- Email: [email protected]
Evolution of classical light statistics in second harmonic generation
Volume
53, number
OPTICS COMMUNICATIONS
4
15 March
1985
EVOLUTION OF CLASSICAL LIGHT STATISTICS IN SECOND HARMONIC GENERATION P. CHMELA Join1 Laboratory of Optics of Czechoslovak Academy of Sciences and Palackj University, Leninova 26, 77146 Olomouc, Czechoslovakia Received
4 December
1984
Classical solution of coherent second harmonic generation process based on the averaging of stochastic quantities over the initial photocount distribution in fundamental input radiation is presented in this paper. The evolution of light statistics in second harmonic generation with chaotic input radiation is calculated for arbitrary values of the time or space parameter.
1. Introduction Coherence and statistical phenomena in nonlinear optical interactions have been the matter of interest of many authors in recent years (for review, see refs. [l-71). The process of second harmonic generation (SHG) in a dispersive medium was described classically, without regarding the light fluctuations and coherence or spectral properties of radiation in [8]. The effects of partial coherence, many mode structure and classical field fluctuations in generating fundamental radiation on SHG process were demonstrated in [9-141 and the double enhancement of the rate of SHG with chaotic input radiation, with respect to coherent one, was observed in ref. [15]. The effects of amplitude and phase fluctuations of fundamental radiation possessing a defined spectral width in SHG process, including also dispersion of the nonlinear medium, were studied in [ 1,161. The quantum theory of nonlinear optical interactions was developed [ 171 and the effects of spontaneous break-up of second harmonic photons into fundamental photons [18,19], as well as the emission processes [20] in SHG were studied. The time evolution of the variance of photon number in the second harmonic mode was numerically computed in [ 191. Shorttime solution of nonlinear optical quadratic interaction up to the second order, using quantum description, was 0 030-4018/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
given in [ 13,21,22], and antibunching in fundamental mode [23-251 and second harmonic mode [24,25] at the beginning of SHG was stated. Recently, squeezing effects have been reported in both fundamental [26-281 and second harmonic [27,28] modes in SHG process. Although great effort has been exerted on the development of a consistent theory of SHG process, including statistical and coherence properties of the inter, acting radiation, the majority of treatments provide the descriptions of the time or space evolution of the field in nonlinear optical interaction either in the approximation of the given generating field, or by means of the numerical computations, or in the short-parameter approximation only. No complete analytical solution of the problem in a closed form has been found yet. In this paper we present a simple classical solution of SHG problem in a closed form, based on the averaging of stochastic quantities over the initial photocount distribution in fundamental mode. A similar treatment using short-parameter method up to the eighth order was given in ref. [29]. Such treatment is suitable for the description of coherent nonlinear optical interactions with generating radiation exhibiting strong classical light fluctuations only. Unfortunately, it does not include the fine quantum effects like photon antibunching and squeezing. On the other hand, it enables us to describe the evolution of classical statis279
tical properties of radiation for arbitrary values of the time or spatial parameter in a good approximation. Especially, it is suitable for the description of SHG with chaotic [12-141 or many-mode laser [lo-121 radiation. 2. General method The degenerate nonlinear optical interaction of two modes of radiation at o and 2w in a non-dissipative quadratic medium can be described, from the point of view of classical theory, by means of two coupled firstorder differential equations for the normalized stochastic field amplitudes [1,8,29-321 da,/dg
= -2iga*,a2,
,
The initial conditions amplitudes are a,(O)
= -bv;
,
qw(0)
of SHG for stochastic field
=0 .
(2)
Moreover, the classical description assumes strong fundamental input radiation, i.e. it holds for the mean input photon number that (n w,~) = (au,oaL,o) S 1 ,
(3)
where the angle brackets denote an ensemble average. Solving eqs. (l), with respect to initial conditions (2), we obtain the following solution for stochastic photon numbers (see e.g. [8,30,31]) n,([)
= n,,O sech2(21/2nzfogQ
,
(44
(la) (lb)
where the variable .$ represents either the time (t = t), when considering the time evolution of the field in an optical resonator, or the normal distance from the first boundary (t = z), when considering the interaction of travelling waves by passing light through a nonlinear crystal plate. g is the real coupling constant that is different for the descriptions in the temporal domain and in the spatial domain [29-3 11. The field amplitudes aj (j = o, 2~) are normalized in such a way that the quantity ni = ajaT represents either the stochastic photon number in an optical resonator or the stochastic photon flux in the normal direction to the nonlinear plate boundaries, respectively to the considered model of nonlinear interaction [29,31]. In the further text we shall call the quantity ni simply “photon number”. However, it is necessary to stress here that the normalized amplitudes or photon numbers are classical quantities and they can be simply replaced by usual quantities as follows: at the description in the temporal domain the radiation energy of the jth mode in an optical resonator is given by Uj = Fiiojnj; at the description in the spatial domain the light intensity of thejth mode is $ = fiUinj/cos pi; A being the Planck constant and pj is the angle of refraction for ray direction in an anisotropic medium. The phase matching [30-321, as well as the conditions for “coherent” interaction * [ I,29 ] are assumed. * Small dispersion of nonlinear medium and great coherence time OI coherence length of input radiation are assumed.
