Stretched pulse shaping and measurement

Optics Communications90 ( 1992) 21-26 North-Holland

OPTICS COM MUNICATIONS

Stretched pulse shaping and measurement G.R. Boyer and M.A. Franco Laboratoire d'Optique Appliqu~e, Ecole Polytechnique, Ecole Nationale Sup~rieure de Techniques Avanc~es, 8atterie de l'Yvette, F91120 Palaiseau, France

Received 5 November 1990; revised manuscript received 15 November 1991

We study the temporal structure of spectrally filtered chirped pulses with a view to a quantitative evaluation of the self-phase modulation in chirped pulse amplification lasers. The analysis takes into account diffraction and finite beam size. Temporal diffraction-like effects are theoreticallyand experimentallydemonstrated. These effects are predicted to be harmful to chirped lasers.

1. Introduction

There is a strong current interest in high-power ultra-fast laser for studies of new interactions in very high (solid-like) densities for which chirped pulse amplification (CPA) lasers present the best possibility for generating the required intensities [ 1 ]. In such systems it is critical to control the front edge of the giant pulse, since a relatively low precursor would induce a significant physical effect on the sample. Multiple pulses of variable delays may also be required. The CPA technique uses a pulse stretcher in which the pulses undergo a large positive group-velocity dispersion ( G V D ) , followed by an optical amplifier and a compressor that compensates exactly for the second-order GVD introduced by the stretcher and the amplifier (fig. l ), making the overall system a nondispersive one. A typical pulse stretching system has a region where the Fourier components are

A

Fig. 1. Block-diagramof pulse-synthesis in chirped-pulse amplification lasers.

separated, offering the possibility of a spectral filtering. This suggests pulse synthesis in CPA lasers similar to the spectral filtering that has been done in a temporally nondispersive grating apparatus [2,3]. For CPA pulse synthesis, however, one has to make sure that the nonlinear effects taking place in the amplifier do not induce any significant self-phase-modulation (SPM). To some extent, high intensities in the amplifier are required to achieve again saturation that is desirable for the efficient use of the available pump energy and to help stabilize the pulse-topulse amplitude fluctuation, increasing SPM to a level where it becomes significant. Fast, diffraction-like fluctuations eventually induced in the pulse spectral filtering may prevent complete recompression. To perform an exact evaluation of that process, we need an analysis of the temporal structure of filtered chirped pulses beyond the classical Fourier theory taking diffraction and finite beam size into account. The shaping of chirped pulses was proposed and demonstrated for the first time by Agostinelli et al. [4 ]. Their simplified Fourier analysis gives an incomplete description of the phenomenon and the experiment using a streak camera provides a temporal resolution of one picosecond, which is not sufficient for femtosecond time analysis. More recently, a complete analysis of the optical temporally nondispersive pulse shaping grating apparatus has been carded out by Danailov and Christov [ 5 ]. In this paper, we present an extension of the analysis of ref.

0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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[5] to temporally dispersive systems and the first experimental results at a temporal resolution of 70 fs, opening a sub-picosecond time resolution technique for the measurement and control of the time profile of the stretched pulses in CPA lasers. This technique is also potentially useful to ultra-high bitrate data transmission in single mode optical fibers using pulse expansion-recompression [6 ].

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plane allows the use of a different spatial filter in each path. Using spherical lenses, the same possibility is offered by using a plane mirror tilted in the y direction instead of the retroreflector (fig. 2). In both cases, the beam exits the stretcher above beam-splitter BS.

3. Spectrally filtered stretched pulse amplitude 2. Description of the experimental setup In our stretching system (fig. 2) the pulses are diffracted by grating G~, filtered at plane F, diffracted again at G2, and directed on a second pass by retroreflector M. The lenses have focal length f, the gratings have groove spacing s. Grating Gi is a distance zi from the focal plane of Li. zi is negative when G~ is between the focal plane and Lg. The angle 0o and ~'o, and the coordinates x, x~, x2 are shown in the fig. 2. Dispersion takes place in the x z plane. The y dependence of the amplitude is implicit, and given by propagation in free space. After a complete transit, the beam exits the pulse stretcher at a diffraction angle Yoand suffers lateral walkoff [ 7 ]. The walkoff can be compensated with a dihedral retroreflector that sends the beam on a reverse transit with a different height. If the lenses are cylindrical, the spatial separation of the forward and reverse paths in the F

[6]

ref. pulse ~

1

crosscorrelator

aout(X2, to) = a ( t o ) exp(ikfl2to2z)

