain parameter gth for plane layers with reflecting boundaries and specified optical depth t 0 s L 0 rl .
A.C. Selden r Optics Communications 181 (2000) 379–384
383
Fig. 4. Threshold of random gain medium with reflecting boundaries: comparison of uniform and linear gain profiles for L 0 rl s 9.75 and l s 10 mm.
solution for which l s 10 mm Ž t 0 s 3. between boundaries with reflectance R b s 0.5 w4,5x. 7. Non-uniform gain In general the active medium will have a gain profile determined by the transport of pump radiation and the local level of excitation in the medium. The form of the gain profile will depend on the scattering regime and the energy and duration of the pump pulse: e.g. it will be asymptotically linear in the weakly pumped diffusion regime for single-sided pumping, but symmetric and almost uniform when pumped from both sides w10x. Assuming the gain profile to be determined, we may solve the albedo equation numerically with the gain parameter g as a Žknown. variable. An example is shown in Fig. 4, where the thresholds for uniform Žconstant gain. and linear gain profiles are compared for a plane layer with reflecting boundaries, indicating a slightly lower mean gain for the latter w10x. 8. Conclusions Application of the albedo equation to the analysis of a random gain medium with reflecting boundary
reveals the essential physics involved without recourse to numerical solution. The analysis shows that the critical length LXc , and thus the generation threshold, is uniquely determined by the gain parameter g and the boundary reflectance R b . A perfectly reflecting boundary halves the critical length, as would be expected from symmetry considerations. This can be further reduced by introducing a second reflecting boundary, e.g. by placing a dichroic mirror in the path of the pump beam close to the surface of the gain medium w5x. The reduction in threshold caused by a reflecting boundary depends on the scattering regime: for diffuse scattering Ž g < 1. the effect is small unless 1 y R < 3g , the gain medium itself providing the major part of the feedback, while for an optically thin layer between reflecting boundaries the gain threshold is a sensitive function of boundary reflectance w12x. The albedo of the passive medium is critically dependent on residual absorption, but can provide effective feedback in colloidal dye systems, resulting in a significant lowering of the gain threshold w4x. Comparison of the thresholds for a plane layer with uniform gain and one with a linear gain profile shows a slight advantage in favour of the latter, in agreement with the remarks made in w10x. The applicability of the two-stream analysis to ran-
'
384
A.C. Selden r Optics Communications 181 (2000) 379–384
dom gain media is discussed in detail in a forthcoming paper.
References w1x N.M. Lawandy, R.M. Balachandran, A.S.L. Gomes, E. Sauvain, Nature 368 Ž1994. 436–438. w2x W.L. Sha, C.H. Liu, R.R. Alfano, Opt. Lett. 19 Ž1994. 1922–1924. w3x G. Beckering, S.J. Zilker, D. Haarer, Opt. Lett. 22 Ž1997. 1427–1429. w4x R.M. Balachandran, N.M. Lawandy, J.A. Moon, Opt. Lett. 22 Ž1997. 319–321.
w5x P.C. de Oliveira, J.A. McGreevy, N.M. Lawandy, Opt. Lett. 22 Ž1997. 895–897. w6x R.M. Goody, Y.L. Yung, Atmospheric radiation, 2nd edn., OUP, 1989. w7x C.W.J. Beenakker, J.C.J. Paaschens, P.W. Brouwer, Phys. Rev. Lett. 76 Ž1996. 1368–1371. w8x A.C. Selden, Opt. Commun. 10 Ž1974. 1–3. w9x V.S. Letokhov, JETP 26 Ž1968. 835–840. w10x D.S. Wiersma, A. Lagendijk, Phys. Rev. E 54 Ž1996. 4256– 4265. w11x An identical conclusion follows for the critical thickness of the homogeneous slab, where the flux density is symmetric about the mid-plane: K.M. Case, P.F. Zweifel, Linear Transport Theory, Addison Wesley, 1967. w12x Similar conclusions apply to diffusive transfer in passive scattering media: see A. Lagendijk, R. Vreeker, P. de Vries., Phys. Lett. A 136 Ž1989. 81–88.