280
,
=a,,0
n2,(t) @,ldE
15 March 1985
OPTICS COMMUNICATIONS
Volume 5 3, number 4
= fn,,o
tanh2(21/2nz20gt)
.
(4b)
If the photocount distribution of fundamental input radiation p (n w,o) is k nown, we can calculate arbitrary moments of the type (ns(!$ nf,(Q> as follows [33]
(nE(E)n&(D)
=J
ng(Ln,,u)
0
= 2-4
s
0
n$‘Aq) sech2J’(21/2nzfo ’
g.$)
X tanh2Q(21’2n~~ogUp(n,,0)dn,,0 . (5) For the further treatment it is advantageous to introduce the reduced variable T that is defined for different models of the nonlinear optical interaction in the following way [29-321 7 = 21i2(n w,o)“2g4
=
I )
PI) 312 %,O) lQw cos 62, cos p, EO
112 cos p2* I
(6)
15 March 1985
OPTICS COMMUNICATIONS
Volume 53, number 4
where eO is the electric permittivity and /.+Jmagnetic permeability in SI units, vw and v2w are the indices of refraction in an anisotropic medium, 6, and 62, are the anisotropic divergency angles, 0, and 02” are the angles of refraction for ray directions, xi? (2~) is the effective nonlinear susceptibility and V is the effective volume of an optical resonator containing nonlinear medium. The efficiency of SHG will be described by the relative mean photon numbers
with respect to eqs. (4) (5) and (7) (n, (T))~ = sech2 (7) ,
(loa>
(qw(7))~ = i tanh2(r).
(lob)
All reduced moments Q!“)(r) (j = w, 2~; m > 2) and (’ reduced correlations Q,.$& (7) @ t 4 > 2) are zero, from the point of view ot classical theory, in this case.
2. SHG with chaotic input radiation = owj(T)T))/jww,-J), = 0 COSfij (Ij(r))/j
COSfl,Cl”,u>
2
0 = W, 2W) (7)
and the statistical properties of radiation in SHG process will be characterized by means of the reduced moments
For demonstrating the evolution of classical light statistics in SHG process we consider the chaotic fundamental input radiation with Bose-Einstein photocount distribution [33]. The photocount distribution in fundamental input radiation is taken, in the classical approach, as [33]
(n,,o)nw~o
P(Qo> = lim Td,O + -
(1 t (nw,O))(l+~“,O)
(11)
=&exp(-&).
Using eqs. (4)-(8) and (11) we have found the following classical behaviour of the field in SHG process with chaotic input radiation. The evolution of relative mean photon numbers in fundamental and second harmonic modes is described by The evolution of photocount distributions in the interacting modes in SHG process will not be presented here. Namely, if using the usual generating function method [33,34], the involved series do not absolutely converge for superchaotic radiation and, consequently, it is not possible to state the analytical form of photocount distribution in this way. In a special case of coherent input radiation with Poisson photocount distribution, for which it holds in the classical approach that [33]
(n,(r)>R
&J@))R =
%,O =
400
we obtain for the relative mean photon numbers,
(12a)
J x3 exp(-x2)
tanh2 (rx) dx .
(12b)
The reduced moments in fundamental harmonic modes are given as
.Lx(2m+1)exp(-x2)
sechti(rx)
and second
dx
6
%J,O!
e,,o - N..J,o))>
dx ,
0
exp(-(n,,0))
lim
sech2(rx)
0
(n,,o)“~~o
P&,0)=
= 2 j= x3 exp(-x2)
Qcm)(7) = w (9)
2(“-‘)
i 0
x3 exp(-x2)
sech2(rx)dx
1
-1,
m
(134 281
Volume
5 3,
m x@+l) J-
exp(-x2)
tanh2m(Tx)
m
03
P-
x3 exp(-x2)
tanh2(7x)
dx 1
0
(m = 2,3,4,
and the reduced correlations Q~~~?(~)
and the mutual correlations 20 are characterized by
dx
-1
Q$:‘(T) = ’ _2(” -1)
= r
15 March 1985
OPTICS COMMUNICATIONS
number 4
x[2(Peq)+11
. ..)
of both modes at w and
lim Qz2;A(,) T-+rn ’ =_ 1 >
forp
(13b)
s
are
x(2@+q)-ll
sech2P(x) tanh2q(x)
exp(-x2)
-l
0
-2)
x3 sech2(x) dx
X sech2P(?x) tanh2q (TX) dx
forp=qtl,
co s
4
x3 exp(-x2)
1 P
sech2(?-x) dx
0
x[[
-1
tanh2 (TX) dx
-1, Ii
@,q=1,2,3
,... ).
