Transient Regime of a Dynamic Distributed Feedback ... - IEEE Xplore

533

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. 5 , MAY 1987

Transient Regime of a Dynamic Distributed Feedback Laser: Theory and Numerical Analysis P.A.APANASEVICH,A.A.AFANAS'EV,

AND

M. V. KOROL'KOV

Abstract-A semiclassical treatment of transient behavior of a gaincoupled, distributed feedback laser is developed. The nonlinear response of the lasing medium to the spatially periodic pump and the effect of self-induced grating are involved in the theory. The influence of a noise signal value on the generation kinetics is demonstrated. The possibility of practically unidirectional lasing is also shown.

ence has been obtained and solved numerically. Although these equations are more exact as compared to the averaged equations of the efficient Q-factor model [3]? [4], the pumping power is supposed to be small herein. This considerably restricts the scope of applicability of the results obtained in [ 111, since excess over the threshold by several times is typical for DDFB dye lasers [6]? [ 121. ThereI.INTRODUCTION fore, the construction of a theory free from the above reONSIDERABLEadvanceshave been made in the strictions seems an urgent problem [3], [4], [ll]. area of creating and developing dynamic distributed In the present paper, a system of nonaveraged integrofeedback (DDFB) dye lasers. These lasers are of special differential equations is obtained and numerically solved interest for obtaining frequency-tuned, ultrashort pulses for DDFB laserpulse amplitudes, which more completely (USP) of light. In early works [l], [2] on USP generation take into account the effect of radiation on the active meby DDFBdyelasers picosecond pulses were used for dium. The equations obtained describe the transient repumping. Later the possibility of generating light USP by gime of generation in the three- and four-level active mesuch lasers at nanosecond pumping was proved theoreti- dia with the DDFB on the light-induced grating of the cally [3], [4]and supported experimentally [ 5 ] , [7]. The inverse population at arbitrary pump powers and lasing shorter duration of generation pulses as compared to pump frequency w and Bragg frequency wB detuning from the pulses of DDFB lasers is due to relaxation oscillations of gain line center wo. of thelasing, causing DFB Qtheinversepopulation switching. The possibility of generating frequency-tuned 11. EQUATIONS FOR A DDFB LASER high-power USP of light holds much promise for practical The most commonly used method of creating DDFB is application of DDFB dye lasers [6]-[9] and offers an in- by pumping the active medium by two light beams concentive to investigation of different regimes of their op- verging at an angle 213. By varying the angle 28 smooth eration. tuning of the lasing frequency is achieved. In this case the Thetheoreticaltreatment of the transient regime of pumping rate W ( z , t ) = ( ap3 p / A w p ) is spatially moduDDFB lasers consistsin analyzing a system of differential lated equations in partial derivatives with boundary conditions given at the edges of the active medium. We managed to obtain analytical estimates only for the efficient Q-factor model of the active medium [3], [4]. This model is valid for small excess over the threshold of single-mode gen- where A = a / k p sin 8, kp = w p n ( w p ) / cand 7 are the eration and for characteristic times of laser pulse ampli- period, the wavenumber, and the visibility parameter of tude change much longer than the time tL the laser light pumping, respectively; n ( w ) is the refractive indexof the transit along the length of the active medium ( tL = L / u, active medium, ap is the absorption cross section of active where L is the active medium length and z, is the velocity molecules at the pumping frequency up,=Ip is the pump of light in the medium). Interms of this model the system intensity. To describe the process of lasing we use well-known of equations in partial derivatives describing the transient semiclassical equationsfor polarization 6 , theinverse DDFB lasing may be reduced to ordinary differential population N , and the electromagnetic field strength E. equations analogous to the balance equations of a FabryThe field is assumed to consist of two counter-propagatPerot cavity laser [lo]. In [l 13, to describe the DDFB ing waves lasers,a set of three-coupled equations for generation eikZ + E. e-'Wf + C.C. (2) wave amplitudes and lasing transition population differE(z, t) = [E,

