Saturation of second harmonic spectral intensity with increase in frequency half-width of exciting radiation

Volume 41A, number I

PHYSICS LETTERS

28 August 1972

SATURATiON OF SECOND HARMONIC SPECTRAL INTENSITY WITH INCREASE IN FREQUENCY HALF-WIDTH OF EXCITING RADIATION V.D. VOLOSOV, S.G. KARPENKt~,N.E. KORNIENKO and V.L. STRIZHEVSKII Department of Radiophysics, Kiev State University, Kiev, USSR Received 1 May 1972 It is shown that the growth of second harmonic spectral intensity with the increase of the frequency half-width ~ of the incident collinear nonmonochromatic laser radiation of fixed maximum spectral intensity is saturated when ~ is greater than the synchronism width for frequency addition.

The problem of second harmonic (SH) spectral density of light intensity (or in short spectral intensity) 1(w) is of essential importance in SH generation by slightly divergent (practically collinear) but nonmonochromatic radiation. Non-triviality of this problem is caused by the fact that SH field with the given frequency w is formed in general not_only by doubling processes but also by addition processes of side frequencies equidistant with respect to ~2[1]. These addition processes cause an increase of 1(w). The aim of the present article is a theoretical investigation of this increase and its maximum possibilities depending on laser radiation bandwidth, index-matching bandwidth and other parameters. The analysis is carried out in the fixed laser field approximation based on the methods described in [2,3]. Following notations are used: k and kc are the wave vector lengths of forced and free harmonic waves respectively [4], x = is the frequency deviation of laser radiation with respect to its central frequency ~ 1 the thickness of parallel plate nonlinear crystal. Superscripts o, e denote ordinary and extraordinary waves respectively, and subscript o denotes the central frequencies of fundamental or harmonic waves (w0 = 2~~); v1,2 are the group velocities of the central frequency components, ‘1,2 the spectral intensities, ‘i ,2 the integral over frequency intensities; subscripts I and 2 correspond to fundamental and harmonic fields respectively. The radiation is assumed to spread along the surface normal which coincides with the index-matching direction for SHG for exciting frequency ~ The aperture length [5] is considered to greatly exceed I. Finally, G is a known [2—4] function describing SHG by monochro-

matic fundamental radiation at frequency ~ in the index-matching direction, in this case G = Numerical calculations are made for KDP assuming SHG excited by a broadband Nd-glass laser radiation (the index data were from ref. [6]). According to refs. [2, 3] ~

=

2G

f sinc2zl1(w

-_

f2)I (~2)d~,

2

(1) sinc z

z

=

-‘(k

—

k) I

z 2 c The value ‘20 = 12(w0) in the center of harmonic line is most interesting, and we shall concentrate our attention to its investigation. For a linear approximation of kc k with respect to frequency deviation x we have z = a’x, where n = 0 and a = 4l[(v~)~—(v~)] _l] for on-e and oe-e types of interaction respectively. Consider at first the case of oo—e interaction. For the sake of definiteness we assume a Gaussian freq~uencydistribution for the fundamental intensity (in fact the quantities investigated here depend weakly on line shape), i.e., ‘

—

1/2

=

I

~/

exp [_

L

~ ~2

ç~)21 0

J

.

(2)

Then after substituting (2) in (1) and taking into account that sinc z = 1, we find ‘20 = (2ln2/1r)h/2GI~0& where 110 = 11 ~ If fundamental spectrum is broadened and spectral intensity in its center ~ is fixed, the value ‘20 increases proportionally to ~- The physical reason of this growth is in the broadening of frequency band in which addition processes may take 31

Volume 41A. number 1

PHYSICS LETTERS

place with the formation frequency w0, all generated forced waves being synchronous. However, when ~ is sufficiently large the approximation linear over x becomes invalid. Inly,the 2, where = y isquadratthe secic approximation z = jIx ond derivative of fundamental wave vector length with respect to the frequency in the point f7 = f2 0. Now the integral (1) has the form ~

—~

28 August 1972

The maximal accessible value of ‘20 corresponds p ~ I and forms I~. Now consider the case of oe—~einteraction. Here ct * 0, ~and frequency addition oe--ethe is synchronism already finite width in the for linear over x approximation. Then

~2o= ‘T0~<”~~)~

jSX

2) -_

‘21) = ~)O R(p), R(p) = {l + [(4p4 —

p

4p2(p =

(I

J~ff = 8 ln 2/(3~.~/fi~) GI~06, 1)(p + 1)1/2 (3)

P(v) =

1(p) + { exp (—p l.39/(2ln2)2~/6.