(13c)
At the beginning of SHG, when considering 7 < 1, the generating fundamental radiation is chaotic and the corresponding reduced moments are QkTinit = m! - 1 ,
(144
the generated weak second harmonic superchaotic with reduced moments Q$;)init
= (20~)!/2~
radiation
- 1
is
(14b)
and the mutual correlations of the interacting at o and 2w are characterized by
QEl$i,init
4
I
P0
x3 exp(-x2)
dx
=0
= [@J+ 2q)!1/2' - 1.
modes
forp>qtl.
(15c)
The classical evolution of relative mean photon num. bers in SHG with coherent and chaotic input radiations is shown in fig, 1. At the beginning the SHG with chaotic input radiation runs about 21i2 time quicker than that with coherent input radiation, but later this enhancement is diminished and the curves describing the evolutions of relative mean photon numbers for coherent and chaotic input radiations approach each other, respectively, in the far advanced process. The evolutions of two-, three- and four-order classical light statistics (reduced moments and correlations) in SHG with chaotic input radiation are shown in figs. 2-4, respectively. The fluctuation level in fundamental radiation at w first decreases up to the value of parameter 7 = 7min A 1.3, corresponding to about 80 per cent depletion of fundamental radiation energy. At 7min
(I4c)
In the far advanced SHG, if nearly total depletion of fundamental radiation occurs, when considering 7 + m, the residual fundamental radiation is extremely superchaotic, lim Qc”)(~) -+ 03 , w r-*m
(15a)
the generated strong second harmonic otic, lim Q’;“,‘(+=m! T-+-J 282
- 1,
radiation
is cha 0
Wb)
1
Fig. 1. Evolution of with coherent (---)
2
r
3
4
relative mean photon numbers in SHG and chaotic ( -) input radiation.
5
1.5 March 1985
OPTICS COMMUNICATIONS
Volume 53, number 4
500 100 ;
50
b 4 G?. soi a a qj
10 5 1 0.5
0.1 -03 - 0.5 0
1
2
3
4
5
6
7
6
9
10
T
0
Fig. 2. Evolution of second-order light statistics in SHG with chaotic input radiation.
the light fluctuations in fundamental radiation are considerably reduced and the radiation approaches the coherent state to the greatest extent. In the later course of SHG the fluctuation level in fundamental radiation increases and the radiation becomes extremely superchaotic at the end of the process. The fluctuation level in the initially superchaotic second harmonic radiation at 20 systematically decreases and the radiation approaches the chaotic state at the end of SHG process. The initially positive correlations of all orders between both interacting modes at w and 2w decrease at the beginning of SHG and, later (7 2 rmin), all order anticorrelations between the modes at w and 2w are gener100 50
-p;”; 0
12
3
4
5
6
7
8
9
r
Fig. 3. Evolution of thirdarder chaotic input radiation.
1
light statistics in SHG with
IO
12
3
4
Fig. 4. Evolution of fourtharder chaotic input radiation.
5T6
7
8
9
IO
light statistics in SHG with
ated. Only the moments Q$‘$‘A, for which p > 4 + 1 holds, exhibit a minimum and, consequently, respective order positive correlations are generated in the far advanced SHG process.
4. Discussion In principle, it is not possible to discuss separately the evolution of photon statistics in individual interacting modes and their mutual correlations in SHG, because all moments of the same order are connected by means of the conservation laws, as shown for second order moments in [29]. However, for characterizing the mutual correlations of two modes at w and 2w, the most roper quantities are the moments of the type Q$& P and, thus, in the further qualitative discussion, we &all mention the mutual correlations of the type Qz$‘A only. At the beginning of SHG (T 4 7min) the superchaotic behaviour of the generated second harmonic mode is simply the consequence of quadratic projection of light fluctuations in fundamental mode into the second harmonic mode. The initial positive correlations between the interacting modes can be explained by the fact that the second harmonic photons are preferably generated in the temporal or spatial regions with greater photon densities (photon clusters) of fundamental radiation due to greater values of nonlinear quadratic polarization there. In the further course of SHG the fluctuation level 283
Volume 53, number 4
OPTICS COMMUNICATIONS
in fundamental mode is successively smoothed by withdrawing photons from fundamental photons clusters and the process becomes more uniform; this causes decreasing both the fluctuation level in second harmonic mode and the mutual correlation level. An inve;sian in fundamental light distribution occurs for 7 e 7min. Namely, the photon clusters in generated second harmonic mode stimulate the further process so that in the regions possessing initiahy fundamental photon clusters practically all fundamental photons have been depleted and, on the contrary, the regions initially occupied by small fundamental photon numbers, which have been affected by SHG process to a very small extent only, represent been affected by SHG process to a very small extent only, represent the photon clusters in the weak residual fundamental radiation now. This also explains anticorrelations generated between the interacting modes in the far advanced SHG that are a consequence of the competition between both the interacting modes. A comparison of our results with the previous classical ones [ 1,12,14,29] manifests very good agreement, and also a good analogy with the quantum behaviour of photon statistics at the beginning of SHG [7,19,23-251 is apparent. However, the oscillations of the variance of photon number in second harmonic mode that was numerically calculated in [ 191 using quantum model of nonlinear optical interaction has no classical analogy. The classical description of evolution of photon statistics in SHG with chaotic input radiation presented here is not only of academic importance, but it can also have practical aspects for treatment of SHG with manymode laser radiation [lo-121. Moreover, general tendencies in evolution of photon statistics in SHG might be revealed on the basis of our results. Concering the dependence of efficiency of SHG on statistical properties of interacting radiation a special paper will be devoted to this problem. The author would like to thank Dipl. Eng. J. Kfepelka for performing the numerical calculations.
