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Phase matching for parametric amplification in a single-mode birefringent fiber: influence of the non-phase-matched waves Eric Lantz, Denis Gindre, Herve´ Maillotte, and Jacques Monneret Laboratoire d’Optique P. M. Duffieux, Unite´ de Recherche Associe´e 214, Centre National de la Recherche Scientifique, Universite´ de Franche-Comte´, Route de Gray, 25030 Besanc ¸on Cedex, France Received March 13, 1996; revised manuscript received June 10, 1996 In a single-mode birefringent fiber, phase matching in four-photon parametric amplification implies linear terms, which are due to dispersion and birefringence, and nonlinear terms, which are due to self- and crossphase modulations. In the normal-dispersion regime combinations of these terms lead to three phasematched regimes. When the pump is equally divided between the two polarizations one must take the nonphase-matched waves into account to obtain accurate values of the phase-matched wavelengths and of the gain. To calculate these parameters we analytically develop corrected models of both the four-wave mixing formalism and of the modulation instability formalism and show the correspondence between these two approaches. Then we experimentally show that the parametric gain increases with the value of the birefringence. © 1997 Optical Society of America. [S0740-3224(97)00901-6]

1. INTRODUCTION 1

Since the paper of Stolen and Bjorkholm, it has been well known that nonlinear terms must be taken in account in the expression of phase matching for parametric amplification by four-wave mixing (FWM) in a fiber. Moreover, the link between parametric amplification and modulational instability (MI) was recognized as early as 1980 by Hasegawa and Brinkman.2 The formalism of modulation instability was developed for negative group-velocity dispersion (GVD), in which nonlinear terms of dispersion cancel the GVD, leading to phase matching. It was extended to the region of positive GVD by use of a birefringent fiber and a pump wave either polarized along one axis of the fiber3,4 or equally divided between both axes.5–8 MI has been used in recent years in studies of competition between Raman and parametric amplification,7–9 amplification near the zero-dispersion wavelength,9 and strong coupling with depletion of the pump.10,11 Although the formalism of MI includes nonlinear terms and although its equivalence to FWM has been recognized, phase matching in parametric amplification had been described for a long time with these terms neglected.12,13 However, in the case of a divided pump the amplified sidebands were experimentally14 and theoretically1,14 proved to experience a frequency shift that depends on the pump power. More surprisingly, it was numerically shown that the maximum gain of MI, given in Refs. 5 and 6, increases with the birefringence of the fiber, although this gain, which corresponds to a perfect phase-matched process with a divided pump, depends only on the pump power in the FWM formalism.1 As explained in Ref. 7, coupling between Stokes and antiStokes waves on both axes must be taken in account for low birefringence. In this case the Stokes wave on the 0740-3224/97/010116-10$10.00

fast axis and the anti-Stokes wave on the slow axis cannot be neglected, although these waves are not phase matched. They lower the gain and change phasematching conditions for the two other sidebands. In this paper we make the appropriate corrections to the FWM formalism to get analytical expressions for the phasematching conditions and gain values that take into account the non-phase-matched waves. The MI formalism takes into account all four sidebands and yields correct values of the gain. With this formalism we calculate analytical expressions for phase matching and gain and show that expansions of successive orders of these expressions correspond to the linear, the nonlinear, and the corrected nonlinear models of phase matching in the FWM formalism. Our analysis is restricted to the positive GVD region, far from the zero-dispersion wavelength, and we assume that Raman amplification can be neglected. We use a plane-wave formalism, well adapted to a single-mode fiber. The link between the plane wave and the real field uses the notion of effective area, which was discussed in Refs. 1 and 15. In Section 2 we extend the study of phase matching in a birefringent single-mode fiber, given in Ref. 12, by including the nonlinear terms. Section 3 is devoted to a further analysis of the process of phase matching with an equally divided pump. We show that FWM model must be corrected by properly taking into account the nonphase-matched waves. We use the MI formalism in Section 4 to obtain expressions of phase-matching conditions and of the gain. The results of Section 3 appear to be accurate approximations of these expressions. To verify that our analysis can be transposed to short pulses, Section 5 is devoted to the simulation of the propagation by a nonlinear Schro¨dinger equation. The results © 1997 Optical Society of America

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Vol. 14, No. 1 / January 1997 / J. Opt. Soc. Am. B

are in full accordance with those in the preceding sections. Finally we present in Section 6 experimental results that confirm the dependence of the gain on the value of the birefringence and the ability of our model to link accurately birefringence, power, and the determined phase-matched wavelengths.

Px 5

117

3« 0 ~ 3 ! x xxxx $ @ E px E px * ~ E px 1 2E sx 1 2E ax ! 4 1 E px E px ~ E sx * 1 E ax * !# 1 ~ 2/3!@ E py E py * 3 ~ E sx 1 E ax 1 E px ! 1 E px E py * ~ E sy 1 E ay ! 1 E px E py ~ E sy * 1 E ay * !# 1 ~ 1/3! 3 @ E py E px * ~ 2E sy 1 2E ay 1 E py !

