Laser pulsewidth compression by coherent wave mixing in silicon

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1 April 1997

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OPTICS COMMUNICATIONS

ELSEVIER

Optics Communications 136 (1997) 419-422

Laser pulsewidth compression by coherent wave mixing in silicon Renzhong Hua, Liejia Qian, Ximing Deng Shanghai Institufe of Oprics and Fine Mechanics. P.O. Box 8211,2OI800

Shanghai, China

Received 25 June 1996; revised 14 October 1996; accepted 7 November 1996

Abstract We propose a novel method to compress laser pulsewidths by coherent wave mixing in silicon. According to our calculation, a compressed and amplified signal beam which pulsewidth approaches 70% of the input can be obtained with a nanosecond to picosecond laser pulse within the damage threshold of the material. The influence of parameters such as pump beam intensity and wave mixing angle on the compressed pulsewidth of the signal beam is discussed.

1. Introduction

It has since long been discovered that semiconductors such as Si [l], GaAs [2], CdS, CdSe and ZnSe [3] can be used for light amplification of Q-switched or mode-locked laser pulses by coherent wave mixing. The amplification is due to energy transfer from a strong pump beam to a weak noncollinear signal beam by the free-index carrier grating formed by the interference of the pump and probe beams in the media [I]. The light amplification is usually used also for the construction of self-pumped phase-conjugating reflectors and potentially for low noise amplification [4,5]. In earlier works, both analytical [6] and numerical methods [7,8] were used to fit the gain of the signal beam obtained from the experiments to that from the theory. Eichler and Khoo emphasized the influence of the multiwave mixing process on the gain but assumed the pump and signal beams to be constant-intensity square pulses [6,7]. Valley assumed that the laser pulses were of Gaussian temporal shape but he ignored the temporal term in his equation describing the light field [8]. Therefore, we find that little attention was paid to the temporal shaping of the amplified signal pulse in the coherent wave mixing process. In this paper, we study such temporal shaping of nanosecond to picosecond laser pulses by simply using a two-wave mixing model which is justified when the wave mixing angle is relatively large [6-81. We study particularly silicon because its free-carrier index grating is formed by onephoton absorption and its two-wave mixing experimental results are abundant. Our study reveals that the pulsewidth 0030~4018/97/$17.00 Copyright PII SOO30-4018(96)00724-9

of the amplified signal beam becomes quite shorter than that of the input beam when the pump beam is strong enough. As an application, the above pulse shaping can be used as an extracavity compression method. Since in the temporal region from nanosecond to tens of picosecond the methods of both intracavity [9] and extracavity pulse compression are scarce and complicated, this extracavity pulse compression can be applied by users to get the particular pulsewidth they want.

2. Dynamics of pulse compression Although a strict theory of coherent wave mixing should adopt a multiwave mixing mode1 and use numerical computation, we adopt only a two-wave mixing model for the reason which is mentioned in the preceding section and is to be elucidate more in the following. Furthermore, we solve analytically the coupled nonlinear equations describing the two-wave-mixing process by a successive approximation method of order one. We find that the gain and the pulse shape of the amplified signal beam obtained by our calculation agrees with those reported by others quite well [6-8,101. The wave mixing scheme adopts a conventional one (see e.g. Ref. 171) with the pump Ep and signal beam Es propagating in the same direction. The silicon slice is assumed to be antireflection coated on both faces. According to Refs. [6,7], when 8 is greater than 3.9” by a certain value, the high order diffracted beams are quenched by

0 1997 Elsevier Science B.V. AI1 rights reserved.

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interference and we can use a two-wave mixing model including only E, and ES instead of a multiwave mixing model. Let the total light field E and the induced dielectric constant be in silicon be expressed as E=E,exp[i(wt-k:~+k,x)] +E,exp[i(ot-k;z-k,x)],

(I)

A~=~~(z,1)exp(i2k,x)+~;(z,t)exp(-i2k,x), (2) where Ed and l z* are the Fourier components of A E; k.+, k,, are the projection of the wave vector in silicon on the z- and x-axis, respectively. The electric fields of the input pump E, and signal ES are

[=z,

7)=t-u/z

and used the well known relationship I = in\/EO/CLO EE * . After some lengthy but straightforward manipulation, we get

.!&,=Aexp[-21n2(t-z/U)2/r~],

(3)

E,, = fiA

(4)

exp[ - 2ln2(r - z/u>‘/Ti],

IS = I,” 1 + G’ exp( - 2r]/~)

where me is the initial pulsewidth, u is the group velocity, M is the ratio of the pump beam energy to that of the signal beam. Following Refs. [I$], the equations goveming the electrical field of the pump and signal beams and e2 are , 1 “Ep C?E 2 + --= -iS,Ese2FE,,

x {I + @[2m

~0’ xexp

JZ

I

1 c?E,

a6

az+---= de2

F=pEpEs*

-iSoEpe;

-IL.

