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On the saturation in a Čerenkov-type optical parametric oscillator
Volume 78A, number 4
PHYSICS LETTERS
18 August 1980
ON THE SATURATION IN A ?~ERENKOV-TYPEOPTICAL PARAMETRIC OSCILLATOR V.N. LUGOVOI P.N Lebedev Physical Institute of the Academy of Sciences of the USSR, Moscow, USSR Received 17 April 1980
The saturation and radiation dynamics in a Cerenkov-type optical parametric oscillator are considered under the conditions that the ~erenkov radiation losses are negligible and the saturation is due to the Kerr nonlinearity and stimulated Raman emission in the medium.
Besides the well-known parametric interactions in nonlinear optics with total phase matching of the light waves [1—3] ~erenkov-type interactions with partial phase matching (along a plane or an axis) has been considered [4—9]and demonstrated [7—9]. The ~erenkov-type interactions are effective provided the transverse size of the interaction volume does not exceed the inverse value of the transverse momentum detuning. Here, the ~erenkov-type interactions are usually studied in thin optical waveguides [8,9] where waveguide waves interact with usual (freely propagating) light waves, In particular, besides the usual optical parametric oscillators, ~erenkov-type optical parametric oscilators have recently been proposed and considered [10—12] Here, a freely propagating pumping wave with a frequency w~ and a wave vector k~is incident from outside at an angle 0 on an optical waveguide (the is supposed be terminated by mirrors). If thelatter substance of the to waveguide (substrate) exhibits an appreciable second-order susceptibility described by the tensor ~ and the pumping light intensity exceeds the threshold value, self-excitati6n of waveguide light waves occurs with frequencies wS, and propagation constants ks, k~,satisfying the relations ,
.
~-‘~
~erenkov effect [11]. For I iO—7 esu (for exampie, for GaAs, LiNbO 3) the saturation in the oscillator can be due to the nonlinear ~erenkov losses of light waves excited in the optical waveguide. This case is considered in refs. [11,121. In the present paper we note that for ~I~ 10—8 esu (for example, for ADP, KDP, a-Si02) the ~erenkov losses of the waves excited in the waveguide seem to be negligible and the saturation can be due to the Kerr nonlinearity and stimulated Raman emission (the first Stokes component oscillation) in the substance. In what follows the saturation and dynamics of the optical parametric oscifiation under these conditions is considered. Here, we assume the mirrors terminating the waveguide (and forming a Fabry—Pérot resonator at frequencies wS > w~)to exhibit an appreciable reflection at the first Stokes frequency = ~‘s WR as well (WR is the Raman shift) and to reflect negligibly 2wR. Also, at second frequency = ‘i’s wethe assume thatStokes a single mode of as” the resonator is allowed for excitation in the limits of the spontaneous Raman scattering linewidth. Here, using a procedure similar to that of ref. [11], one can derive the following equations for the complex amplitudes Y 1 of the corresponding three resonator modes (with frequencies w. z~,j= 5, S’, ~
—
—
—
ks~±k~+k~cos0. (1) The second of these relations is a ~erenkov-type condition, and the parametric oscillation can be considered as a nonlinear optical analogue of the inverse Wp~W~+W,,,
338
~s+ws~~_w):
/
~,
~
Volume 78A, number 4
dYs
-~—
+
=
~S
PHYSICS LETTERS
a
3s~lY
—
.