2. PHASE-MATCHED FOUR-PHOTON MIXING PROCESSES IN A SINGLE-MODE BIREFRINGENT FIBER

1 E py E py ~ E sx * 1 E ax * !# % .

A. Derivation of the Nonlinear Terms in the Propagation Constants In what follows, we assume a phase-matched continuous wave process in which the field on each polarization can be expressed as the superposition of three monochromatic waves of pulsation, vs , vp , and va , corresponding to the Stokes, the pump, and the anti-Stokes waves, respectively: E ~ z, t ! 5 1/2~ xˆ E x 1 yˆ E y ! 1 c.c.,

(1)

with E x 5 E sx 1 E px 1 E ax 5 A sx exp@ i ~ b sx z 2 v s t !# 1 A px exp@ i ~ b px z 2 v p t !# 1 A ax exp@ i ~ b ax z 2 v a t !# and v p 2 v s 5 v a 2 v p 5 D v . 0. The same expression holds for the y polarization. The nonlinear dielectric polarization created by this field can be written as P NL~ z, t ! 5 1/2~ xˆ P x 1 yˆ P y ! 1 c.c.

(2)

P i , i 5 x or y, denotes the nonlinear polarization on the principal axis i and can be expressed, in an isotropic medium, as a function of the complex amplitudes E x and E y of the field16: Pi 5

3« 0 4

( @x j

~3! xxyy E i E j E j *

3! 1 x ~xyyx E jE jE i*# ,

3! 1 x ~xyxy E jE iE j*

(3)

3) 3) 3) 3) where i, j 5 x or y and x (xxxx 5 x (xxyy 1 x (xyxy 1 x (xyyx . In Eq. (3) the term corresponding to third-harmonic generation has been omitted, inasmuch as no phase matching occurs for this process. Note that only elements of the third-order susceptibility tensor x(3) for which the indices are equal in pairs are nonzero. As a direct consequence of this property of an isotropic medium, waves interacting in a FWM process must be also polarized in pairs (either all on one axis or two on each axis). In the case of silica fibers, for which the dominant contribution is of electronic origin, the elements of the three-component tensor in Eq. (3) have nearly the same magnitude and can be assumed to be equal. In the undepleted-pump approximation, only terms that include at least two pump fields are retained in the expression for the polarization. Hence we obtain from Eqs. (1) and (3)

(4)

We obtain the expression for P y from Eq. (4) by interchanging x and y. Let ba i be the real part of the propagation constant of wave a on polarization i. We have

b a i 5 b a i L 1 b a i NL 5 ~ n a i L 1 n a i NL! K a ,

(5)

where K a 5 v a /c is the free-space wave vector for the wave a. L and NL designate, respectively, the linear and the nonlinear parts of the propagation constants and of the refractive indices n a i . The nonlinear real part of the index can be derived from the first terms of self- and cross-phase modulation within the first two sets of brackets in Eq. (4). Indeed, these terms act as pure dephasing terms on the waves, and the other terms imply energy exchange between the waves. We have n a x NL 5 n 2 ~ r a 1 E px E px * 1 r a 2 E py E py * ! ,

(6)

3) with n 2 5 3 x (xxxx /8n, where n is the mean linear refractive index of silica in the considered spectral range; r a1 5 r s1 5 2, r p1 5 1, r a 2 5 r s 2 5 r p 2 5 2/3. The same expression holds for n a y NL when x and y are interchanged. In other words, the Stokes and the anti-Stokes waves are shifted in phase twice more by a pump wave on the same axis than by the pump by itself. If the pump is on the other axis the shift is identical for the three waves, and its value is the value of the shift of the pump by itself multiplied by 2/3. In what follows, we represent the total intensity of the pump waves as I 5 E px E px * 1 E py E py * .

B. Phase-Matched Processes Here we consider the conditions that permit four waves to be phase matched. We neglect the influence of nonphase-matched waves. This influence is discussed in Section 3. The phase-matching condition is O 5 D b 5 b a 1 b s 2 b p1 2 b p2 5 D b L 1 D b NL. (7) In a spectral region far from the zero-GVD wavelength, b L can be approximated13 as L b aLi > b pi 1

]bx 1 ] 2b x ~Ka 2 Kp! 1 ~ K a 2 K p !2. ]K 2 ]K2 (8)

In what follows we assume that ( ] b x )/( ] K 2 ) 5 ( ] 2 b y )/( ] K 2 ) 5 ( ] 2 b )/( ] K 2 ), we denote by DK the positive difference K a 2 K p 5 K p 2 K s , and we use K for the pump wave vector K p . Equation (7) can be rewritten as 2

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] 2b

O 5 D b 5 D xy 1

]K

2

Lantz et al.

