T " < 0 approximation vs semi-classical approachto active media

A. V. Dorofeenko1, A. A. Zyablovsky1,2, A. A. Pukhov1,2, A. P. Vinogradov1,2

1Institute for Theoretical and Applied Electromagnetics RAS, Moscow, Russia2Moscow Institute of Physics and Technology, Russia

Abstract: We analyze applicability of the " < 0 approximation to wave propagation in active material, comparing the results to the FDTD-based calculation employing more general Maxwell-Bloch model. It is shown that the slab thickness should be less than some critical value, corresponding to the divergence of the Airy series and to the start of the laser generation.

Recently the problem of light propagation through a gain media has attracted attention in connection with attempts to decrease losses in metamaterials [1]. In particular, one is interested in retrieval of effective constitutive parameters from measured values of reflection R and transmission Tcoefficients. In this connection it is desirable to be able to describe a gain medium in terms of permittivity. The propagation of light through a gain medium is accompanied by growth of field. Therefore, the effective imaginary part of permittivity should be negative (we use time exp( )i tdependence). The growth of the field’s intensity is limited bydecrease of the inverse population due to light emission, which brings nonlinearity in the problem. Though the possibility to describe light propagation in unbounded system in terms permittivity was studied in [2-4] the problem of light propagation through a slab of a gain media is an intriguing problem till now [5,6].

Below, employing semi-classical approach (the Maxwell-Bloch system of equations) [7]:

2 2 2

2 2 2 2 2

4E E Pc t c t

,

220 02

2 2P

P P P nEt t

, (1)

*0

0

1 1 Ren

n Pn n Et t

,

we study the propagation of a semi-infinite pulse through a slab of gain medium. In Eq.(1) E stands for the electric field, Pfor the polarization, is the permittivity of the matrix materialthat surrounds quantum dots (QDs) with population inversion density n . The latter is pulled by the pump towards the value

0n and is suppressed by the emission, which causesnonlinearity. The values of , 0n and interaction constantare dependent on the coordinate . The system (1) accounts for the mentioned nonlinearity, transient processes and laser generation. In this paper, for description of wave propagation through a 1D slab of gain medium we employ the FDTD

method. We calculated transmission of semi-infinite pulse through a finite slab of active medium. The 0n is chosen to be less than the values, at which the chaotic dynamics appears [8].The values of field having approached the stationary value arecompared to the ones obtained from the Fresnel formula and the Airy scheme with complex permittivity ' ''i ,

'' 0 . In the stationary case , ~ expE P i t , constn the variables P and n can be eliminated, and the system (1) reduces to equation

2 2 20/ '' 0d E d i E , (2a)

where

2''1 | |E

, (2b)

04 Pn , P n .The denominator in Eq. (2b) describes suppression of the inverse excitation by the field.

Seemingly, in the limit of small intensity 2| | 1E , (3)

the Fresnel formulae should give results coinciding with ones obtained by solving the system (1), even if no nonlinearity (2b) is taken into account in the Fresnel scheme, e.g., '' .Below we will show that in the general case this assumption is wrong and widely used simplified calculation, based solely on the Fresnel formula, can significantly distinguish from the exact ones. In the both approaches the same parameter and the amplitude of the incident wave are the same.

If the layer thickness is small (the Airy series converges),our calculations have shown good agreement between the result of the Fresnel formulae application and the Maxwell-Bloch model (Fig. 1a)

With growth of the layer thickness the situation changes. The Fresnel formula approach gives nearly single decaying wave inside the layer (solid line in Fig. 1b). The intensity of this wave is small, so that the condition (3) holds. However, the exact solution of the system (1) has much larger amplitude (dotted line in Fig. 1b). As a consequence, the influence of nonlinearity is strong, since the value of n is considerablysuppressed below 0n (Fig. 2).

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LFNM*2010 International Conference on Laser & Fiber-Optical Networks Modeling, 12-14 September, 2010, Sevastopol, Ukraine

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a

b

Fig. 1. Field intensity, calculated from the Maxwell-Bloch equations (dotted lines) and from the Fresnel formula (solid lines) in the case of a thin (a) and thick (b) active layer.

Fig. 2. The distribution of the population inversion n in the system, shown in Fig. 1b by the dotted line. The value of n is suppressed, compared to the value of 0n , shown by the dashed line.

Let us determine the condition when the system changes its behavour from the Fresnel-like to the strongly nonlinear one.The Fresnel-like result usually can be obtained by summation of the Airy series, when the transmission of wave is considered through its consequent reflections from the slab’s edges.According to this approach, the transmitted wave amplitude is

given as 00

n

nt t , where 0 02

4 exp1

t i d ,

2

01 exp 21

i d , ' ''i is a complex

permittivity, 0 / c is the reduced frequency and d is the layer thickness. In the case of active medium the series can diverge. It occurs when | | 1 , i.e., when the amplification in the slab exceeds the energy radiation through the boundaries of the slab. The latter condition means the laser generation. This condition determines a specific thickness *d .

Thus, the application of the '' 0 approximation with the Fresnel formula to the description of active media is strongly restricted by the condition *d d . Otherwise, amplification of field in a thick layer suppresses the population inversion,making the medium effectively inhomogeneous. Even if the Fresnel formula predicts small values of field, from which follows the linear behavior, the real system can demonstrate very distinct result.

It is worth emphasizing that the Fresnel formula gives afinite result even for *d d , despite the divergence of the Airy series. Really, the Fresnel’s result is a continuation of the Airy series and has no physical sense. It corresponds to the Airy series that starts with a backward wave that decays inside the layer. The Poynting vector of this wave is directed towards the incident wave.

REFERENCES

[1] A. Fang, T. Koschny, C. M. Soukoulis, “Lasing in metamaterial nanostructures”, Journ. of Optics, vol. 12, p. 024013, 2010.

[2] A.A. Kolokolov, “The plane wave reflection from gain medium”, hETF Pis. Red., vol. 21. No. 11. .660-662, 1975.

[3] L.A. Vainshtein, “The propagation of the pulses”, Sov. Fi . spe hi, vol. 118. No. 2. . 339-367, 1976.

[4] A.A. Kolokolov, “Fresnel formulas and the principle of causality”, Phys. sp., vol. 42. No. 9. . 931-941, 1999.

[5] J. Skaar, “Fresnel equations and the refractive index of active media”, Phys.Rev. E, vol. 73, 026605, 2006.

[6] B. Nistad, J. Skaar, “Causality and electromagnetic properties of active media”, Phys.Rev. E, vol. 78,036603, 2008.

[7] P. Meystre, M. Sargent III, Elements o Quantum Optics, Berlin, Springer-Verlag, 1999.

[8] Y.I. Khanin, Principles o Laser Dynamics,Amsterdam, North-Holland, 1995.

LFNM*2010 International Conference on Laser & Fiber-Optical Networks Modeling, 12-14 September, 2010, Sevastopol, Ukraine

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