C

Manuscript received June 18, 1986; revised November 11, 1986. The authors arewith the Institute of Physics, B.S.S.R. Academy of Sciences, Minsk, U.S.S.R. IEEE Log Number 8613517.

with slowly varying amplitudes E,(z, t ) , frequencies w , and wavenumbers k resonant to the periodic structure of (1)

0018-9197/87/0500-0533$01.00O 1987 IEEE

534

IEEE JOURNAL OFELECTRONICS, QUANTUM

(./A)

-

k = A, I A

I

+ 1. Hr . (I,,,+ 1

( ~ ~ e""' )

0

is)

[ E +eiki + E-e-"]

. e-iW'+ C.C.

-

(4)

Z,,,-~(S')

e-'\l'+)] e -Si'+

irnil.7'

dr'].

Cat + ( W + i ) - N - ( W - E ) * N o

+ 2 Re [ ( E T E - ) e-""]}

*

(5)

N

where 5 = 0 or 1 for the four- and three-level active media, respectively, p , T I and w0 are the dipole moment, the population relaxation time and the lasing transition frequency,respectively; No is theactive particle density, a ( $ , W ) = [ 4 I p I2T1 T 2 / ( 2 - '3) * ( 1 S 2 ) ] , 6 = T2

Z,,, is the m's order-modified Bessel function. Expansion (8) explicitly contain analytical expressions for the amplitudes y,,, of the mth spatial harmonics of the active medium population. Substitution of (8) into (4) and then into the wave equation

+

*

(0

- wo).

Equation (5) differs from the analogous equation of [ 111 in term N W , which takes into account saturation of the lasing transition by pump radiation (see formulas (6) and (9) of [ 111). Taking into account (l), the solution of ( 5 ) with the initial conditions y ( t = 0 ) = N ( t = 0 )/ N o = - 5 may be written as

-

$1

,d(Z,T')dTl

-

(6)

3

where x = z / L is the dimensionless coordinate, G = 2a k I p l2N0 Tz / A is the unsaturated gain, + = tL/ T I , e is the medium permeability.

-

where

r2(z,T )

makes it possible to obtain reduced equations for generation wave amplitudes, which correctly takes into account the nonlinear response of the medium to the spatial-periodic pumping of (1) [ 131:

+

= ~ H ( T ) 2 Re [ € ~ 8 - e 2 i A Z 1 / (+1

r 3 ( ~ ,T ) =

2

(7b)

(12)

~ r n [ € ~ ~ - e ~+ 'a2). ~ ~ ] / (( 7~4

So, unlike work [11] where for the spatial harmonic amplitudes of populations ym an infinite chain of equations (see addition to [ 113) is obtained, the present approach makes it possible to obtain equations for generation wave amplitudes in a closed form. In solving (10) numerically, the missing noise Sources of generation fields are modeled by nonzero boundary and

We have introduced here normalized values of the dimensionless pump power H ( t ) = W ( t ) TI/2, the time T = t / T I andthegeneration amplitudes 8, = E , a l l 2 . Let US further express formula (6) as expansion in terms of spatial harmonics

-

535

APANASEVICH et al.: TRANSIENT REGIME OF A DFB LASER

2

I

z

3

(a) 3*h!

30

t i

3b

= 2’3 20

io

3

I

Fig. 1. Intensity 3* (7).= I G , ( x = k0.5, 7 )(* as a function of time 7 = t / T , for 9 = 1, 8: = lo-*, 5 = 1, 7p = 1-(a)-(e), and 7p = 0.5-(f).

initial conditions

€,(x

=

~ 0 . 57,) =

E+(x, 7 = 0) =

Et

7 2

0

I

5

Ix

(134

0.5.

(13b)

Note that the values of the fields €1may vary over a relatively wide rangeaccordingto experimental conditions. In [ l 11, E t is equivalent to one photon in the excited region. Comparing oscillograms of DFB-laser generated pulses [ 6 ] ,[ 141 with those calculated for different = low2as typical valvalues of E t , we have chosen ues. An analysis of (10) for the steady-state regime where the processes of the development of lasing may be neglected is carried out in [ 151.

