THEORY OF A RANDOM FIBER LASER
I. V. Kolokolova* V. V. Lebedev11**, E. V. Podivilovh, S. S. Vergelesa
"Landau Institute for Theoretical Physics, Russian Academy of Sciences bInstitute of Automation and Electrometry, Siberian Branch of Russian Academy of Sciences
Received July 3, 2014
We develop the theory explaining the role of nonlinearity in generation radiation in a fiber laser that is pumped by external light. The pumping energy is converted into the generating signal due to the Raman scattering supplying an effective gain for the signal. The signal is generated with frequencies near the one corresponding to the maximum value of the gain. Generation conditions and spectral properties of the generated signal are examined. We focus mainly on the case of a random laser where reflection of the signal occurs on impurities of the fiber. From the theoretical standpoint, kinetics of a wave system close to an integrable one are investigated. We demonstrate that in this case, the perturbation expansion in the kinetic equation has to use the closeness to the integrable case.
Cwitribvtiwi for the JETP special issue in honor of A. F. Andrew's 75th birthday
DOI: 10.7868/S0044451014120153
1. INTRODUCTION
Wo consider the theory of random fiber lasers. The concept of random lasers exploiting multiple scattering of photons in an amplifying disordered medium in order to generate coherent light without a traditional laser resonator has attracted much attention in recent years. This research area lies at the interface of the fundamental physics of disordered systems and laser science. The idea of a random laser was originally proposed in the context of astrophysics in the 1960s by V. S. Lotokhov, who studied scattering with "negative absorption" of the interstellar molecular clouds. Research on random lasers has developed into a mature experimental and theoretical field. A simple design of such lasers would be promising for potential applications.
In traditional random lasers, the properties of the output radiation are typically characterized by complex features in the spatial, spectral, and temporal domains, making them less attractive than standard laser systems in terms of practical applications. Recently, an interesting and novel type of random lasers that operate in a conventional telecommunication fibers without any
* E-mail: kolokolov'flitp.ac.ru
**E-mail: lebede'flitp.ac.ru
predesigned resonator mirrors was demonstrated. The feedback required for laser generation in the random fiber laser is provided by Rayloigh scattering from the inhomogonoitios of the refractive index that are naturally present in silica glass. In the proposed laser concept, the randomly backscattorod light is amplified through the Raman effect, providing distributed gain over distances up to 100 km. Although an effective reflection due to the Rayloigh scattering is extremely small, the lasing threshold may be exceeded when a sufficiently large distributed Raman gain is supplied.
The random distributed feedback fiber laser has a number of interesting and attractive features. The fiber waveguide geometry provides transverse confinement, and the effectively one-dimensional random distributed feedback leads to the generation of a stationary beam with a narrow spectrum. The random distributed feedback fiber laser has efficiency and performance that are comparable to and even exceed those of similar conventional fiber lasers. The key features of the generated radiation of random distributed feedback fiber lasers include a stationary narrow-band continuous modeless spectrum that is free of mode competition, nonlinear power broadening, and an output beam with a Gaussian profile in the fundamental transverse mode (generated both in single-mode and multi-mode fibers). Details of the random laser performance can be found in recent review [1].
_4'- <_
pump pump ->- •-• -«-
z = 0 z = L
Scheme illustrating arrangement of a fiber laser
2. BASIC DYNAMIC EQUATIONS
The random fiber laser is a piece of optical fiber of length L that is optically pumped from the fiber ends. As a result, randomly backscattered light in the fiber is amplified through the Raman effect, and the system starts to lase at some level of the amplification (see Ref. [1]). Two electromagnetic waves propagating to the right and to the left are generated in the fiber. A schematic distribution of the generated waves along the fiber is presented in the figure. Due to pumping, their amplitudes increase during the propagation and achieve maxima near the ends of the fiber, before passing outside the fiber. We stress that the nonlinear interaction of the generated waves propagating to the right and to the left is weak because their maxima are achieved at the opposite ends of the fiber. Therefore, they can be considered independently.