DK 2 1 D b NL,

(9)

where Dxy is a group birefringence term obtained from the two first terms in relation (8). We assume in what follows that the x and the y axes correspond, respectively, to the slow and the fast axes of the fiber. Because the number of waves on each polarization must be even, processes 2–4 of Ref. 12 cannot actually lead to parametric amplification. Hence three cases are to be considered, and they are treated in the three next subsections. 1. All Waves on the Same Polarization Dxy is zero and D b NL 5 2n 2 IK is positive. Hence ( ] 2 b )/( ] K 2 ) must be negative, yielding the first phenomenon, called modulation instability.2 We mention this situation for completeness, but it will not be treated further. 2. Pump Equally Divided, Stokes Wave on One Polarization and Anti-Stokes Wave on the Other (Process 1) The case of interest is the positive-GVD region, where the dispersion and the nonlinear terms are both positive: D b NL 5 n 2 IK. Hence Dxy must be negative. The fast Stokes wave must propagate on the slow x axis, whereas the slow anti-Stokes wave propagates on the fast y axis. This situation, which corresponds to process 1 of Ref. 12, is characterized by the equations D xy 5

S

D

]by ]bx 2 DK ]K ]K

Db 5 0 5

] 2b ]K 2

D

]by ]bx 2 DK 2 1 DK 1 n 2 IK. ]K ]K (10)

Equations (10) are of second degree in DK, with the roots DK

5

S

D FS

]bx ]by 2 6 ]K ]K

]by ]bx 2 ]K ]K 2

] 2b ]K 2

D

3. Pump on One Axis, Stokes and Anti-Stokes on the Other Only when the pump is on one axis and the Stokes and anti-Stokes waves are on the other is the nonlinear term negative: D b NL 5 22/3n 2 IK. Hence solutions exist in the positive-GVD region, at least in theory, for both polarizations of the pump: (a) Pump polarized on the slow x axis (process 2) We have4 D xy 5 2 ~ b Lpy 2 b Lpx ! , 0, DK 5 $ @ 2 ~ b Lpx 2 b Lpy ! 1 ~ 2/3! n 2 IK # / ~ ] 2 b / ] K 2 ! % 1/2. (12) This situation corresponds to process 5 of Ref. 12, the second MI process in Ref. 3, and first experiments of parametric amplification in birefringent fibers.18 There is no power threshold for the pump, unlike for the following process. (b) Pump polarized on the fast y axis (process 3) When the pump is polarized on the y axis, the situation is called polarization instability,3,19 and the nonlinear birefringence must compensate for both linear birefringence and dispersion; DK stills obeys Eqs. (12), but Dxy is positive, inducing a threshold on the minimum power that permits exact phase matching: I . 3~ b y 2 b x !/~ n 2K !.

D xy , 0,

S

clude that the coupling among all four sidebands inhibits the growth of the waves corresponding to the smallest spectral shift.

2

2 4n 2 IK

] 2b ]K 2

G

1/2

.

(11)

It results from Eq. (11) that • Positive real solutions exist only if the group birefringence is sufficient to overcome the nonlinear term, unlike in Ref. 12 where there were always a solution because the nonlinear term was neglected. • If solutions exist, there are two of them. This consequence is questionable, because two amplified spectral bands were never observed either in experiments17 or in the analysis of modulation instability.5–7 However, Ref. 7 presents results obtained with a truncated model of MI that does not include the non-phase-matched waves. With such a model a spurious gain curve appears, corresponding to the smallest spectral shift solution of Eq. (11). Hence, according to the analysis of Ref. 7, we may con-

(13)

Actually, the gain is nonzero for a power higher than half that of the perfect phase-matching threshold.3 In practice, the power is so high that a high-birefringence fiber would be destroyed. Polarization instability can be observed only in a low-birefringence fiber.20 Table 1 summarizes the results of this section. It should be noted that Dxy is proportional to wave vector K for processes 2 and 3 [see Eqs. (12)], whereas the group birefringence is proportional to the wave-vector shift DK in process 1 [see Eqs. (10)]. Hence the phase-matched wavelength shift DK is much more important in processes 2 and 3 than in process 1.