(2) /
~I
+~+~
2wsPs’s r X [P~~’ +
/
Lr
— —
~
2)
~
dYs~ ut
(fl5?1~)
=
1i
1
dY* -~-~
18 August 1980
2~ /
3ss’
(~vv~Sv)]~
+
(5)
i(0) ~~s’)j ~
+
t
P/rnP/rn~
1=U5’t,
21 —
PS’S
~
j
5’
P’SS’
‘
Im P5~’ (Ps’s >0 P~~’ <0), ,
the initial amplitudes being represented as
where
I Y5
2irw1N~f~(K)(W) :E/*E/ErnE,~~
=
~jrn
=
27riw1N~f~(R)(w)
=
=
:
Ei*EJEmE7n
(3)
~,
4~2~sc~.vN2 f~(Ws): E~EOEU
2
~ (K)(~.) ~ (K)(~.+ ~ Wm) is the Kerr susceptibiity tensor (we assume the Kerr effect to be electronic; the I3jm are real), ,~(R)(w 1)is the Raman susceptibiity tensor due to a vibronic (polariton) resonance in the substance, N = f n/ E1EI dr, E1 the coordinate part of the jth resonator mode, n1 the linear part of the refractive index of the medium for the polarization E1, E0 the complex amplitude of the unperturbed pumping field, p1 (c/2Ln1) (I r,) the frequency half-width of the corresponding resonator mode (r1 is the mirror reflectivity at the frequency w1, L the spacing between the mirrors). For example, for ~-SiO2one has I~i 2 X i0~ esu, ~(K) I 0.7 x l0~~ esu, ~(R) I~1.3 X 10—13 esu (at the exact resonance = ~s’ consideration for WR = 467 the 1), and as is seen in the ws’ following cm— effect of the Kerr and Raman nonlinearities on the —
—
parametric oscillation can be appreciable when the pumping intensity slightly exceeds its threshold value. In this case, proceeding from (2) in a way similar to that of refs. [11,12] one can deri~rethe following systern of equations (at p5 = = ‘~~s = = z~): dx d = (a + bx + cy)x, ~ = (—1 + x)y, (4) 1
where
1
2
~
2~
IYs’i2
iYvi = t.~v1S’l~~SPS’S, ~(O)arbitrary. We shall consider the usual case c< 0 assuming the pumping intensity I~to exceed its threshold value ~ i.e. we shall assume a >0.
According to eqs. (4), for b <0 the saturation in the optical parametric oscillator can be due to the Kerr nonlinearity only (which is realized for a/lb I < 1) or can be due to both the Kerr and Raman nonlinearities (which is realized for at b I> 1). Here, the light oscillation intensities reach the stable limiting value x y tn~ (in the first case = alibi, 5 0 and in the second one = 1, ~ = (a + b)l cl). The threshold pumping intensity ~ required for stimulated Raman emission is given by the condition a = I b I, which can be written in the form j(l) ij ~ 2 ~2 6 0, thi’ 0,th~ ~s’I iIV.L + ) Using the above values of I ~ (K) I and ~(R) i one can ) see that for o~-SiO 2the corresponding coefficient in eq. (6) is of the order of unity. Here, for I I ‘~i.t, jig’ p or for I I j.t, p~’ p the threshold of the stimulated Raman emission is reached when the pumping intensity slightly exceeds the parametric oscilation threshold I~, Note also that according to eqs. (2) and (4) for b > 0 a hard self-excitation of the parametric oscillation can be observed in the system. Of interest may also be transient processes in the oscillator. To study these processesfor ~s’ ~ ~ eqs. (4) ~‘
—
.
~.
339
Volume 78A, number 4
PHYSICS LETTERS
10~ X O5~
2
5
\
0
2
/_
—
4
7
4t
1 I
Fig. 1. The time-dependence of the intensity of the parametric oscillation (x) and of the Stokes component (y) during the coupled pulse (solid and dashed curve, respectively).
were solved numerically with an ES-1022 computer for a 1 and c = —0.5. At low values of the initial intensitiesx0,y0, transient pulsation of x,y is observed as a series of coupled pulses of the parametric oscillation (x) and the Stokes component oscillation (y). A typical form ofa separate coupled pulse obtained forx0 Y0 l0—~,b = —3 X 10~in the region t1 105 is represented fig. 1. The times at which the pulse and theincorresponding values OfXm/Xi(Xi is appears the maximum value of x in the first pulse) are given in fig. 2 where the envelopes (l)—(4) correspond to different values of b. As is seen for b <0 the intensity pulsation range decreases with time giving the limiting values ~ For b >0 the pulsation range increases with time, a limitation not being reached in eqs. (4) [the limitation is reached in eqs. (2)]. We note also that the pulsation increase and the corresponding increase of the intervals between the pulse [see, for example, curve (4)] is followed by an exponential decrease of the intensitiesx,y in these intervals to extremely low values. Such a decrease can really be limited by spontaneous noise in the system and the pulsation can be practically periodic and somewhat irregular. I am indebted to A.T. Matachun, L. Ya. Trendeleva and E.A. Zubova for the computer calculations used in the work.
