l. Introduction.....................

I,I E

W

I i..i

ST

F{

UM

E

T\g

TS

A N D M EA$ U A E M

HN

T

Ti'X

ET

HOM

$

phase phensrnena in parametric anrplifiers and generator$ of ultrashac't tight pulsesl) A. Piskarskas, A. Stabinis, and A' Yanka"uskas Y. Kapsukas State Uniuersity, Vilnius fir. Nauk 150, 127-143 (September 1986)

Utp,

in the dev-eloprnent of This article reviews recent theoretical and experimental investigatiorrs respoirsible t'or Phenomena oscillators. parametric femtosecond lasers hased on optical quadratic susceptibility mediarvith pulses in producing tunable high-poweifemtosecond pulies xith of self-compression parametric time, real in (purur..tii" chirp, cliirp reversal strong eorrg] exchange) are discussed'

CONTENTS

l.

Introduction.....................

.................869

2. Parametric amplification in a phase-modulated pumping

3. Parametric amplification of time. '.. '.. "...'... 4. Pulse self compression

field""""'""""""""""

phase-mod:Ti:::::.::1.:::::.,::."::1::

with stiong energy exchange

""""""""""""v""""""""'

5.I Experiments or pafametric amplification with phasc'modulated pumping'

870

;?; 873

5.2

Experiments on parametric amplification of phase-modulated pulses and chirp reversal in real time.

Conclusions..............'..........'......'........... References..... 6.

l.INTNODUCTION '

The development ofsrrurces of high-power tunable femil

!, i.

! l

tosecond light |ulses is a problem of great importance in laser technology;. Its solution opens up entirely nerv prrssibilitics in the physics oifast prcce;ses' Ai the present time the tbr:hniques of generating extremely short light pulses (a few femtoseconds at the visible wavelengths) arc based mostly on the formation of phase-modulated pulses in rnedia with cubic nonlinearity, for instatice in optical fibers.r and the subsequent compression of these pulses in devices with nega-

tive group veiocity dispersion or by self-compression in an opticai fiber.2r Compressors based on optical Iibers make it possible to compress pulses to 8 fs.s At the same time, mode' locked dye lasers have recently been developed that produce 30 fs pulses.6 However, the energy irr the pulse s produced-by these systems is usually small, not exceeding l0-8-10-e J, so that it is necessar:y to amplify these pulses, a task that is beset with great diffrculties.T We would prefer, in this review, to refer our readers to putrlicationsr-ro devoted to the methods mentioned above, and concentrate on'novel rnethods ofcontrclling the phase of light ivaves, methorls based on parametric interaction of the light waves. The most recenr work has shown that these methods are very promising for the purpose of generating high-power femtosecond tunabie pulses. Substantial progress has been made over the past ten years in tire developrnent of picosecond and ferntosecond redratica sources based on parametric amplification and

859

Scv. Phys. Usp. ?9(S), September 1986

""""""''""877

""""""""'

878

generation of light waves in quadratically nonlinear media, the so-called optical parametric oscillators. Optical parametric oscillators combine a broad trrning range (from the ul' traviolet to the infrared ) , a broad frequency band, and a high gain, rvith vilrious possibilities of changing rhe arnplitucle aud phasc cl;rauteiistics ofthe puises. The physical principles of optical pararnetric aEciiiators and possible schemes for tuning were outlined in 1962 by Akhmanov and Khokhlovrr and also by Kroll12 and Kings-

tort.I3 Optical parametric oscillators

in the nanosecond

pulse-length range were devetroped for the first time in the work reported in Ref. 14. An important stage in the development of parameffic generators of picosecorrd and femiosecond pulses was the appearance of theoretical investiga-

tionsri'r6 which introduced the idea of paramttric corlpretsion oflight pulses under conditions ofstiong energy exchange and group velocity mismatch. Subsequentiy, larametric g.r.ruiin, Lf tignt pulses in the picr:secondrT'rB iange, and iomewhat later in the femtosecondre'ro pulselength range \Yas attained. 1Ve stress that the pttise conipression observed in these studies was the result of anrpiitude modulation and group velocity mismatch of the interacting light pulses. Irt ac.cord rvith the theory, the maximum degree oicompression did not exceed one order of magnitude, and rs'2o ths pulie leugths varied Irorn 300 to 900 fs In receni investigationtzl-21 11 ryas shorvn that subsiantial progress in the femtosecond pulse iength range was posof siUle Uy the use rrf three'wave parametric interactions investigathese of phase-modulated liSht pulses. The results