3. INFLUENCE OF NON-PHASE-MATCHED WAVES ON THE GAIN AND THE PHASE-MATCHED WAVELENGTHS FOR A PUMP POLARIZED AT 45° OF THE PRINCIPAL AXES The frequency shift involved in process 1 is much smaller than in processes 2 and 3. In this section we show that non-phase-matched waves cannot be neglected for such a small frequency shift. Roughly speaking, these nonphase-matched waves cancel the amplification for the smallest frequency shift solution of Eq. (11), whereas they lower the gain and modify the phase-matching conditions for the biggest root of Eq. (11). In process 1, E sx , E px , E ay , and E py are the four phase-matched waves; E ax and E sy are not phase

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119

Table 1. Summary of the Three Phase-Matching Processesa Polarization of Individual Waves Reference First Experiment

Process

Anti-Stokes

Stokes

P1

P2

16

1

S

F

S

F

Phase-Matched Frequency Shift

DK 5

S

Dxy 5

S

D

]by ]bx 2 DK , 0, ]K ]K

D FS

]bx ]by 2 6 ]K ]K

]by ]bx 2 ]K ]K

a

2

F

F

S

S

19

3

S

S

F

F

S and F denote, respectively, the slow and the fast axes of the fiber.

matched. We can classify the terms within the first two sets of brackets of Eq. (4) into four categories: • The first terms in each set of brackets are pure dephasing terms and were analyzed above. • The terms E px E px E sx * , (2/3)E px E py * E ay , and (2/3)E px E py E sy * generate the non-phase-matched wave E ax . Note that E px E py E sy * is small in comparison with the two other terms inasmuch as E sy is not phase matched. The other non-phase-matched wave E sy is generated by similar terms in P y . • The terms E px E px E ax * and (2/3)E px E py * E sy are responsible for the modification of the phase-matched wave E sx by the non-phase-matched waves E ax and E sy . E ay is modified by similar terms in P y . • Last, (2/3)E px E py E ay * and its counterpart on the y polarization are terms that generate exponential gain for the waves E sx and E ay , provided that phase matching occurs. In what follows, we neglect the last term in brackets in Eq. (4). This term is responsible for energy exchanges between polarizations involved in processes 2 and 3. However, it can be neglected in process 1 because the associated coherence length (i.e., the length at which energy exchanges occur in one sense) is of the order of the beat length, i.e., much smaller than the fiber length in a highbirefringence fiber.15 A. Amplitude of the Non-Phase-Matched Waves We now develop a first-order model of the interaction between the phase-matched and the non-phase-matched waves. First, the amplitudes of the non-phase-matched waves are determined when phase matching occurs, in the usual sense of Eqs. (10). Then we show that taking these non-phase-matched waves in account leads to a correction of the phase-matching conditions. This model is based on the following assumptions: • The pump waves remain undepleted. Let P 5 I/2 5 E p j E pj * be the pump intensity on one polarization, j 5 x or y.

D xy DK D xy DK

5 5 5 5

2

2 4n 2 IK

] 2b ]K 2

G

1/2

] b 2

2 17

D

]K 2

2( b y 2 b x ) , 0, $ @ 2( b x 2 b y ) 1 (2/3)n 2 IK # /( ] 2 b / ] K 2 ) % 1/2 2( b y 2 b x ) . 0, $ @ 2( b x 2 b y ) 1 (2/3)n 2 IK # /( ] 2 b / ] K 2 ) % 1/2

P1 and P2 denote the pump fields.

• The amplitudes of the non-phase-matched waves are small with respect to the phase-matched waves. Hence only the phase-matched waves generate the nonphase-matched waves. • The phases wa j of the six interacting waves E a j permit a pure exponential amplification. Actually, because the last bracketed term in Eq. (4) has been neglected, the interaction between polarizations is incoherent, and only phase differences on one polarization make sense. Moreover, the assumption of a pure exponential amplification means that, in the reference (z, t) of the laboratory, these phase differences evolve during propagation at a common speed proportional to the mean group velocity ¯v : ¯ !, w sx 2 w px 5 ~ w 0 ! sx 2 D v ~ z/v ¯ !, w ax 2 w px 5 ~ w 0 ! ax 1 D v ~ z/v ¯ !, w sy 2 w py 5 ~ w 0 ! sy 2 D v ~ z/v ¯ !, w ay 2 w py 5 ~ w 0 ! ay 1 D v ~ z/v with

S

(14)

D

1 ]by 1 ]bx 1 3 1 5 . ¯v c 2 ]K ]K The four values (w0)a i are constants as long as the amplification is purely exponential. It is convenient to express the fields in a frame of reference moving with the pulse ¯ ) # in which the second terms in Eqs. (14), @ z, t 5 t 2 (z/v i.e., the propagation terms, vanish. Moreover, we can take the pump phase as the phase origin on each polarization: w px 5 w py 5 0. These conventions are used in the following discussion. Equations (4) and (14), the first undepleted-pump assumption, and these conventions give the equations of propagation of the four Stokes and anti-Stokes waves:

F

G

] E aj sgn~ a ! sgn~ j ! D xy ] 2b 1 5i 1 DK 2 1 n 2 PK E a j ]z 2 2 ]K 2 1 in 2 PK @ E s j * 1 ~ 2/3!~ E a m 1 E s m * !# ,

(15)