340
18 August 1980
2
0
50
100
150 t1
Fig. 2. The times at which the pulse appears and the relative 3,thenx Curve corresponds — 102, = 15; (1) curve (2) tob to —3b X= i0~, Xo Yo = i0~, thenx1 16; curve (3) to b = 3 x2,x iO~, x0 3,yo~ Yo = l0~, thenx1 curve (4) to b = 10 0 i0~, thenx~ 17;24). peak intensities x0 =y0 10 Xrn/Xl. 1
=
l0
References [1] N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965). [2] Y.N. Lugovoi, Vvedenie v teoriyu vynuzhdennogo kombinatsionnogo rasseyaniya (Introduction to the theory of stimulated Raman scattering) (Moscow, 1968). [3] Y.R. Shen, Rev. Mod. Phys. 48 (1976) 1. [4] A. Szöke, BulL Am. Phys. Soc. 9 (1964) 490. [5] V.N. Lugovoi and 1.1. Sobel’man, Zh. Eksp. Teor. Fiz. 58 (1970) 1283. [6] V.N. Lugovoi 69(1973) 84. and A.M. Prokhorov, Zh. Eksp. Teor. Fiz. [7] A. Zembrod, H. Puell and J.A. Giordmaine, J. Optoelectron. 1(1968) 64. [8] P.K. Tien, R. Ubich and P.J. Martin, AppL Phys. Lett. 17(1970) 447. [9] B.U. Chen, C.L. Tang and J.M. Telle, Appl. Phys. Lett., 25 (1974) 495. [10] V.N. Lugovoi, Zh. Eksp. Teor. Hz. Pjs’ma 25 (1977) 563. [11] V.N. Lugovoi, Opt. Acta 25 (1978) 337. [12] V.N. Lugovoi, Phys. Stat. SoL (b) 94(1979) 79.
PHYSICS LETTERS
18 August 1980
ON THE SATURATION IN A ?~ERENKOV-TYPEOPTICAL PARAMETRIC OSCILLATOR V.N. LUGOVOI P.N Lebedev Physical Institute of the Academy of Sciences of the USSR, Moscow, USSR Received 17 April 1980
The saturation and radiation dynamics in a Cerenkov-type optical parametric oscillator are considered under the conditions that the ~erenkov radiation losses are negligible and the saturation is due to the Kerr nonlinearity and stimulated Raman emission in the medium.
Besides the well-known parametric interactions in nonlinear optics with total phase matching of the light waves [1—3] ~erenkov-type interactions with partial phase matching (along a plane or an axis) has been considered [4—9]and demonstrated [7—9]. The ~erenkov-type interactions are effective provided the transverse size of the interaction volume does not exceed the inverse value of the transverse momentum detuning. Here, the ~erenkov-type interactions are usually studied in thin optical waveguides [8,9] where waveguide waves interact with usual (freely propagating) light waves, In particular, besides the usual optical parametric oscillators, ~erenkov-type optical parametric oscilators have recently been proposed and considered [10—12] Here, a freely propagating pumping wave with a frequency w~ and a wave vector k~is incident from outside at an angle 0 on an optical waveguide (the is supposed be terminated by mirrors). If thelatter substance of the to waveguide (substrate) exhibits an appreciable second-order susceptibility described by the tensor ~ and the pumping light intensity exceeds the threshold value, self-excitati6n of waveguide light waves occurs with frequencies wS, and propagation constants ks, k~,satisfying the relations ,
.