I

× e x p { - i k x 2 / [ 2 q ( d + 2aez) ] },

stretched pul,'~ltering d)

plane F

T~

G 1

~.-

J

M

fz f -

Fig. 2. Experimental setup for chirped pulse measurement by crosscorrelation. 22

We now present an analysis of the filtered stretcher. Our analysis makes use of an expression for an expanded pulse emerging from a stretcher without filtering [7]. We propagate the pulse back to plane F, filter it, and then propagate back to the output. This "backwards propagation" method may avoid considerable amount of algebra if one can use simple optical imaging relations, or previously established results. To simplify further, we make the assumption that the spatial filtering takes place only in the second transit of the stretcher and the G~ is in the front focal plane of L~, so that zj =0, and all the group-velocity dispersion comes from the position of G2, placed at a distance z2 of the back focal plane of L2. We consider a gaussian input pulse in time and space of complex amplitude a (x2, m ) = a (x2) a ( to ), of spatial and frequency dependent parts a(x2) and a(to), respectively, and of 1/e amplitude time halfwidth Zo. The angular frequency shift to the central frequency too is to. After a double transit without any filtering, the outgoing stretched pulse amplitude is

( 1)

where fl= m22/2ncscos Oo is the angular dispersion o f G t , m the diffraction order, c the velocity of light, 2 the wavelength. With our assumption of Zl = 0, z reduces to z2. The complex radius of curvature is q(d) = d + i k a 2 / 2 , where a is the waist size and d is the distance between G~ and the beam waist. The argument of the function q ( d + 2a 2z) that accounts for the double transit is unchanged through this paper and will not be represented from now on. The wavenumber is k. In eq. (1), an unsignificant multiplicative complex constant has been dropped. The grating G~ is the second disperser that the pulse encounters in the reverse transit. The angle of inci-

dence of the central beam on Gt is 0o. After dispersion on G~, the beam exits the stretcher at the angle 70. The relation between the incident amplitude a (x~) on Gl and de diffracted amplitude a' (x2) is given by

[7] a' (x2, to) =boa(to) exp(-ikfltox2/ot)

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a(x2/ot),

Assuming a gaussian envelope for the input pulse amplitude, we express the spectrum as a ( t o ) = e x p ( - t o z / 4 a o ) , where Oto= 1/3o2. The pulse temporal profile is the inverse Fourier transform of eq. (6) with respect to 09. Performing the integration over to, and dropping the complex constant gives the complex amplitude of the filtered output pulse

(2a)

where a = -cosTo/COSOo, and the set of constants {bi}, (i=0, 4) is introduced through this paper to conserve pulse energy and to simplify the expressions [5,81. The inverse of transfer operator defined in eq. (2a) serves to deduce the incident pulse amplitude knowing the output pulse collimated by GI:

2

anlt(X2, t,=exp[- ~--~(t-kfl~) ] +oo

X J F(Z) e x p ( - v x 2 + w z ) dz. The parameters p, v, w are defined as

(2b)

P= 4ao

1

ikfl2(z +

Inserting eq. (2b) in eq. ( 1 ), we get for the pulse amplitude just before G~ (and for the reverse pass)

iq

(1

a(xl, to) =boa(to) exp(ikftoxl) a' (otxl) .

a, (x,, to) =a(to)

×exp[i(kflZtoZz+kfltoxl-kxEctZ/2q) ] .

(3)

The pulse amplitude at the back focal plane F is the Fourier transform of eq. (3). We take the spatial filtering into account by multiplying the result by F(Z), the filter function to find

a(z, to)=b~a(to)F(z) ×exp{i[kflEtoEz+q(kflto-X)2/2ka2]} .

(4)

In the Fourier plane, z=kx/f is the spatial frequency. Propagating the beam to Gt is equivalent to an inverse Fourier transform a~ (xl, to) = bEa(to)

exp(ikfl2to2z)

W=

~

q)

'

_ikfl2z )

qflt + 2akflZx2z + ix2 a/2Oto 20t2p

(8)

Setting z= 0 in eq. (7) reproduces eq. (6) of ref. [5]. We can recover eq. ( l ) for the unfiltered stretched pulse by setting F(X)= 1 and z ~ 0 in eq. (7). To analyze the filtering of a slit of full-width ~, we set: F(Z) = 1 for IzI ~<~/2 and F(Z) = 0 for IzI > ~/2. Integrating eq. (7) gives O~0/2

aslit (X2, t) =b4 exp X[erf(~

+oo

(7)

-co

--1 -~q

1-4ikotofl2z

2~v) +erf(~

+ 2~v)] .