References [l] S.A. Akhmanov and A.S. Czirkin, Statisticheskiye yavleniya v nelineynoy optike (Moskovkiy Universiteit, Moscow, 1971).
284
15 March 1985
[2] R.B. Andreev and V.D. Volosov, Kvantovaya elektronika l(1974) 1355. [3] M. Schubert and B. Wilhelmi, Progress in optics, Vol. 17, ed. E. Wolf (North-Holland, Amsterdam, 1980) p. 163. [4] J. PeHna, Progress in optics, Vol. 18, ed. E. Wolf (NorthHolland, Amsterdam, 1980) p. 127. [S] S. Kielich, Molekulyarnaya nehneynaya optika (Nauka, Moscow, 1981). [6] H. Paul, Rev. Mod. Phys. 54 (1982) 1061. [ 7) J. Pefina, Quantum statistics of linear and nonlinear optical phenomena (Reidel, Dordrecht, 1984). [8] J.A. Armstrong, N. Bloembergen, J. Ducuing and P.S. Pershan, Phys. Rev. 127 (1962) 1918. [9] A. Ashkin, G.D. Boyd and J.M. Dziedzic, Phys. Rev. Lett. 11 (1963) 14. [lo] J. Ducuing and N. Bloembergen, Phys. Rev. 133 (1964) A1493. [II] G.E. Francois, Phys. Rev. 143 (1966) 597. [ 121 S.A. Akhmanov, A.S. Chirkin and V.G. Tunkin, Optoelectronics 1 (1969) 196. [13] P. Dewael, J. Phys. A8 (1975) 1614. [ 141 J.A. Churnside, Optics Comm. 51(1984) 207. [ 15 ] M.C. Teich, R.L. Abrams and W.B. Gandrud, Optics Comm. 2 (1970) 206. [16] N.K. Dutta, Opt. Quant. Electron. 11 (1979) 217. [17] D.F. Walls and R. Barakat, Phys. Rev. Al (1970) 446. [ 181 D.F. Walls and C.T. Tindle, Nuovo Cimento 2 (197 1) 915. [ 191 D.F. Walls and C.T. Tindle, J. Phys. A5 (1972) 534. [20] M. Orszag, P. Carrazana and H. Chuaqui, Optica Acta 30 (1983) 259. [21] G.P. Agrawal and CL. Mehta, J. Phys. A7 (1974) 607. [22] J. PeNna, Czech. J. Phys. B26 (1976) 140. 1231 L. MiSta and J. Peiina, Acta Phys. Ponon. A52 (1977) 425. [24] S. KieIich, M. Kozierowski and R. Tanas’, Coherence and quantum optics IV, ed. L. Mandel and E. Wolf (Plenum, New York, 1978) p. 511. [25] M. Kozierowski and R. Tanas’, OpticsComm. 21 (1977) 229. [26] L. Mandel, Optics Comm. 42 (1982) 437. 1271 M. Kozierowski and S. Kielich, Phys. Lett. A94 (1983) 213. [28] L.A. Lug&o, G. Strini and F. De Martini, Optics Lett. 8 (1983) 256. [29] P. Chmela, Czech. J. Phys. B31 (1981) 977 and 999. [30] H. Paul, Nichtlineare Optik I and II (Akademie-Verlag, Berlin, 1973). [ 3 l] P. Chmela, Uvod do nelinearni optiky I (Introduction to nonlinear optics I, in Czech) (Palacky University, Olomouc, 1982). [32] F. Zernicke and J.E. Midwinter, Applied nonlinear optics (Wiley, New York, 1973). [33] J. Pefina, Coherence of light (Van Nostrand Reinhold, London, 1972). [34] F.T. Arecchi and V. Degiorgio, Laser Handbook, Vol. 1, eds. F.T. Arecchi and E.O. Schulz-Dubois (NorthHolland, Amsterdam, 1972) p. 194.