~-‘~
~erenkov effect [11]. For I iO—7 esu (for exampie, for GaAs, LiNbO 3) the saturation in the oscillator can be due to the nonlinear ~erenkov losses of light waves excited in the optical waveguide. This case is considered in refs. [11,121. In the present paper we note that for ~I~ 10—8 esu (for example, for ADP, KDP, a-Si02) the ~erenkov losses of the waves excited in the waveguide seem to be negligible and the saturation can be due to the Kerr nonlinearity and stimulated Raman emission (the first Stokes component oscillation) in the substance. In what follows the saturation and dynamics of the optical parametric oscifiation under these conditions is considered. Here, we assume the mirrors terminating the waveguide (and forming a Fabry—Pérot resonator at frequencies wS > w~)to exhibit an appreciable reflection at the first Stokes frequency = ~‘s WR as well (WR is the Raman shift) and to reflect negligibly 2wR. Also, at second frequency = ‘i’s wethe assume thatStokes a single mode of as” the resonator is allowed for excitation in the limits of the spontaneous Raman scattering linewidth. Here, using a procedure similar to that of ref. [11], one can derive the following equations for the complex amplitudes Y 1 of the corresponding three resonator modes (with frequencies w. z~,j= 5, S’, ~
—
—
—
ks~±k~+k~cos0. (1) The second of these relations is a ~erenkov-type condition, and the parametric oscillation can be considered as a nonlinear optical analogue of the inverse Wp~W~+W,,,
338
~s+ws~~_w):
/
~,
~
Volume 78A, number 4
dYs
-~—
+
=
~S
PHYSICS LETTERS
a
3s~lY
—
.
(2) /
~I
+~+~
2wsPs’s r X [P~~’ +
/
Lr
— —
~
2)
~
dYs~ ut
(fl5?1~)
=
1i
1
dY* -~-~
18 August 1980
2~ /
3ss’
(~vv~Sv)]~
+
(5)
i(0) ~~s’)j ~
+
t
P/rnP/rn~
1=U5’t,
21 —
PS’S
~
j
5’
P’SS’
‘
Im P5~’ (Ps’s >0 P~~’ <0), ,
the initial amplitudes being represented as
where
I Y5
2irw1N~f~(K)(W) :E/*E/ErnE,~~
=
~jrn
=
27riw1N~f~(R)(w)
=
=
:
Ei*EJEmE7n
(3)
~,
4~2~sc~.vN2 f~(Ws): E~EOEU
2
~ (K)(~.) ~ (K)(~.+ ~ Wm) is the Kerr susceptibiity tensor (we assume the Kerr effect to be electronic; the I3jm are real), ,~(R)(w 1)is the Raman susceptibiity tensor due to a vibronic (polariton) resonance in the substance, N = f n/ E1EI dr, E1 the coordinate part of the jth resonator mode, n1 the linear part of the refractive index of the medium for the polarization E1, E0 the complex amplitude of the unperturbed pumping field, p1 (c/2Ln1) (I r,) the frequency half-width of the corresponding resonator mode (r1 is the mirror reflectivity at the frequency w1, L the spacing between the mirrors). For example, for ~-SiO2one has I~i 2 X i0~ esu, ~(K) I 0.7 x l0~~ esu, ~(R) I~1.3 X 10—13 esu (at the exact resonance = ~s’ consideration for WR = 467 the 1), and as is seen in the ws’ following cm— effect of the Kerr and Raman nonlinearities on the —
—
parametric oscillation can be appreciable when the pumping intensity slightly exceeds its threshold value. In this case, proceeding from (2) in a way similar to that of refs. [11,12] one can deri~rethe following systern of equations (at p5 = = ‘~~s = = z~): dx d = (a + bx + cy)x, ~ = (—1 + x)y, (4) 1
where
1
2
~
2~
IYs’i2
iYvi = t.~v1S’l~~SPS’S, ~(O)arbitrary. We shall consider the usual case c< 0 assuming the pumping intensity I~to exceed its threshold value ~ i.e. we shall assume a >0.