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where either [a 5 a, s 5 s, sgn(a) 5 21] or [a 5 s, s 5 a, sgn(a) 5 1] and either [ j 5 x, m 5 y, sgn( j) 5 1] or [ j 5 y, m 5 x, sgn( j) 5 21]. In Eq. (15) the two first terms are linear. For example, the linear phase mismatch between E sx and E px can be expressed in the reference (z, t) as

w u sx ~ z, t ! 5

S

D

5

S

D

S

D

]bx ]by 1 z ]K ]K (17)

S

2

D

1 ] 2b D xy 1 DK 2 1 n 2 PK E ax 2 2 ]K 2

1 in 2 PK @ E sx * 1 ~ 2/3! E ay # ,

(18)

with, when we use the phase-matching condition given in Eqs. (10),

(21)

where g is the usual gain per unit length; i.e., g 5 (2/3)n 2 PK because phase matching is assumed to be perfect.1 Figure 1 depicts the conditions of amplification of a non-phase-matched wave. It can easily be shown that Eqs. (20) and (21) imply that i E ax i in 2 PK @ 1 1 ~ 2/3! i # . 5 i E sx i iD xy 1 ~ 2/3 ! n 2 PK

(22) Hence the assumption of a small non-phase-matched wave in comparison with the phase-matched wave is equivalent to 2D xy @ (2/3)n 2 PK. With this assumption, we have the approximation i E ax i n 2 PK ~ A13/3! > , i E sx i u D xy u

w sx 1 w ax > 22.56. (23)

Relation (23) means that the amplitudes of the nonphase-matched waves are proportional to the ratio between the nonlinear part of the index and the group birefringence, in the limit where this ratio remains much smaller than 1 but cannot be neglected. We obtain relations for wsy and i E sy i by replacing E ax with E sy and E sx with E ay in Eq. (22) and relation (23). As a result, we still obtain relation (23) for the amplitude of E sy , whereas the relation for the phases becomes

w sy 1 w ay > 22.56 ⇒ w sy 2 w sx > 2256 2 ~ p /2! .

(19)

Equation (19) means that linear and nonlinear dephasing terms are added to each other for a non-phase-matched wave and that they cancel each other for a phase-matched wave [see Eqs. (10)]. Because E sx and E ay are perfectly phase matched, the sum of their phases (with respect to the pump waves) is equal to p /2.21 Hence, using Eq. (19) and taking i E ay i 5 i E sx i , we can rewrite Eq. (18) as

H

i dE sx i i E sx i

(24)

D xy 1 ] 2b x 2 1 DK 2 1 n 2 PK 5 2D xy . 2 2 ]K 2

dE ax 5

5

5 gdz 5 ~ 2/3! n 2 PKdz,

exp@ i ~ w sx 1 w ax !#

D xy 1 ] 2b 1 z. 2 2 ]K 2

Equation (17) and similar equations for the three other waves lead to the two first terms in Eq. (15). The third term in Eq. (15) is due to the difference between the nonlinear phase shifts of the pump and of the Stokes and the anti-Stokes waves [see Eq. (6)]. Finally, the gain terms within the second set of brackets in Eq. (15) can easily be derived from Eq. (4). Neglecting the influence of the non-phase-matched wave E sy *, we can rewrite Eq. (15) for the non-phasematched wave E ax as

] E ax 5i ]z

i dE ax i i

(16)

We can express this linear phase mismatch in the reference (z, t), using Eqs. (14), as DK 2

dE ax D 1 dE ax' 5 0;

i E ax i

]bx 1 ] 2b DK 1 DK 2 z. ]K 2 ]K 2

w u sx ~ z, t ! 5 w u sx ~ z, t ! 1

propagation, whereas the relative gains on all the involved waves must be identical. Hence we have

B. Relation among the Non-Phase-Matched Waves, the Phase-Matching Conditions, and the Gain We can write Eq. (15) for E sx using Eqs. (10), as

] E sx 5 in 2 PKE ax * 1 ~ 2/3! in 2 PK ~ E sy 1 E ay * ! . ]z (25)

i E sx i ] E ax dz 5 E ax dz 2iD xy 1 in 2 PK ]z E ax

F

3 exp~ 2i w sx ! 1 ~ 2/3! exp i 5 dE ax D 1 dE ax' 1 dE ax i ,

S

p 2 w sx 2

D GJ (20)

where dE ax D is the dispersion term that is due to the first term in Eq. (20), dE ax' is the imaginary part of the second term in braces, and dE ax i is the real part, responsible for the gain, of this second term. Inasmuch as the amplification is assumed to be purely exponential, relative phases do not change during the

Fig. 1. Conditions of amplification of a non-phase-matched wave. The gain term dE ax' 1 dE ax i is almost at right angles to the wave to cancel the dispersion term dE ax D .