53, number
OPTICS COMMUNICATIONS
4
15 March
1985
EVOLUTION OF CLASSICAL LIGHT STATISTICS IN SECOND HARMONIC GENERATION P. CHMELA Join1 Laboratory of Optics of Czechoslovak Academy of Sciences and Palackj University, Leninova 26, 77146 Olomouc, Czechoslovakia Received
4 December
1984
Classical solution of coherent second harmonic generation process based on the averaging of stochastic quantities over the initial photocount distribution in fundamental input radiation is presented in this paper. The evolution of light statistics in second harmonic generation with chaotic input radiation is calculated for arbitrary values of the time or space parameter.
1. Introduction Coherence and statistical phenomena in nonlinear optical interactions have been the matter of interest of many authors in recent years (for review, see refs. [l-71). The process of second harmonic generation (SHG) in a dispersive medium was described classically, without regarding the light fluctuations and coherence or spectral properties of radiation in [8]. The effects of partial coherence, many mode structure and classical field fluctuations in generating fundamental radiation on SHG process were demonstrated in [9-141 and the double enhancement of the rate of SHG with chaotic input radiation, with respect to coherent one, was observed in ref. [15]. The effects of amplitude and phase fluctuations of fundamental radiation possessing a defined spectral width in SHG process, including also dispersion of the nonlinear medium, were studied in [ 1,161. The quantum theory of nonlinear optical interactions was developed [ 171 and the effects of spontaneous break-up of second harmonic photons into fundamental photons [18,19], as well as the emission processes [20] in SHG were studied. The time evolution of the variance of photon number in the second harmonic mode was numerically computed in [ 191. Shorttime solution of nonlinear optical quadratic interaction up to the second order, using quantum description, was 0 030-4018/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
given in [ 13,21,22], and antibunching in fundamental mode [23-251 and second harmonic mode [24,25] at the beginning of SHG was stated. Recently, squeezing effects have been reported in both fundamental [26-281 and second harmonic [27,28] modes in SHG process. Although great effort has been exerted on the development of a consistent theory of SHG process, including statistical and coherence properties of the inter, acting radiation, the majority of treatments provide the descriptions of the time or space evolution of the field in nonlinear optical interaction either in the approximation of the given generating field, or by means of the numerical computations, or in the short-parameter approximation only. No complete analytical solution of the problem in a closed form has been found yet. In this paper we present a simple classical solution of SHG problem in a closed form, based on the averaging of stochastic quantities over the initial photocount distribution in fundamental mode. A similar treatment using short-parameter method up to the eighth order was given in ref. [29]. Such treatment is suitable for the description of coherent nonlinear optical interactions with generating radiation exhibiting strong classical light fluctuations only. Unfortunately, it does not include the fine quantum effects like photon antibunching and squeezing. On the other hand, it enables us to describe the evolution of classical statis279
tical properties of radiation for arbitrary values of the time or spatial parameter in a good approximation. Especially, it is suitable for the description of SHG with chaotic [12-141 or many-mode laser [lo-121 radiation. 2. General method The degenerate nonlinear optical interaction of two modes of radiation at o and 2w in a non-dissipative quadratic medium can be described, from the point of view of classical theory, by means of two coupled firstorder differential equations for the normalized stochastic field amplitudes [1,8,29-321 da,/dg
= -2iga*,a2,
,
The initial conditions amplitudes are a,(O)
= -bv;
,
qw(0)
of SHG for stochastic field
=0 .
(2)
Moreover, the classical description assumes strong fundamental input radiation, i.e. it holds for the mean input photon number that (n w,~) = (au,oaL,o) S 1 ,
(3)
where the angle brackets denote an ensemble average. Solving eqs. (l), with respect to initial conditions (2), we obtain the following solution for stochastic photon numbers (see e.g. [8,30,31]) n,([)
= n,,O sech2(21/2nzfogQ
,
(44
(la) (lb)
where the variable .$ represents either the time (t = t), when considering the time evolution of the field in an optical resonator, or the normal distance from the first boundary (t = z), when considering the interaction of travelling waves by passing light through a nonlinear crystal plate. g is the real coupling constant that is different for the descriptions in the temporal domain and in the spatial domain [29-3 11. The field amplitudes aj (j = o, 2~) are normalized in such a way that the quantity ni = ajaT represents either the stochastic photon number in an optical resonator or the stochastic photon flux in the normal direction to the nonlinear plate boundaries, respectively to the considered model of nonlinear interaction [29,31]. In the further text we shall call the quantity ni simply “photon number”. However, it is necessary to stress here that the normalized amplitudes or photon numbers are classical quantities and they can be simply replaced by usual quantities as follows: at the description in the temporal domain the radiation energy of the jth mode in an optical resonator is given by Uj = Fiiojnj; at the description in the spatial domain the light intensity of thejth mode is $ = fiUinj/cos pi; A being the Planck constant and pj is the angle of refraction for ray direction in an anisotropic medium. The phase matching [30-321, as well as the conditions for “coherent” interaction * [ I,29 ] are assumed. * Small dispersion of nonlinear medium and great coherence time OI coherence length of input radiation are assumed.