According to eqs. (4), for b <0 the saturation in the optical parametric oscillator can be due to the Kerr nonlinearity only (which is realized for a/lb I < 1) or can be due to both the Kerr and Raman nonlinearities (which is realized for at b I> 1). Here, the light oscillation intensities reach the stable limiting value x y tn~ (in the first case = alibi, 5 0 and in the second one = 1, ~ = (a + b)l cl). The threshold pumping intensity ~ required for stimulated Raman emission is given by the condition a = I b I, which can be written in the form j(l) ij ~ 2 ~2 6 0, thi’ 0,th~ ~s’I iIV.L + ) Using the above values of I ~ (K) I and ~(R) i one can ) see that for o~-SiO 2the corresponding coefficient in eq. (6) is of the order of unity. Here, for I I ‘~i.t, jig’ p or for I I j.t, p~’ p the threshold of the stimulated Raman emission is reached when the pumping intensity slightly exceeds the parametric oscilation threshold I~, Note also that according to eqs. (2) and (4) for b > 0 a hard self-excitation of the parametric oscillation can be observed in the system. Of interest may also be transient processes in the oscillator. To study these processesfor ~s’ ~ ~ eqs. (4) ~‘
—
.
~.
339
Volume 78A, number 4
PHYSICS LETTERS
10~ X O5~
2
5
\
0
2
/_
—
4
7
4t
1 I
Fig. 1. The time-dependence of the intensity of the parametric oscillation (x) and of the Stokes component (y) during the coupled pulse (solid and dashed curve, respectively).
were solved numerically with an ES-1022 computer for a 1 and c = —0.5. At low values of the initial intensitiesx0,y0, transient pulsation of x,y is observed as a series of coupled pulses of the parametric oscillation (x) and the Stokes component oscillation (y). A typical form ofa separate coupled pulse obtained forx0 Y0 l0—~,b = —3 X 10~in the region t1 105 is represented fig. 1. The times at which the pulse and theincorresponding values OfXm/Xi(Xi is appears the maximum value of x in the first pulse) are given in fig. 2 where the envelopes (l)—(4) correspond to different values of b. As is seen for b <0 the intensity pulsation range decreases with time giving the limiting values ~ For b >0 the pulsation range increases with time, a limitation not being reached in eqs. (4) [the limitation is reached in eqs. (2)]. We note also that the pulsation increase and the corresponding increase of the intervals between the pulse [see, for example, curve (4)] is followed by an exponential decrease of the intensitiesx,y in these intervals to extremely low values. Such a decrease can really be limited by spontaneous noise in the system and the pulsation can be practically periodic and somewhat irregular. I am indebted to A.T. Matachun, L. Ya. Trendeleva and E.A. Zubova for the computer calculations used in the work.
340
18 August 1980
2
0
50
100
150 t1
Fig. 2. The times at which the pulse appears and the relative 3,thenx Curve corresponds — 102, = 15; (1) curve (2) tob to —3b X= i0~, Xo Yo = i0~, thenx1 16; curve (3) to b = 3 x2,x iO~, x0 3,yo~ Yo = l0~, thenx1 curve (4) to b = 10 0 i0~, thenx~ 17;24). peak intensities x0 =y0 10 Xrn/Xl. 1
=
l0
References [1] N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965). [2] Y.N. Lugovoi, Vvedenie v teoriyu vynuzhdennogo kombinatsionnogo rasseyaniya (Introduction to the theory of stimulated Raman scattering) (Moscow, 1968). [3] Y.R. Shen, Rev. Mod. Phys. 48 (1976) 1. [4] A. Szöke, BulL Am. Phys. Soc. 9 (1964) 490. [5] V.N. Lugovoi and 1.1. Sobel’man, Zh. Eksp. Teor. Fiz. 58 (1970) 1283. [6] V.N. Lugovoi 69(1973) 84. and A.M. Prokhorov, Zh. Eksp. Teor. Fiz. [7] A. Zembrod, H. Puell and J.A. Giordmaine, J. Optoelectron. 1(1968) 64. [8] P.K. Tien, R. Ubich and P.J. Martin, AppL Phys. Lett. 17(1970) 447. [9] B.U. Chen, C.L. Tang and J.M. Telle, Appl. Phys. Lett., 25 (1974) 495. [10] V.N. Lugovoi, Zh. Eksp. Teor. Hz. Pjs’ma 25 (1977) 563. [11] V.N. Lugovoi, Opt. Acta 25 (1978) 337. [12] V.N. Lugovoi, Phys. Stat. SoL (b) 94(1979) 79.