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121

neglected with respect to this birefringence. The curve of the actual gain is given in Fig. 3 and is described in Section 4.

4. COMPARISON WITH MI FORMALISM

Fig. 2. Directions of the three vectors that take part in the amplification of the phase-matched E sx vector. The values of angles a1 and a2 are a 1 5 ( p /2) 2 ( w sx 1 w ax ) > 4.13 rad and a 2 5 w sx 1 w ax > 22.56 rad.

The MI formalism, developed in Ref. 6 for the case of process 1, consists in finding the stationary solutions of the system of four equations with four unknowns given by Eq. (15). The unknowns are the complex amplitudes E ax , E sx , E ay , and E sy . A purely exponential amplification corresponds to an imaginary negative eigenvalue of the matrix associated with this system. The corresponding

To show the corrections that are due to the non-phasematched waves, we can rewrite Eq. (25) in the form

] E sx 5 E sx @~ 2/3! n 2 PK 1 Dg 1 ~ iD b corr/2!# , ]z

(26)

where Dg is a corrective term of the gain and Dbcorr is a corrective term of the nonlinear propagation constant. Figure 2 presents the directions in the complex plane of the vectors that appear in Eq. (25). Now we can conclude with respect to the influence of the non-phase-matched waves that • In the phase-matching condition [Eqs. (10)] the nonlinear term n 2 IK must be replaced by DbNL, with D b NL 5 n 2 IK 1 D b corr

S

5 n 2 IK 1 1

i E ax i i E sx i

H F sin

1 ~ 2/3! sin~ w sx 1 w ax !

S

> n 2 IK 1 1

D

p 2 ~ w sx 1 w ax ! 2

JD

G (27)

13 n 2 PK . 9 D xy

(28)

The amplitude of the correction is of the order of the ratio between the phase-matched and the non-phase-matched waves given in relation (23). With this correction the imaginary term in Eq. (26) disappears, and the assumption of a pure exponential amplification is verified. • The gain g per unit length of the fiber in an actual parametric amplification is smaller than the gain given when one takes into account only the four phase-matched waves: g 5 ~ 2/3! n 2 PK 1 Dg 5 n 2 PK

F

i E ax i 2 1 i E sx i 3

S HF DG D cos

p 2 ~ w sx 1 w ax ! 2

GJ

1 ~ 2/3! cos~ w sx 1 w ax !

(29)

S

(30)

> ~ 2/3! n 2 PK 1 1

2n 2 PK . D xy

Because the correction is inversely proportional to the birefringence, the gain attains its FWM value only for high birefringence, where the nonlinear index variation can be

Fig. 3. Values of (a) the gain and of (b) the wavelength shift versus the birefringence by use of the linear model of phase matching12,13 (lighter dotted curve), the FWM theory with a nonlinear part of the propagation constant1 (dotted–dashed curves), and the MI model5–8 with the gain maximum for a given wavelength (dashed curves) and a given birefringence (heavier dotted curves). In (a) the values of the gain are given in both cases versus the birefringence. The corresponding wavelength is shown in (b), where the corrected FWM model (this paper) is shown by the solid curves. The value of the pump intensity is I 5 (108 V/m) 2 .

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Table 2. Comparison between Results of FWM of MI Formalismsa Values in perfect phasematching of

Group Birefringence Index for Phase Matching ( 3 1024 ) Phase-Matched Wavelength Shift

Formalism

Dl 5 3 nm

FWM MI FWM MI

Dl 5 6 nm

Linear Model

Nonlinear without Correction

Nonlinear with Correction

Gain (m21 )

w sx 1 w ax

i E sx i / i E ax i

3.93 – 7.83 –

6.07 – 8.90 –

5.54 5.60 8.81 8.82

3.16 3.26 4.16 4.18

22.67 22.68 22.59 22.59

0.21 0.20 0.072 0.070

a The nonlinear model without correction1,14 does not take in account the non-phase-matched waves, unlike our nonlinear model with correction. The parameters are l 5 532 nm, n 2 5 1.2 3 10222 m2/V2, and I 5 (108 V/m) 2 . The gain from the standard FWM formalism is 4.72 m21 , and the limit value of w sx 1 w ax for high birefringence is 22.56 rad [see relation (23)].

eigenvector gives the complex values of the waves, which can be compared with the approximations given in Eq. (22). Table 2 gives a comparison of the MI model and of the results of our corrected FWM model given by Eqs. (22), (27), and (29) for two values of the birefringence. The agreement between the two models is very good, and the results confirm the link between the gain and the birefringence. We have derived the analytical expression of the gain by unit length g from the eigenequation given in Ref. 6 as

S

D

F

D xy 2 k 22 k 2p k 22 2g 2 5 1 DK 4 2 p 2 DK 4 DK 2 1 4 2 4 9 D xy ~ 2k 2 pDK 4 1 k 2 2 DK 6 ! 4 2

1

G

1/2

.