Et

111. GENERATION KINETICSOF A DDFB LASER In this section the results of the numerical solution of (10) are presented for the case of total frequency resonance ( 6 , A E 0 ) and the Gaussian shape of the pump pulse H ( x , .) = H,(,-) . e - [ ( 7 - 2 T ~ f / T ~ 1 2 (14) where rp = $ / T I , tp is the pump pulse length, H,(x) is the maximum pump power. The generation kinetics of the DDFB laser is largely

determined by the relation of the laser pulse transit time ( t L )along the length of the active medium and the population inversion relaxation ( p parameter). For typical generation intensity oscillationstake values of p 5 place (see Fig. l(a) where 3 + (7) = 1 € , ( x = + O S ; 7 )12). As the parameter p increases, these oscillations are damped. In the near-threshold region of gain [see Fig. l(b)] the regime of single short pulse generation is realized. As the linear gain G ( G L parameter) increases the time dependence of the generation intensity becomes more and GL = 10 complicated [Fig. l(c)].Thus, at p = a train of short spikes is observed at the leading edge of the generation pulse (solid line). The intensity of these spikes may considerably exceed that of the stationary generation (at H , = 4 and tp >> TI). For greater GL values with the same p generation occurs at earlier times, the duration of individual generation spikes decreases with increasing GL. The presence of the extended trailing edge is dueto the generation changeoverto the quasi-stationary regime. To elucidate theeffect of the “self-induced” grating on the generation kinetics of the DDFB laser, we obtained solutions of (8) excludingthe“self-induced”grating, i.e., the coefficients r2and r3were assumed to be equal to r2= q H and r3 = 0, respectively [see (7b) and (7c)l. The results of these calculations are represented in Fig.

536

IEEE JOURNAL OF ELECTRONICS, QUANTUM

l(d) by dashed lines. It is seen from comparison of solid (with “self-induced” grating) and dashed lines that the “self-induced’ ’ grating has a significant influence on the DDFB laser generation kinetics increasing the lasing radiation oscillations. In experimentally feasible DDFB laser schemes the pump power is not always uniform along the length of the active medium, i.e., H , = H, (x). It follows from the general considerations that in this case output pulse intensities at right- and left-hand edges will be different 3+( 7 ) # 3- ( 7 ) (the symbol “ - ” denotes pump inhomogeneity). Fig. l(e) gives the results of the numerical solution of (8) with pumping power changing linearly along the lenglth of the active medium H, ( x ) = H, ( 1 x). In this case the shapes of both pulses are similar, but the output lasing power 3+( 7 )from the side of higher pumping (x = 0.5 ) considerably exceeds the radiation power from the opposite edge of the active medium (x = -0.5 ). The calculations show that as the pulse j H, (x) * & = const), with asymmetry increases (at appropriate parameters there is an increase in the asymmetry of the output lasing powers, and unidirectional output of radiation is realized. It is noteworthy that with the same parameters and homogeneous pumping power H, (x) = const the output lasing powers satisfy the inequality 3+( 7 ) > 3 (7) > 3- (7). The comparison of the solutions of (8) for the threeand four-level active media ( 5 = 1 and 5 = 0, respectively) is shown in Fig. l(f). It is seen from comparison of corresponding curves that the basic laser parameters will have the same qualitative effect on both the three- and four-level media. It should be noted that the generation kinetics of the DDFB laser is essentially determined by the relations between all the parameters characterizing the properties of its active medium and the pump radiation. Therefore, a change in one of the parameters, the values of the others being different, may lead toqualitativechanges in the generation kinetics. The above analysis makes it possible to formulate the conditions of generation of a single USP by a DFB-laser: a shortpump pulse tp < TI, a short length of the active medium L < UT,, modest excess over the threshold gain coefficient GL 5 4, and a high-quality interference pattern of pumping 17 = 1. The number of spikes and the modulation depth of radiation are also very dependent on the amplitude values of the input signals €: (see Fig. 2), simulating noise sources at the lasing frequency. Fig. 3 shows the effect of the lasing radiation on the gain and the wave coupling coefficient (parameters yo and yt ). It is seen that in the range of the parameters under consideration every generation spike causes the appearance of a hole in the time dependence of these coefficients. Moreover, the inverse population of the medium occurs at a later time than 7’ in which the pump power reaches the threshold value H ( 7‘ ) = Hth.Fig. 4 shows spatial dependence of the lasing intensities [Fig. 4(a) and (b)], the “self-induced” grating amplitude ( [ 3 + (7,x)

VOL. QE-23. NO. 5 , MAY 1987

.. ..