(9)

× ~F(z)exp[i(2--~ot2(kflto-Z)2+Zx,)]d)c. --oo

(5) Dispersion on Gl is achieved by using the transformation (2a) in eq. (5), giving for the complex amplitude of the spectrally filtered emerging stretched pulse a'l (xl, to) = b2a(to) exp (ikfl2to2z -

The exponential contains a real and imaginary term representing the time expansion and chirp of the unfiltered pulse, respectively, while the expression within brackets is the slit-filter dependent partf(z), where z is defined below. The time-expansion factor G can be seen to be [ 1 + (4kaofl2z) 2] l/2,~4kaofl2z,

ikfltox2/ot )

for G>> 1 .

+oo

X f F(Z)exp[i(2-~2 (kPto-X)2+X~)]dz. --oo

(6) 23

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4. Experimental In our experiment, the 70 fs pulses are delivered by a dispersion c o m p e n s a t e d C P M oscillator a n d m o n i t o r e d with an autocorrelator a n d a spectrometer coupled with an Optical Multichannel Analyser. At the output o f the oscillator, the b e a m splitter BS (fig. 2) separates the reference pulses and the pulses to be stretched. A delay line on the reference b e a m permits us to overlap the two pulses within the 0.1 m m thick BBO crystal o f a crosscorrelator. Lenses L, and L2 are spherical and plane m i r r o r M is tilted to provide separation o f the two spectra in plane F. The experimental p a r a m e t e r s are as follows: 2 = 620 rim, n u m b e r o f l i n e s / m m 1800, f = 0 . 5 m, a = 2 × l0 -3 m, ~= 1 . 1 X l 0 -3 m a n d 0 . 1 < z < 0 . 2 8 m. The order o f magnitude o f the p a r a m e t e r s o f interest can be specified as k ~ 1 0 7 m -m, f l ~ 3 . 2 × 1 0 -m6 s, o r ~ l ,

OtO~ 1026 S-2, i q ~ 2 0 m, so that eqs. ( 8 ) simplify to p ~ _ i q k f l E / 2 a 2, v ~ i z / k , w,~ it/kfl. We choose to use the dimensionless variables u = ~ / ( k r A a l ) and z = 2 t / ( G r o ) , where Ato is the full-width-at-half-maximum o f the intensity spectrum o f a(oJ) as defined above, that is Ao~=2.35/ro. After elementary algebra, we express the slit width d e p e n d e n t part o f ( 9 ) as f(z) =erfIx/~

(0.59u+r) ]

+erf[x//~ (0.59u-z) ] ,

(10)

a n d use I f ( r ) 12 for our c o m p a r i s o n to experimental data. Multiple slit-filtering has been handled similarly by introducing the p a r a m e t e r r that measures the center-to-center distance between two slits in the same dimensionless units as p a r a m e t e r u. Fig. 3 shows a c o m p a r i s o n o f experimental and theoretical curves o f the outgoing pulse spectrally filtered by a single slit for G = 170 a n d u = 1.30. The

(B) (B)

1.0 i

1.0

.5

(A)

.5

(A) 0.0

-lO0 O

0.0

t00.0 ,

,

.

.

.

.

.

,

0.0

Normalized time

(t/%)

Fig. 3. Theoretical (A) and experimental (B) temporal profiles of a spectrally slit-filtered stretched pulse. The stretching factor G is 170, the u parameter 0.18. Curve (A) is obtained by crosscorrelation with the CPM laser pulse output. The theoretical result is for a gaussian input pulse. The asymmetric profile of the experimental curve is due to an asymmetric spectrum of the input pulse. 24

-100.0

0.0

Normalized time

100.0

(t/Xo)

Fig. 4. Theoretical (A) and experimental (B) temporal profiles of a double-slit filtering with G= 170, u=0.18, r=0.38. Discrepancy in the symmetry of curves (A) and (B) can be explained similarlyto fig. 3.

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horizontal axis is the normalised time t/Zo. The two curves show that diffraction-like fringes are formed by the edges o f the slit and may induce unwanted selfphase modulation for CPA applications. Fig. 4 shows the effect of double-slit filtering. The relevant parameters are G = 1 7 0 , u = 0 . 1 8 , r=0.38. The corresponding output shows two subpulses, as previously mentioned [4], but also considerable ripple due to diffraction-like effects, as depicted in the theoretical plot (fig. 4A). Fig. 5 illustrates the mapping of five spectral bands into five subpulses for G = 3 5 6 , u--0.18, r--0.36, again with a significant diffraction-like structure. We are now able to estimate the effect of such intensity fluctuation on the spectral broadening occurring in the amplifier. As an example, an initially gaussian pulse of 100 fs, o f beam size 2 mm, stretched to 100 ps and amplified to 250 mJ by a solid-state amplifier of self-focusing coefficient 1.8 X 10-16 cm2/W,has an output intensity of 83 G W / c m 2. Assuming that the maximum gain takes

(B)

L

,,.,.~Ul,,~dL..am~

u

IO

(A)

0.0 -2000

0.0

2000

Normalized time F i g . 5. T h e o r e t i c a l ( A ) a n d e x p e r i m e n t a l

1 J u n e 1992

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(t/Xo)

(B) temporal profiles

....... .....~i.

iv ~ i.......ir ~ i.......ir"-"~ i.......ir ! i ~ !i

i......