(31)

In Eq. (31) two new notations have been adopted to improve the readiness: k2 5

] 2b ]K 2

,

p 5 n 2 IK.

Phase matching corresponds to a maximum of gain with respect to either the birefringence or the wavelength shift of the signal. We can analytically determine such maxima by equating to zero the partial derivatives of Eq. (31). For a given birefringence the derivation of Eq. (29) with respect to the wavelength shift leads to an equation of the third degree. The analytical solution of the equation is straightforward but is not given here because it cannot be expressed in a reasonably compact form. On the other hand, the birefringence that leads to a maximum gain can be expressed for a given wavelength shift as

D xy 5 22

S

8 2 k 2 3 DK 4 p k 2 1 pk 2 2 DK 2 1 9 4 2p 1 k 2 DK 2

D

1/2

. (32)

If 2p/(k 2 DK 2 ) ! 1, Eq. (32) can be expanded to the successive orders of 2p/(k 2 DK 2 ). Results are as follows: Zero order:

D xy 5 2k 2 DK 2 .

(33)

Equation (33) is the linear phase-matching condition given in Ref. 12.

First order:

S

D xy 5 2k 2 DK 2 1 1

p k 2 DK 2

D

. (34)

Equations (34) and (10) are equivalent. They express the nonlinear phase-matching condition obtained by neglecting the non-phase-matched waves.1 Second order:

F

D xy 5 2k 2 DK 2 1 1

p k 2 DK 2

2

~ 13/9! p 2

2k 2 2 DK 4

G

.

(35)

Starting from Eq. (9), we can obtain Eq. (28) from Eq. (35) by replacing k 2 DK 2 by its zero-order approximation 2D xy in the denominator of its second-order corrective term. Hence the physical meaning of this corrective term is to take in account the non-phase-matched waves when they remain small with respect to the phasematched waves. The same approach has been taken for the gain. The first step is to obtain an analytical expression of the maximum of this gain by substituting Eq. (32) into Eq. (31). The usual constant gain g 5 p/3 appears as a secondorder expansion of this exact analytical expression, whereas relation (30) is obtained by a third-order expansion. Figure 3 shows the curves of the gain and of the phasematched wavelength shift versus the birefringence for the five models that have been proposed. In the nonlinear model of phase matching without correction [Eqs. (10)], two wavelength shifts give perfect phase matching for a single value of the birefringence, whereas there is only one phase-matched wavelength in the MI model. For weak birefringence, two different phase-matching curves are obtained for the MI model, representing either the maximum of the gain for a given value of the birefringence or the maximum of the gain for a given value of the wavelength shift. For higher birefringence these curves and the approximation derived from Eq. (27) become similar. Hence relation (28) can be considered an accurate determination of the phase-matching conditions if the birefringence is sufficiently high to permit a unique definition of phase matching. Moreover, only such values of the birefringence as are accurately given by relation (30) permit high gain values.

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5. SIMULATION BY NONLINEAR ¨ DINGER EQUATION SCHRO The plane-wave formalism used in the corrected FWM and MI models remains valid for pump pulses if the pump linewidth is less than the parametric bandwidth.1 Indeed, the products of the fields that appear in the expression for the nonlinear polarization [Eq. (4)] become convolution products in Fourier space22 that use all frequency components of the pump, whatever the distribution of these components is. As a confirmation, picosecond pulses interacting in a 3-m-long fiber were simulated by two standard coupled nonlinear Schro¨dinger equations. We found that the maximum of the gain occurs exactly for the birefringence values that give a maximum gain with the MI formalism. Table 2 shows that these birefringence values are also given with excellent precision by our corrected FWM model. Comparison of the gains is more difficult because purely exponential amplification occurs only if the phases of the interacting waves have precise values. Figure 4 shows the evolution of the gain during the propagation in the fiber when only the signal is injected into the fiber. As the idler is not injected, the amplification is phase insensitive, and the gain becomes purely exponential when the idler level has attained the signal level. In this linear part of the curve the gains are in good agreement with the values shown in Table 2.