.. .. .. t

.

... ...

+

-E:;

-

I

085

1.5

z

Fig. 2. The dependence of the generation kinetics on the boundary condition for I” = 0.01, GL = 4, H , = 4, T~ = 0.5, 5 = 0, and 9 = 1.

005

I

~5

Fig. 3. Time-dependence of the functions y 0 ( 7 ) , y * , ( T ) , 3* ( T ) at the H ( T ) for GL = 4, boundaries of the DDFB-laser and pumping power H , = 16, p = 0.01, 5 = 1, 9 = 1 , E t = lo-’, and T,, = 0.5.

3-

(7,x

) ] ’ / 2- ( c ) and yo and y+l factors Fig. 4(d) and

(e). IV. CONCLUSIONS We have investigated the transient regime of a laserwith DFB on the population grating formed in the three- and

APANASEVICH e iREGIME al. : TRANSIENT

537

OF A DFB LASER

z

2.0

1.0

0

(e) Fig. 4. Dependence of space- andtime-intensityof one of the coupled waves 3+ (7,x ) , the integrated intensity 3+ ( 7 ,x ) + 3 - ( 7 ,x ) , the selfinduced grating amplitude [ 3 + ( 7 , x ) * 3- (7,x)]’”, and the parameters of gain y o ( 7 , x ) , and coupling y 5 , (7,x ) for GL = 4, H , = 16, p = 0.01, 5 = 1, = 1, E $ = lo-*, and T,, = 0.5.

four-level active media by the pulsed space-periodic field of pump radiation. A system of two nonlinear integrodifferential equations has been obtained for slow generation field amplitudes describing the generation kinetics of the DFB laser with pic0 and nanosecond duration of pump radiation (t,, >> T , ) . The possibility has been shown of practically unidirectional output of radiation from the laser pumped by radiation with intensity varying along the length of the active medium. The generation conditions of single USP have been analyzed.

REFERENCES [ l ] V. A. Zaporozhchenko, A. N. Rubinov, and T. Sh. Efendiev, “Ultrashort pulse generation in a distributed feedback dye laser,” Pis ’ma Zh.Tekh. Fiz., vol. 3, pp. 114-116, 1977. [2] B. A. Bushuk, V. A. Zaporozhchenko, A. L. Kiselevskii, A. N. Rubinov, A. P. Stupak, and T. sh. Efendiev, “Single ultrashort pulse generation o f radiation in a dye laser with light-induced DFB,” Pis ’ma Zh. Tekh. Fiz., vol. 5, pp. 880-882, 1979. [3] G . A. Lyakhov and N. N. Suyazov, “A calculation of the DFB laser output parameters (a four-level efficient Q-factor model),” in Proc. 2nd All-Union Conf., Lasers Based on Complex Organic Compounds, Appl., IFAkad. Nauk B.S.S.R.., 1977, Minsk, U.S.S.R., pp. 39-41. “A theory of unsteady distributed-feedback laser action: A model [4] -, of efficient Q-factor,” Kvant.Elektron., vol.10, pp. 1572-1580, 1983. [5] 2s. Bor,“A novelpumpingarrangementfortunablesinglepico-