10

...... •~....... i------~....... .L.....,.'....... .L.....,.' .... .L.....i......~ ....... i.......

...... -~....... ;-.---.~....... L- --i....... i-...-.-i....... ~-.-.-.i....... E....... i....... -5

-10 614

616

618

620

622

624

626

Wavelength (nm) F i g . 6. C h i r p m e a s u r e m e n t

o f a 170 t i m e s e x p a n d e d p u l s e . T h e

curve shows the time delay of a narrow band-pass filtered pulse a s a l i n e a r f u n c t i o n o f t h e slit p o s i t i o n s c a l e d in w a v e l e n g t h s .

place over a length of one cm in the amplifier, fig. 5 shows that a half-full modulated pulse time variation o f 0.83 ps induces a frequency shift of 110 GHz, representing 3% o f the initial pulse spectrum. This spectral broadening prevents exact recovery of the synthesised pulse after recompression. A direct chirp measurement where the phase of the spectral components was obtained by measuring the time position of the correlation curve. Following ref. [9 ], we have used a translatable slit in the Fourier plane, and plotted the time T(2) of the m a x i m u m o f the cross correlation as a function of the slit position scaled in wavelength. Fig. 6 shows a very linear variation o f T(2), and the slope was measured to be 3.7× l0 -2s s 2, a value that compares well with the 3 . 4 × l0 -25 s 2 deduced from the experimental data, i.e.: z-- 0.11 m; G = 170. One limitation to such cross correlation measurement is the decrease of the signal to noise ratio with increasing stretching factor. We obtained reasonably noiseless profiles o f pulses stretched o f a factor 1600, thus demonstrating that the present experimental setup is relevant to CPA lasers technology. 5. Conclusion

of a stretched pulse spectrally filtered be five equally spaced ident i c a l slits G = 3 5 6 ,

u=0.18,

r=0.36.

In conclusion we have considered, in some detail, 25

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the t e m p o r a l structure o f spectrally filtered pulses. The basic features agree well with experimental data. Significant diffractionlike effects have been demonstrated to be harmful to pulse synthesis in chirped lasers. The limitations o f spectral filtering on the recompression process are now u n d e r study. Finally, we have shown that the experimental m e a s u r e m e n t o f the t e m p o r a l profile and the chirp o f the e x p a n d e d pulses by crosscorrelation is useful for the control o f the stretching and recompression process in CPA lasers.

Acknowledgements Laboratoire d ' O p t i q u e A p p l i q u r e is Unit6 de Recherche Associre 1406 o f the CNRS. This work was partially supported by the European Economic C o m m u n i t y Stimulation Action under contract no. SC1-0103-C. We thank Dr. J. W h i t e for helping us to present a better manuscript. We thank the referee for helpful suggestions.

26

l June 1992

References [ 1] M Pessot, P. Maine and G. Mourou, Optics Comm. 62 (1987) 419. [2] C. Froehly, B. Colombeau and M. Vampouille, Progress Optics XX, ed. E. Wolf (North-HoUand, Amsterdam, 1983 ) p. 65. [3 ] A.M. Weiner, J.P. Heritage and E.M. Kirschner, J. Opt. Soc. Am. B 5 (1988) 1563. [ 4 ] J. Agostinelli,G. Harvey, T. Stone and C. Gabel, Appl. Optics 18 (1979) 2500. [ 5 ] M.B. Danailov and I.J. Christov, J. Modem Optics 36 (1989) 725. [6] G.R. Boyer, M.A. Franco, M.K. Jackson and A. Mysyrowicz, Nonlinear guided-wave phenomena topical meeting, Cambridge, UK, 1991, p. 150. [7] O.E. Martinez, IEEE J. Quantum Electron. QE-23 (1987) 59. [8] O.E. Martinez, J. Opt. Soc. Am. B 3 (1986) 929. [9] J.L.A. Chilla and O.E. Martinez, Optics Len. 16 ( 1991 ) 39.