280
,
=a,,0
n2,(t) @,ldE
15 March 1985
OPTICS COMMUNICATIONS
Volume 5 3, number 4
= fn,,o
tanh2(21/2nz20gt)
.
(4b)
If the photocount distribution of fundamental input radiation p (n w,o) is k nown, we can calculate arbitrary moments of the type (ns(!$ nf,(Q> as follows [33]
(nE(E)n&(D)
=J
ng(Ln,,u)
0
= 2-4
s
0
n$‘Aq) sech2J’(21/2nzfo ’
g.$)
X tanh2Q(21’2n~~ogUp(n,,0)dn,,0 . (5) For the further treatment it is advantageous to introduce the reduced variable T that is defined for different models of the nonlinear optical interaction in the following way [29-321 7 = 21i2(n w,o)“2g4
=
I )
PI) 312 %,O) lQw cos 62, cos p, EO
112 cos p2* I
(6)
15 March 1985
OPTICS COMMUNICATIONS
Volume 53, number 4
where eO is the electric permittivity and /.+Jmagnetic permeability in SI units, vw and v2w are the indices of refraction in an anisotropic medium, 6, and 62, are the anisotropic divergency angles, 0, and 02” are the angles of refraction for ray directions, xi? (2~) is the effective nonlinear susceptibility and V is the effective volume of an optical resonator containing nonlinear medium. The efficiency of SHG will be described by the relative mean photon numbers
with respect to eqs. (4) (5) and (7) (n, (T))~ = sech2 (7) ,
(loa>
(qw(7))~ = i tanh2(r).
(lob)
All reduced moments Q!“)(r) (j = w, 2~; m > 2) and (’ reduced correlations Q,.$& (7) @ t 4 > 2) are zero, from the point of view ot classical theory, in this case.
2. SHG with chaotic input radiation = owj(T)T))/jww,-J), = 0 COSfij (Ij(r))/j
COSfl,Cl”,u>
2
0 = W, 2W) (7)
and the statistical properties of radiation in SHG process will be characterized by means of the reduced moments
For demonstrating the evolution of classical light statistics in SHG process we consider the chaotic fundamental input radiation with Bose-Einstein photocount distribution [33]. The photocount distribution in fundamental input radiation is taken, in the classical approach, as [33]
(n,,o)nw~o
P(Qo> = lim Td,O + -
(1 t (nw,O))(l+~“,O)
(11)
=&exp(-&).
Using eqs. (4)-(8) and (11) we have found the following classical behaviour of the field in SHG process with chaotic input radiation. The evolution of relative mean photon numbers in fundamental and second harmonic modes is described by The evolution of photocount distributions in the interacting modes in SHG process will not be presented here. Namely, if using the usual generating function method [33,34], the involved series do not absolutely converge for superchaotic radiation and, consequently, it is not possible to state the analytical form of photocount distribution in this way. In a special case of coherent input radiation with Poisson photocount distribution, for which it holds in the classical approach that [33]
(n,(r)>R
&J@))R =
%,O =
400
we obtain for the relative mean photon numbers,
(12a)
J x3 exp(-x2)
tanh2 (rx) dx .
(12b)
The reduced moments in fundamental harmonic modes are given as
.Lx(2m+1)exp(-x2)
sechti(rx)
and second
dx
6
%J,O!
e,,o - N..J,o))>
dx ,
0
exp(-(n,,0))
lim
sech2(rx)
0
(n,,o)“~~o
P&,0)=
= 2 j= x3 exp(-x2)
Qcm)(7) = w (9)
2(“-‘)
i 0
x3 exp(-x2)
sech2(rx)dx
1
-1,
m
(134 281
Volume
5 3,
m x@+l) J-
exp(-x2)
tanh2m(Tx)
m
03
P-
x3 exp(-x2)
tanh2(7x)
dx 1
0
(m = 2,3,4,
and the reduced correlations Q~~~?(~)
and the mutual correlations 20 are characterized by
dx
-1
Q$:‘(T) = ’ _2(” -1)
= r
15 March 1985
OPTICS COMMUNICATIONS
number 4
x[2(Peq)+11
. ..)
of both modes at w and
lim Qz2;A(,) T-+rn ’ =_ 1 >
forp
(13b)
s
are
x(2@+q)-ll
sech2P(x) tanh2q(x)
exp(-x2)
-l
0
-2)
x3 sech2(x) dx
X sech2P(?x) tanh2q (TX) dx
forp=qtl,
co s
4
x3 exp(-x2)
1 P
sech2(?-x) dx
0
x[[
-1
tanh2 (TX) dx
-1, Ii
@,q=1,2,3
,... ).