0038-5670i 86/C90859-1

1

$01 .80

@ 1987 American lnst:tute o{ Physics

..i:q+r-*iF$ -{.J.-:sw}ilFryl

869

***-,r"-*T*'aa=

JIr.i4it{:'!r

tioos and the ideas formulated in them make it possible to approach in a new way the problem of fast control of the phase of the light wavcs. To the present time studies have been maile of the efect of phase modulation of the pumping

light pulses on parametric amplificatiofl,2o'2S and processes sueh as conyersion of quadratically phise modulated pump-

ing (linear frequency variation-the frequency .,chirp,,1zr

iato phase modulation of the generated pulses,zt'22 parametric generation of broad-band pulses (a continuum).2c2e sign'reversal of the frequency modulation, "chirp reversal'.' ir parametric amplifiers,23 parametric pulse compression under conditions ofsecond order dispersion,3&32 soliton for-

mation,3 r'33'31 and others. The results that have been obtained leave no doubt as to the high degree of promise inherent in this direction of investigation. The balance of this review is devoted to a specific discussion of these results. :

i:[

i

We shall assume thar /(1,r, and yrlz/(Lr&*) 41, where I length of the nnnlinear meclium. In this approximation

is the

:

df:

-io.l,,t('r) Bi,

- l/Lbro)"'oro7

11

vgr

:6-

-af, i

pulses with quadratic phase modulation At: qro exp[( - (t'/4)(l + iy) ]. Let us introduce the dimensionless time q : t /r3, the group interaction lengths Z, . : rAvzrl, and Lr, : rr/lv3l, and change to the new functions

+:

: I - pt andp, -

-io,a,

(r-=;)

Az.*p

[;pr7,

(r-#)']

,

"

(2)

vur/vp. We have

.*p

as a

result

(-iy,q,- iy,*)

AB.,

dB":

t

-Lozas (tt

B;,

and

Lr,) /. We note that the value of

a,

:

#r*p & [-L1r * q,) -,p,y,q,] xI

exp Iip,1, {,e ,u1q-ao1

-t-

* ) "' rl, (n .

(

ao) exp [

-

i

pztt (q

870

exp

(-iT,'I,- ,h *)

Sov. Phys. Usp. 29 (9), September 1996

Bi.

-

(q*"r)r] or),]i

e-

, Az:

(*)"'exp ( - i.7r1z).Ai; where a : rr/a3; where r. : (lLn1)',,lvrrlrfl

x' dx, (s)

Let us note some regularities that can be inferred from

tance of this fact later. Here rve point out that the phase conjugation is maintaiRed with accuracy to quantities of the order yrl also for puarping with phase-modulated pulses. The quantity r. determines the parametric amplification bandwidth Aou for monochromatic pumpjng. In particulal,

setting Tt:0, Arc-6(t), and Azo:0, we have l1-exp( --t'/1 ) and the amplification ga6rlu,iclth. Aa;.

:4$ni/lv2l 1/(lL^t), rvhich is analogous ro the resulrs of Ref. 35. If the fact tirat the amplificatiorr bandwidtlr is bounded is not relevant (a-0), then in the case Azo:0 it follows from (5) thar Ar: (Arc(T)/2) expUt /L)"1 (l - n') l. Phase modulation of the pumping tras no efect whatever or the signal (copropagating) wave. Moreover, it

is ndteworthy that phase modulation of the pumping is entirely superimposed on the idler (outgoing) wave, a result that agrees rvith the conclusions of Ref. 2*. lf it is of basic importance that the amplification band is Lounded (a*0), then the effect of phase modulation of the pumping on wave generation in the optical parametric oscillator is more complicated. Using the Fourier transformation.

A$(t):#

J tr(o:)exp(iorr)

dor,

-6

we have

:*.-o[*(t*+)] &

x

(4)

caliulated for a'pump pulse length rr. Recovering the functions l, and A2,we have

ri(,)

z\

-,')

Bi.