6. EXPERIMENTAL RESULTS Gaussian pulses of 35-ps duration were delivered by a mode-locked Nd:YAG laser and frequency-doubled by a KDP crystal. The second-harmonic output, at l 5 532 nm, was launched in a 3.4-m-long high-birefringence fiber and was polarized at 45° to the principal axes. The spec-

Fig. 5. (Left) Spectra of the pump and the amplified waves at the output of a 3.4-m-long birefringent single-mode fiber for increasing pump power. (Right) Magnified spectrum of the SPMbroadened pump. The values of the power can be deduced from the number of peaks in this spectrum.

trum at the output was analyzed by a Littrow grating spectroscope. A more comprehensive description of the experimental setup can be found in Ref. 23. In the experiment described in the present paper no signal was injected, and phase-matched amplification from the noise occurred. Figure 5 shows the amplification for increasing pump power in the spectral plane of the spectroscope. This increase can be calibrated by the width of the pump spectrum, broadened by self-phase modulation (SPM). We can clearly see that both the Stokes and the anti-Stokes bands move toward the pump when the pump power increases. Analogous results, with smaller spectral shifts, were obtained by Park et al.14 Let us recall that this shift is due to the change in the ratio between the nonlinear dephasing term and the dispersion term, whose sum cancels the birefringence to yield phase-matched waves. For the highest powers, mixing terms between the pump and the signal appear, resulting in a quasi-continuum. The number N of peaks in the SPM-broadened pump spectrum, magnified at the right in Fig. 5, gives the nonlinear phase shift of the central part of the pump pulse24: N p 5 n 2 IKL,

Fig. 4. Gain for the signal during the propagation of picosecond pulses through two 3-m-long fibers whose birefringences give wavelength shifts Dl 5 3 nm (solid curve) and Dl 5 6 nm (dashed curve) for perfect phase matching. The slopes of the straight lines obtained after the first 0.5 m correspond to a gain coefficient of 3.1 m21 (Dl 5 3 nm) and 3.9 m21 (Dl 5 6 nm). The pump intensity is I 5 (108 V/m) 2 . The values of the birefringence are 5.60 3 1024 (Dl 5 3 nm) and 8.82 3 1024 (Dl 5 6 nm), corresponding to perfect phase matching (see Table 2).

(36)

where L is the fiber length. From this value and from the experimental mean spectral shift of the Stokes wavelength, the linear birefringence that ensures perfect phase matching can be calculated with relation (28) and Eqs. (10). Results are given in Fig. 6 and are compared with the value deduced from the beat length of the fiber. The figure also shows the birefringence values obtained with the linear model of phase matching12,13 or with a nonlinear model without correction1,14 [Eqs. (10)]. It appears that the corrected FWM model permits an accurate calculation of the birefringence, especially for the lowest values of the pump power. For all values of the pump power our model provides a birefringence value that is much more accurate than those from the two other models. These experimental results are also in good agreement with curves of Fig. 3. Indeed, the pump power used

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Lantz et al.

weakest value of the birefringence, whereas the amplification is limited by the pump depletion for the highest value.

7. CONCLUSION We have given precise conditions for phase matching in four-photon parametric amplification with a divided pump in a birefringent fiber, taking in account the nonlinear dephasing terms and the influence of the nonphase-matched waves. Simple analytical formulas to foresee the phase-matched wavelengths accurately have been given and experimentally verified. In accordance with the formalism of modulational instability, the model that we developed exhibits the influence of the fiber birefringence on the actual maximum gain value. This influence has been confirmed by experience, for the first time to our knowledge.

ACKNOWLEDGMENT Fig. 6. Comparison of the birefringence values calculated from both the experimental pump power and the experimental Stokes wavelength shift by use of the beat length (dashed–dotted curve), the linear model of phase matching12,13 (crosses), the FWM theory with a nonlinear part of the propagation constant1,14 (circles), and the corrected FWM model of this paper (asterisks). The value of the pump intensity, I 5 (108 V/m) 2 , used in Table 2 and in Figs. 3 and 4, corresponds to 15 peaks in the SPMbroadened pump spectrum.

We thank Jean Botineau (Laboratoire de Physique de la Matie`re Condense´e, Universite´ de Nice) for his helpful comments about this research.

REFERENCES 1. 2. 3. 4.

Fig. 7. Spectra of the pump and the amplified Stokes waves at the exit of two different 3-m-long birefringent fibers: fiber 1, birefringence 5 3 1024 ; fiber 2, birefringence 2 3 1024 . The pump power, calibrated by the width of its SPM spectrum, is identical in both fibers. Dl represents the mean wavelength shift.

5. 6.

7.

in Fig. 3 corresponds to a number of peaks N 5 15 for a 3.4-m-long fiber. From the linear birefringence value calculated from the beat length as being 4.54 3 1024 , it is possible to evaluate in Fig. 3 the phase-matched wavelength shift. The result of 1.95 nm corresponds rather well to the experimental value of 2.1 nm given in Fig. 6. Note that, with the nonlinear FWM model without correction and with the pump power used in Fig. 3, there is no wavelength shift that corresponds to phase matching with a linear birefringence value smaller than 5.8 3 1024 . Figure 7 shows amplification in two different 3-m-long fibers, whose linear birefringence index differences are approximately 2 3 1024 and 5 3 1024 . The pump powers, measured by the width of the spectral broadening that is due to SPM, are the same in both experiments. The difference between the gains appears clearly in this figure: The amplified signal is scarcely visible for the

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