We begin with the dynamic equation describing the evolution of the envelope of the generation electromagnetic field, over the evolution coordinate i within the fiber, at 0 < z < L, where L is the fiber length. The equation for the generation wave propagating in the fiber to the right is
i(d, = + (i)
where t is the time, 7 is the Kerr nonlinear coefficient, and 3 is the quadratic dispersion coefficient. We consider the generation processes high above the generation threshold and therefore neglect noise terms in Eq. (1). An equation analogous to Eq. (1) can be formulated for the signal propagating to the left, the only difference being in the sign of the derivative d-.
The gain operator g is determined by an interplay of the pumping and the attenuation of the signal. In the frequency domain, it is a frequency-dependent factor
9 = 9RP(z) - a 1,
where gu the Raman gain coefficient, P(z) is the power of the pumping wave, and a:/ is the linear attenuation coefficient in the fiber. The distribution of the pumping over the evolution coordinate i is defined by the factor P(z), which is assumed to be known. The lasing
is realized for frequencies near the frequency where the gain g achieves a maximum. We take the frequency as the carrying frequency for the envelop Then we obtain
g{u) = go - km2, (2)
which is an expansion of the gain coefficient near its maximum. Here, ui is the frequency shift from the carrying frequency. We note that above the generation threshold, the condition g0 > 0 has to be fulfilled.
We stress that in reality, the power P of the pumping wave is dependent on the generation wave they are related via the balance equation fl, 2]. Therefore, the problem should be solved in two steps. First, we have to solve the balance equations to find P(z). Then, P(z) can be involved in calculating Here, we concentrate on the second step.
In a random fiber, almost all generated radiation is coupled out from the fiber end. Only a small part of the energy is reflected back via Raylcigh backscattering processes. Because the amplitudes of generated waves increased during evolution, the scattering process is effective only at the end of the fiber. This implies an effective initial condition for the generation wave propagating to the right, in terms the amplitude of the
generation wave _, propagating to the left. Formally,
the initial conditions for the waves have the form
4'-(LJ) = R,.(^)4'+(L,t),
where f?/ and Rr are reflection coefficients on the left end and on the right end of the fiber, defined in the frequency domain. They have different ui-depcndcnces in different situations. In the case of the random-fiber laser, \R\ -C 1. The reflection smallness leads to the conclusion that the signal is weakly disturbed by the reflection, thus justifying conditions (3).
The spectrum of the generated wave in the random-fiber laser is relatively broad (compared to traditional lasers) and consists of a high number of spectral components near the carrying frequency (see fl]). The main challenge here is to describe the influence of nonlinear-ity on the generation spectrum. For this, we use the standard kinetic approach dealing with averaged quantities. We assume that the dispersion length (/?A2)-1 (where A is the spectral width) is small in comparison with the fiber length L. Then the harmonics with different frequencies possess essentially different phases and therefore, under averaging over a length larger than the dispersion length, the harmonics can be treated as approximately independent.
The main object in the kinetic theory is the pair correlation function
f du)
J
-iíüt
(4)
where angular brackets mean averaging over a distance larger than the dispersion length and "*" denotes complex conjugation. Due to the assumed time homogeneity, the average in (4) depends solely on the time difference t and is independent of However, in examining real fibers, it is useful to average over time (integrate over ti) to eliminate effects related to different fluctuations (noises) neglected in our formalism. We stress that due to the ¿-dependence of the generation wave, the system is not homogeneous in space, in contrast to the time behavior. The function F is no other than the spectrum of the generated signal. We note that the signal intensity I can be expressed via the spectrum as the integral
I
ivf > =
(5)
Boundary conditions (3) lead to the following relations for the averages:
F+( 0,u) = |Í?./(uí)|2F_(0,U;), F_(L,uj) = \Rr(ui)f F+(L,ui)
(6)
where F+ and F_ correspond to the respective generation waves propagating to the right and to the left. In what follows, we consider the symmetric stationary situation where f?/ = Rr and F+(z) = F^(L-z). Then we obtain the condition
F(0,u) = |F(U;)|2F(L,U;)
(7)
for the signal propagating to the right. The condition relates values of the correlation function F taken at different ends of the fiber.
3. KINETICS
We assume weak nonlinoarity of the system. Then a perturbation theory has to be developed to examine nonlinear effects in the random laser. The starting point for the theory is the basic equation (1) for
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