second pulse generation with a NZlaser pumped distributed feedback dye laser,” Opt. Commun., vol. 29, pp. 103-108, 1979. [6] A. N. Rubinov, T. Sh. Efendiev, “A dye laser withlight-induced distributed feedback,” Kvant. Elektron., vol. 9, pp. 2359-2366, 1982 (english translation: Sov. J . Quantum Electron., vol. 12, pp. 15391543,1982). [7] Zs. Bor, “Picosecond pulse generation dye to self-Q-switching in a DFB dye laser,” Izv. A N S S S R , vol. 48, pp. 1527-1533, 1984. [8] A. D. Das’ko, V . M. Katarkevich, A. N. Rubinov, S . A. RyzhechIzv. AN kin, and T. Sh. Efendiev, “Holographic DFB dye lasers,” SSSR, V O ~ .48, pp. 1522-1526, 1984. [9] A. Piskarskas, V. Smilgyavichyus,A.Umbrasas, and P. Chesnulyavichyus,“ADFBdyelaserpumped by subnanosecondand nanosecond light pulses,” Pis’ma Zh. Tekhn. Fiz., vol. 10, pp. 526529, 1984. [lo] Calculation Methods f o r Optical Quantum Generators, B. I. Stepai Tekhnika,1968,vol. 11, p. nov, Ed.Minsk,U.S.S.R.:Nauka 656. [ l l ] I. N. Duling 111 and M. G . Raymer, “Time-dependent semiclassical theory of gain-coupled distributed feedback lasers,” IEEE J . Quanfum Electron., vol. QE-20, pp. 1202-1207, Oct. 1984. generationof a [12] E. I. Zabello and E. A.Tikhonov,“Singlemode dynamic DFB dye laser,” Pis’ma Zh. Tekhn. Fiz., vol. 5, pp. 10541056, 1979. [13] V. N.Abrashin, A. A. Afanas’ev, T. V. Veremeenko, and M. V. Korolkov, “Unsteady regime of laser generation with dynamic distributed feedback,” preprint IF AN BSSR N 266, p. 28, 1985. “Calculationmethodsandmathematical modeling,” in Proc. All-Union School Young Scientists, Moscow, 1984, pp. 160-162. [14] V. I. Bezrodnyi, E. I. Zabello, and E. A.Tikhonov, “Stimulated emission of frequency-tunable ultrashort pulses from a dynamic dis-

538

JOURNAL IEEE

tributed feedback laser,” Kvunt. Elektron., vol. 11, pp. 2438-2442, 1984. [15] A.A. Afanas’evand M. V . Korol’kov,“Steady-statestimulated Kvant. Elekemission from a dynamic distributed-feedback laser,” tron., vol. 11, pp. 739-744, 1984 (English translation: Sov. J . Quantum Electron, vol. 14, pp. 500-503, 1984).

OFELECTRONICS, QUANTUM

VOL. QE-23, NO. 5, MAY 1987

A. A. Afanas’ev was born in 1942 in Maly Uzen’, Saratovregion. He graduatedfromtheSaratov State University in 1967. For his work on quantum electronics he received the degree of Candidate ofSciences(physicsandmathematics) in 1973. His current research interests are in nonlinear optics, laser physics, and dynamic holography.

P. A. Apanasevich was born in 1929. In 1954 he graduated from the Byelorussian State University, Minsk. Forhis work on the quantum-electrodynamic theory of radiation-matter interaction hereceived the degree of Candidate of Sciences (physics and mathematics) in 1958. In 1974 he received the doctoral degree in physics and mathematics for fundamental work in the field of nonlinear optics and quantum electronics. Since 1954 he has been a member of the Institute of Physics of the B.S.S.R. Academy of Sciences. He became a Professor in 1977. His current research interests are in nonlinear optics, laser physics, and theoretical spectroscopy. of Sciences. Dr. Apanasevich is a member of the Byelorussian Academy He received the Science and Technology Prizes from the government of the B.S.S.R. (1978) and U.S.S.R. (1982).

M. V. Korol’kov was born in 1954 in Khotimsk, Byelorussian S.S.R. He graduated from the Byelorussian State University in 1976. In 1985 he received the degree of Candidate of Sciences (physics and mathematics) for his work in the theory of the DFB laser. His current research interests are in nonlinear light scattering, distributed feedback lasers, and dynamic holography.