(13c)
At the beginning of SHG, when considering 7 < 1, the generating fundamental radiation is chaotic and the corresponding reduced moments are QkTinit = m! - 1 ,
(144
the generated weak second harmonic superchaotic with reduced moments Q$;)init
= (20~)!/2~
radiation
- 1
is
(14b)
and the mutual correlations of the interacting at o and 2w are characterized by
QEl$i,init
4
I
P0
x3 exp(-x2)
dx
=0
= [@J+ 2q)!1/2' - 1.
modes
forp>qtl.
(15c)
The classical evolution of relative mean photon num. bers in SHG with coherent and chaotic input radiations is shown in fig, 1. At the beginning the SHG with chaotic input radiation runs about 21i2 time quicker than that with coherent input radiation, but later this enhancement is diminished and the curves describing the evolutions of relative mean photon numbers for coherent and chaotic input radiations approach each other, respectively, in the far advanced process. The evolutions of two-, three- and four-order classical light statistics (reduced moments and correlations) in SHG with chaotic input radiation are shown in figs. 2-4, respectively. The fluctuation level in fundamental radiation at w first decreases up to the value of parameter 7 = 7min A 1.3, corresponding to about 80 per cent depletion of fundamental radiation energy. At 7min
(I4c)
In the far advanced SHG, if nearly total depletion of fundamental radiation occurs, when considering 7 + m, the residual fundamental radiation is extremely superchaotic, lim Qc”)(~) -+ 03 , w r-*m
(15a)
the generated strong second harmonic otic, lim Q’;“,‘(+=m! T-+-J 282
- 1,
radiation
is cha 0
Wb)
1
Fig. 1. Evolution of with coherent (---)
2
r
3
4
relative mean photon numbers in SHG and chaotic ( -) input radiation.
5
1.5 March 1985
OPTICS COMMUNICATIONS
Volume 53, number 4
500 100 ;
50
b 4 G?. soi a a qj
10 5 1 0.5
0.1 -03 - 0.5 0
1
2
3
4
5
6
7
6
9
10
T
0
Fig. 2. Evolution of second-order light statistics in SHG with chaotic input radiation.
the light fluctuations in fundamental radiation are considerably reduced and the radiation approaches the coherent state to the greatest extent. In the later course of SHG the fluctuation level in fundamental radiation increases and the radiation becomes extremely superchaotic at the end of the process. The fluctuation level in the initially superchaotic second harmonic radiation at 20 systematically decreases and the radiation approaches the chaotic state at the end of SHG process. The initially positive correlations of all orders between both interacting modes at w and 2w decrease at the beginning of SHG and, later (7 2 rmin), all order anticorrelations between the modes at w and 2w are gener100 50
-p;”; 0
12
3
4
5
6
7
8
9
r
Fig. 3. Evolution of thirdarder chaotic input radiation.
1
light statistics in SHG with
IO
12
3
4
Fig. 4. Evolution of fourtharder chaotic input radiation.
5T6
7
8
9
IO
light statistics in SHG with
ated. Only the moments Q$‘$‘A, for which p > 4 + 1 holds, exhibit a minimum and, consequently, respective order positive correlations are generated in the far advanced SHG process.
4. Discussion In principle, it is not possible to discuss separately the evolution of photon statistics in individual interacting modes and their mutual correlations in SHG, because all moments of the same order are connected by means of the conservation laws, as shown for second order moments in [29]. However, for characterizing the mutual correlations of two modes at w and 2w, the most roper quantities are the moments of the type Q$& P and, thus, in the further qualitative discussion, we &all mention the mutual correlations of the type Qz$‘A only. At the beginning of SHG (T 4 7min) the superchaotic behaviour of the generated second harmonic mode is simply the consequence of quadratic projection of light fluctuations in fundamental mode into the second harmonic mode. The initial positive correlations between the interacting modes can be explained by the fact that the second harmonic photons are preferably generated in the temporal or spatial regions with greater photon densities (photon clusters) of fundamental radiation due to greater values of nonlinear quadratic polarization there. In the further course of SHG the fluctuation level 283
Volume 53, number 4
OPTICS COMMUNICATIONS
in fundamental mode is successively smoothed by withdrawing photons from fundamental photons clusters and the process becomes more uniform; this causes decreasing both the fluctuation level in second harmonic mode and the mutual correlation level. An inve;sian in fundamental light distribution occurs for 7 e 7min. Namely, the photon clusters in generated second harmonic mode stimulate the further process so that in the regions possessing initiahy fundamental photon clusters practically all fundamental photons have been depleted and, on the contrary, the regions initially occupied by small fundamental photon numbers, which have been affected by SHG process to a very small extent only, represent been affected by SHG process to a very small extent only, represent the photon clusters in the weak residual fundamental radiation now. This also explains anticorrelations generated between the interacting modes in the far advanced SHG that are a consequence of the competition between both the interacting modes. A comparison of our results with the previous classical ones [ 1,12,14,29] manifests very good agreement, and also a good analogy with the quantum behaviour of photon statistics at the beginning of SHG [7,19,23-251 is apparent. However, the oscillations of the variance of photon number in second harmonic mode that was numerically calculated in [ 191 using quantum model of nonlinear optical interaction has no classical analogy. The classical description of evolution of photon statistics in SHG with chaotic input radiation presented here is not only of academic importance, but it can also have practical aspects for treatment of SHG with manymode laser radiation [lo-121. Moreover, general tendencies in evolution of photon statistics in SHG might be revealed on the basis of our results. Concering the dependence of efficiency of SHG on statistical properties of interacting radiation a special paper will be devoted to this problem. The author would like to thank Dipl. Eng. J. Kfepelka for performing the numerical calculations.