-$Es,

i-

In the above equations

p=

mind that we usually will make use of the amplified signal beam for other applications where a certain pulse energy is needed. Under the above assumption, we have E, = Ez. In addition, we assume Es = E$ + E: when solving Eq. (6). Here EF and Ef are the solutions of Eqs. (5) and (6) without taking account of the nonlinear coupling between E, and Es. To obtain the term E,’ resulting from the nonlinear coupling we use l2 obtained from Eq. (7) by approximating E, and Es in that equation with E: and Ef. Furthermore, to get E,‘, we have adopted the transform

-2naN,_,/hv,

a’=~(1

7-l = 4D,kj,

+a,.,E,,/hv),

where k,, kZ, k, are the vacuum wave vector and its projection on the z- and x-axis respectively, 8 is the wave mixing angle between the pump beam and the probe beam in air, A, is the vacuum wavelength, h is Planck’s constant, v is the optical frequency, n is the refractive index of silicon, LYis the linear absorption coefficient, (Y’ is the absorption coefficient including the electron-hole plasmaattenuation effects of all beams [6]. Ne_,, is the electronhole dispersion volume, D, is the ambipolar diffusion coefficient, T is the lifetime of the free-carrier index grating, a,., is the free-carrier cross section, Edp is the energy density of the pump beam. To solve Eqs. (5)-(7), we assume that MB- 1 which means that the pump to probe ratio is large and pump depletion can be neglected. However, for practical use we should not let the signal beam be too weak by keeping in

i 8T21n2

(71- r~/8rln2)/ro]}2),

[l -exp(-a,Qj*

1

2 al

where cy, = 3a’/2, d is the thickness of the silicon slice, I,, is the peak intensity of the pump beam. Since we have obtained implicitly the expression of the amplified signal beam intensity, we will give some numerical examples of both a 20 ns and a 50 ps laser pulse. For the parameters related to silicon, we adapt after Refs. [&II], (Y= 10 cm-‘, NC_+,= -lo-*’ cm3, hv= 1.875 X 1O-‘9 J, n = 3.5, d = 0.4 mm, D = 10 cm’/s and o-., = 5 x IO-‘* cm’. As the wave mixing angle, we use 8= 6” (T= 2.5 ns) for the 20 ns pulse and 0 = IO” (r= 900 ps) for the 50 ps pulse. We take the above values of laser pulsewidths, silicon thickness, and wave mixing angle for the purpose of comparing our calculated gain to that from the references. Another parameter to be determined is the pump to probe ratio. We adopt M = 50, which ensures that when the pump energy is several mJ, the signal beam can be amplified to a useful level as one-tenth of several mJ. Fig. l(a) shows the curves of the intensity gain or of the energy of the signal beam (gain is defined as [signal(presence of pump) - signal(n0 pump)]/signal(no pump)) and the compressed pulsewidth versus pump beam intensity. Estimating from Fig. l(b) of Ref. [6], a 16 ns signal beam can get a gain of about 3.5 at an input pump energy of 80 mJ/cm2. This value is just what we get from Fig. 1 of our paper. As the pump energy density becomes larger than 80 mJ/cm2, the gain from our model deviates from that from the experimental results because of the simplicity of this model. As we have shown, our model

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(4 --gain -gain

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the gain occurs when the nanosecond or picosecond pump beam energy density approaches 100 mJ/cm2, we can conclude that our calculation is already instructive for the application of the amplified and compressed signal beam. To have an idea of the energy of the input and output signal beam energy which is crucial in an application, we assume that the spot size of the spatially Gaussian distributed input beams are 4 mm and the pump beam energy is 10 mJ (i.e. pump intensity 80 mJ/cm*). According to our calculation, when M equals 50, the amplified signal beam energy approaches 1.2 mJ which can be useful in many applications. If higher pulse energy is preferred, we need simply to put the compressed signal beam into an ordinary laser amplifier.

14 0

10

20

30

40

Pump Intensity

50

60

70

80

@J/cm’)

50

48

‘;: 4

36 0

10

20

30

40

Pump Intensity -40

-30

-20

-10

0

10

20

30

40

can predict

60

70

80

(mJ/cm’)

1

tw Fig. 1. (a) Dependence of the compressed pulsewidths and gains of intensity and energy of the 20 ns signal beam on pump beam intensity. The wave mixing angle is 0 = 6” and pump to probe beam ratio is 50. (b) Compressed pulse shapes widths at the pump intensity of 0, 20 mJ/cm’, 40 mJ/cm*, 80 mJ/cm2, respectively.