@

(3)

-;+T;;,I

(q)

waves at the output of the optical parametric oscillatoi are phase-conjugate, We shall discuss the fundamental,impor-

u, arethe group velocities ofthe waves, and o, and a, are the coupling consrants.'A coordinate system is chosen in which the first pulse is stationary. We notethat Eqs. ( I ) are written in the approximation of a specified pumping field. The solution of ( I ) in general form involveithe integrals of the Riemann function (see Ref. 25) and its ahalysis is difficult. We shall analyze ( 1) for the case of Gaussian pun:p

Ar: Ar.exp (-iprr;rri2), ,4:

-do,d,

(5). In the absence of phase moclulation of the pumping; the

*: -io,/, (t,-i,z) Ai, (l) aA. aA^ # + v*:# : - iazAz $ - ttrrz) Al, whereA, ( j:1,2,3) are the complex wavi amplitudes, ,tlLl

,*+* #:

is"

'

'tx:T -;; '

are important oniy when the

ary conditions at z: O: Rp(n) : Ap(rt) exp( ip2yrTr). The salution of Eqs. (4) is known (see Ref. 35). We shall assume that the arrplification is large, /y.I., where I,,

J

ory20:

v,

The system of equations (4) are to be solved with the bound-

PUMPING FIELD

Let us examire horv.new features arise in parametric generation of light waves in a phase.modulated pumping field. The equations that describe the three-photon interaction of plane light waves in a nonlinear medium have the following form in the first approximation of dispersion the-

and

thus obtain

2. PARAMETFIC AMPLIFICATIOI.I IN A PHASE.MODULATED

wherep,

z,

the rnismatches

phase modulation of the pumping is taken into account, We

x

J

s,o1toy.*p

[i,r- *

(.o

*4#)'] Piskarqkas

a,. ror

elal

870

"

le8sFlil

I

phase modulation).

i

a2h

@to

2 ffil0G l- Frcquency shift ofthe amplification band ofthe signal wave a;, and in an optical parametric oscillator pumpea''Uy u puf." *rif,

rye. *1I.

r,

firlar chirp.

(D

ft c.n be seen that amplification with,pumping by phase_ @cdulaied pulses is equivalent to amplification with u ti*e r&ift in the amplificdtion frequency band (Fig. l ). We note th*t the physically similar effect of a signal of changing fre_ lpency on a resonant system has been studied.previously {c-9., Ref. 36). The central frequency in the band contour of tte signal wave amplification varies fls d), : etro _ Zpryrt / rl. Taking into account th &t@r: atro Zyrt /ir,*. oUiuir, -

fs the frequency deviation Aator:prAdt and Aa;., :pzLoy If lp,l(l and lprl Tv'pipulse' but onlv for the signal piisio" it ,g;in ;;ssible,

shown in Fig' 2d'lf Ldt/Lar) l' then ifr. .pr""aing of the signal pnlse becomes unimportant and of the pul*i do*t not depencl on the sign of ti.

relatedtothatofthecurvesinFig'2;Themiaimumlengths

I"ipuft. prJfilo"t.

-

"o*pt.tJion k',i,rk 'l,t'

'-'

frirrt.

(curves + shows the Iength of the signal pulse

1

)

point of maximum and thJpumping-a pulse (curves 2) at the k i'k as iun"tio, of the ratio L,o, /Lo, for

""*p*Jti..

pulses arc obtained in the region of mirxithe interact"o*ptessed mum inten;ity. It can be seen that the lengths of

Liii.

i'

r"srlt of compression can be.an order of ma[pulse' niiuae smaller than the length of the initial pump compulse for conditions In conclusion we note that for obtained be can pr.rrlo.rri, their three'wat-e interaction

;;;;;i;"t;.

u

and can be exa wide range ofdispersion spreading lengths

proi,"a fotiubstaniial

elecrease

in the lengths and increase in

the intensities of the Pulses5. EXPERI[IEHTAL RESULTS

Experiments on parametric conve'rsion of phase-mod-utated pumping, parametric amplification of phase-modula'

parametric ehirp and Block diagram ofthe apparatus for studying lascr' 2) second c.moression of optical 4) compressor' 5) harmonic senerator, pulse lengih measuring device, 6) dynamic interleromeler'

FIG.

(2) pulses at the point ofmaxi' FIG.4. Length ofthe siinal ( I ) and pump for k irk'i' > 0 (solid tatio Lor/Lo3 lhe of function mu* comprxsion as a curves) and

874

k';t k';t