References [l] S.A. Akhmanov and A.S. Czirkin, Statisticheskiye yavleniya v nelineynoy optike (Moskovkiy Universiteit, Moscow, 1971).
284
15 March 1985
[2] R.B. Andreev and V.D. Volosov, Kvantovaya elektronika l(1974) 1355. [3] M. Schubert and B. Wilhelmi, Progress in optics, Vol. 17, ed. E. Wolf (North-Holland, Amsterdam, 1980) p. 163. [4] J. PeHna, Progress in optics, Vol. 18, ed. E. Wolf (NorthHolland, Amsterdam, 1980) p. 127. [S] S. Kielich, Molekulyarnaya nehneynaya optika (Nauka, Moscow, 1981). [6] H. Paul, Rev. Mod. Phys. 54 (1982) 1061. [ 7) J. Pefina, Quantum statistics of linear and nonlinear optical phenomena (Reidel, Dordrecht, 1984). [8] J.A. Armstrong, N. Bloembergen, J. Ducuing and P.S. Pershan, Phys. Rev. 127 (1962) 1918. [9] A. Ashkin, G.D. Boyd and J.M. Dziedzic, Phys. Rev. Lett. 11 (1963) 14. [lo] J. Ducuing and N. Bloembergen, Phys. Rev. 133 (1964) A1493. [II] G.E. Francois, Phys. Rev. 143 (1966) 597. [ 121 S.A. Akhmanov, A.S. Chirkin and V.G. Tunkin, Optoelectronics 1 (1969) 196. [13] P. Dewael, J. Phys. A8 (1975) 1614. [ 141 J.A. Churnside, Optics Comm. 51(1984) 207. [ 15 ] M.C. Teich, R.L. Abrams and W.B. Gandrud, Optics Comm. 2 (1970) 206. [16] N.K. Dutta, Opt. Quant. Electron. 11 (1979) 217. [17] D.F. Walls and R. Barakat, Phys. Rev. Al (1970) 446. [ 181 D.F. Walls and C.T. Tindle, Nuovo Cimento 2 (197 1) 915. [ 191 D.F. Walls and C.T. Tindle, J. Phys. A5 (1972) 534. [20] M. Orszag, P. Carrazana and H. Chuaqui, Optica Acta 30 (1983) 259. [21] G.P. Agrawal and CL. Mehta, J. Phys. A7 (1974) 607. [22] J. PeNna, Czech. J. Phys. B26 (1976) 140. 1231 L. MiSta and J. Peiina, Acta Phys. Ponon. A52 (1977) 425. [24] S. KieIich, M. Kozierowski and R. Tanas’, Coherence and quantum optics IV, ed. L. Mandel and E. Wolf (Plenum, New York, 1978) p. 511. [25] M. Kozierowski and R. Tanas’, OpticsComm. 21 (1977) 229. [26] L. Mandel, Optics Comm. 42 (1982) 437. 1271 M. Kozierowski and S. Kielich, Phys. Lett. A94 (1983) 213. [28] L.A. Lug&o, G. Strini and F. De Martini, Optics Lett. 8 (1983) 256. [29] P. Chmela, Czech. J. Phys. B31 (1981) 977 and 999. [30] H. Paul, Nichtlineare Optik I and II (Akademie-Verlag, Berlin, 1973). [ 3 l] P. Chmela, Uvod do nelinearni optiky I (Introduction to nonlinear optics I, in Czech) (Palacky University, Olomouc, 1982). [32] F. Zernicke and J.E. Midwinter, Applied nonlinear optics (Wiley, New York, 1973). [33] J. Pefina, Coherence of light (Van Nostrand Reinhold, London, 1972). [34] F.T. Arecchi and V. Degiorgio, Laser Handbook, Vol. 1, eds. F.T. Arecchi and E.O. Schulz-Dubois (NorthHolland, Amsterdam, 1972) p. 194.