50

0.9 the the the and and

the gain of coherent wave mixing quite well, we can expect that it can also be effective in predicting the pulsewidth of the amplified signal beam. From Fig. 1 we conclude that the width of the compressed signal pulse decreases rather fast with increasing pump intensity. A pulsewidth approaching 70% of the input pulse is easy to be obtained. An experimental verification can be found in Fig. 3 of Ref. [12], where the input signal pulsewidth was 20 ns, and it became about 14 ns after coherent wave mixing. When the pump intensity approaches 50 mJ/cm’, the compression of the pulse begins to saturate and we can get only a little more compression by further increasing the pump intensity. In the experiment, because saturation of

0.8 2 .z s2

0.7 0.6

2 0.5 S E 0.4 L? ‘;i 0.3 0.2 0.1 J

0

-100 -80 -60 -40 -20

0

20

40

60

80

100

t(ps) Fig. 2. (a) Dependence of the compressed pulsewidths and gains of intensity and energy of the 50 ps signal beam on pump beam intensity. The wave mixing angle is 8 = 10” and pump to probe beam ratio is 50. (b) Compressed pulse shapes widths at the pump intensity of 0, 20 mJ/cm*, 40 mJ/cm*, 80 mJ/cm’, respectively.

the the the and and

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136 (1997) 419-422

large scale to avoid damage and make full use of the amplification effect of the signal beam in a coherent wave mixing process under mJ of pump beam energy. We have also investigated the influence of r (which can be varied by changing the wave mixing angle as shown in Fig. 3) on the pulse compression. From Fig. 4, we find that r changes the gain quite strongly but changes the compressed pulsewidth very little, therefore we prefer a small mixing angle in practice. But we should not allow the angle to be too small because multiwave mixing processes will dominate and our two-wave mixing model will break at small wave mixing angle.

0.01

‘,“““““‘,““““““““““,J 0

5

10

15

3. Summary 20

2.5

30

35

eo Fig. 3. Dependence of the grating decay time caused by ambipolar diffusion on the wave mixing angle in air.

Fig. 2 shows the results of the 50 ps input pulse. The are very similar to that of the 20 ns pulse except that the gain of the picosecond pulse is somewhat greater than that of the nanosecond pulse. Since the damage threshold is about 300 mJ/cm’ for the 50 ps laser pulse [6] and gain saturation occurs at a pump energy density of 80 mJ/cm*, the spatial beam size should be expanded at a results

We have proposed a novel method to compress the laser pulsewidth by two-beam coupling in silicon. A compression factor up to 0.7 can be obtained easily for both nanosecond to picosecond pulses. This effect can be used to generate short pulses at circumstances where shorter pulsewidth is preferred. It can also be used in a regenerative amplifier to obtain more short and powerful laser pulses. Although our analysis is primary related to silicon, we think our theory can be extended after some adaptation to other semiconductors which show two-beam coupling effects.

References 9

II] V.L. Vinetskii, N.V. Kukhtarev, S.G. Odulv and M.S. Soskin, -gain of intensity --..gain of eye

8

Usp. Fiz. Nauk. 129 (1979) 113. [2] M.B. Klein, Optics Lett. 9 (1984) 350. [3] K. Jarasiunas and H.J. Gerritsen, Appl. Phys. Lett. 33 (19781 190. [4] A. Krumins and P. Gunter. Appl. Phys. 19 (19791 153. [5] H.J. Eichler, P. Gunter and D. Pohl, Laser-Induced Dynamics Gratings, Springer Series in Optical Science, Vol. 5 (Springer, Berlin, 1986). [6] H.J. Eichler, M. Glotz, A. Kummrow, K. Richter and X. Yang, Phys. Rev. A 35 (19871 4673. [7] I.C. Khoo, Ping Zhou, P.G. Lindquist and P. Lopresti, Phys. Rev. A 41 (1990) 408. [S] G.C. Valley, J. Dubrad and A.L. Smirl, IEEE J. Quantum Electron. 26 (1990) 1058. 191 A. Penzkofer, Appl. Phys. B 46 (1988) 43. [lo] I.C. Khoo, Appl. Phys. Lett. 52 (1988) 525. [ll] T.F. Bogges, K.M. Bohnert, K. Mansur, S.C. Moss, I.W. Boyd and A.L. Smirl, IEEE J. Quantum Electron. QE-22 (1986) 360. [12] H.J. Eichler, J. Chen and K. Richter, Appl. Phys. B 42 (1987) 215.

__..______ ...-. ---.-+ __. __ ./-.-z._.__ __ ..._.,.

7 FI ‘2 6 i? 5

_/

‘:: _..__._.....---=j

._:

,.,I

pump:80mJ/cm*

;.’

4

3 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.x

Fig. 4. Dependence of the compressed pulsewidth and the gains of intensity and energy of the signal beam on the grating decay time. The pump to probe beam ratio is 50 and the pump intensity is 80 ml/cm’.