(2 In 2/1 .39) GI~06,

=

l} /~f~

=

(4)

l)21/(p2112))/(2p~),

+ 4p4)112

p

=

(l.39/(21n2)~2~/6 ~/6

;

o represents the half-width of the function sinc2z and is given by 0 = ~ = 3.33/(/Iyj)1/2. The decrease of the function sinc2z when x increases leads to the rise of phase mismatching of forced waves which are generated by the side frequency addition. Therefore the magnitude 8°°~ may be called the width of synchronism (index-matching) for frequency addition when SH is generated by nonmonochromatic radiation in the case of oo—e interaction (by analogy with the width of synchronism for frequency doubling [2, 3}). We can say that 0 represents the maximal fundamental spectral width, for which the phase mismatching may yet be neglected. If one accounts for the dependence of z on x, it means taking into account the finite of magnitude 5, while in the linear over x approximation 0 The behaviour of the function 120(P) at ~ fixed is shown in fig. 1. -*

~.

Here 1 is the probability integral and the magnitude 6 is6 = 60e-e = 7.561 hI(v?)~I-~(v’I~. In fig. I the behaviour of the function P(v) is also shown. Thus in both cases as ~ increases the magnitude ‘20 also increases, reaching the saturation in the region ~ > 0. Therefore the broadening of the fundamental spectrum (when ‘10 is fixed) up to ~ 6 may be used as a method of SH spectral intensity increase. The maximal accessible values I9J~are seen from (3), (4) to be proportional to 5. Usually the value of 500 ~e greatly exceeds 50e~~ For KDP these values are I 770A and 25A (I = 1 cm). Therefore when ~ ~ ~ it is preferable to use the oo—e interaction for increasing 120 by incident spectrum broadening. If ~ < Ooe---e both types of interaction are practically of the same efficiency.

~o-~ e 350

_________

________

_________

Gee

________ 0

____

___

-

___

2~,5

_______

o

_________

—---

___

_________

~~—--~_

20~

2

3

Iig. I.

32

~

4

5

Fig. 2.

__________

Volume 41A, number I

PHYSICS LETTERS

In fig. 2 the relative gain of the spectral intensity ~20(~)/~20(~= SA) is shown for KDP. (For oo—e interaction in the region of large i~the results have the illustrative nature.) it is obvious that the accessible values of ‘i are large enough (up to some tens) and apparently are of practical importance. It should be noted the effect discussed can also produce the saturation of the conversion efficiency J2/J1 when the increase in J1 is due to the increase in ~ with fixed 110 and when i~exceeds 0, but is greatly less than the width of synchronism for frequency doubling. =

28 August 1972

The authors acknowledge the referees for paying attention to the above discussed possibility of conversion efficiency saturation and for helpful comments. References Ill V.D. Volosov and 121 131

RB. Andreyev, Opt. i Spectroskopiya 26 (1969) 809. V.L. Strizhevskii, Opt. i Spectroskopiya 20 (1966) 516. V.L. Strizhevskii, S.G. Karpenko and A.V. Bugayev,

141

Opt. i Specktroskopiya 29 (1970) 953. S.A. Akhmanov and R.V. Khokhlov, Problemy nyelineynoy optiki (Akad. Nauk SSSR, Moscow, 1964).

151

S.A. Akhmanov, A.P. Sukhorukov and AS. Chirkin, Izv. Vuz’ov, Radiofiz. 10 (1967) 1639.

(61

F. Zernike Jr., J. Opt. Soc. Am. 54(1964)1215.

33