3.4 The Order Parameter Equation for Photorefractive Oscillators

59

∂A

∂y1

∂y2

∂y1

= ε2

+ ε3

+

∂t

∂T 1

∂T 1

∂T 2

= ε2

−i

γ1γ2

(˜a1 − a˜2) Θy1

+ ε3

∂y2

+

∂y1

.

(3.46)

γ1 + γ2

∂T 1

∂T 2

Finally, the evolution equation of the order parameter can be written in terms of the original parameters as

1 ∂A				= (E − 1) A − A |A|2 − id ω − 2 A −	1	ω − 2	2	A , (3.47)
Γ		∂t			2

where Γ = γ1γ2/(γ1 + γ2) is the decay rate of the order parameter, and d = (˜a1 − a˜2)/2 is a di raction coe cient.

Equation (3.47) is a complex Swift–Hohenberg (CSH) equation, formally identical to the order parameter equation derived for lasers in the preceding chapter.

A multiple-scale expansion is also possible in the case of large pump detuning, leading to

1 ∂A = (E − 1) A − A |A|2 − id ω − 2 − ω0 |A|2 A Γ ∂t

−12 ω − 2 − ω0 |A|2 2 A + 12ω0 A 2A − A 2A A , (3.48)

which is the CSH equation with a nonlinear resonance, as derived in [6].

3.4 The Order Parameter Equation for Photorefractive Oscillators

3.4.1 Description and Model

A photorefractive crystal is a nonlinear medium that responds to the light intensity via the electro-optic e ect, where spatial variations in the refractive index are induced according to the light profile. When the crystal is placed inside a resonator and subjected to an optical pump, this nonlinear optical system is called a photorefractive oscillator. The pump wave, when scattered by the imperfections of the crystal as it passes along the optical axis, initiates an oscillation process, generating a signal wave. During the process, both the pump and the generated waves are present in the resonator.

The total optical field inside the resonator is given by

¯ − −

E(r, t) = Ap(r, t) exp(ikpr iωpt) + As(r, t) exp(iksr iωst) + c.c. . (3.49)

where the indices p and s attributes to the pump and signal waves, respectively. This induces a spatial modulation of the refractive index,

60 3 Order Parameter Equations for Other Nonlinear Resonators

n¯(r, t) = n(r, t) exp(iqr − iΩt) + c.c. ,

(3.50)

where Ω = ωp − ωs and q = kp − ks.

In the mean-field limit, the equations describing the evolution of the signal wave and the refractive index are (the details of the derivation can be found

in [7])
∂A	= κ − (1 + iβ) A + in + ia 2A ,	(3.51a)
∂t

∂n	= −γ	n − in − ns	A		(3.51b)
				,
∂t			1 + |A|2

where A = As/Ap is the normalized signal field, ns is the saturation value of the index grating, β is the detuning of the resonator frequency from the center of the gain line, and κ and γ are the decay rates of the photon and index gratings, respectively. Usually, the condition γ κ holds, which allows the adiabatic-elimination of the optical field. In the next section, the adiabatic elimination technique is used to derive an order parameter equation for PROs.

3.4.2 Adiabatic Elimination and Operator Inversion

The envelope of the refractive-index grating can be expressed in terms of the optical field by assuming that the field is a fast-relaxing variable, i.e. ∂A/∂t = 0. In this case

n = 1 − iβ + ia 2 A .

(3.52)

Substituting (3.52) in (3.51), and letting the di erential operator act on

both sides of (3.51) yields
	∂A		= −(1 + i)A +	ins		A		(3.53)
							,
	∂τ			1 + iβ − ia 2		1 + |A| 2

where τ = γt is a normalized time.

The stationary solution of (3.53) is A (r, t) = A0 exp (ikr), where A0 = ns/2 − 1 and ak2 = 1 − β. This means that certain spatial modes with wavenumbers k proportional to the resonator detuning are favored. Note the presence of a constant frequency shift β = 1, which is di erent from the case of a laser.

The di erential operator in (3.53) can be expanded in a Taylor series,

ins

=

ins

(3.54)

1 + iβ

ia

2

(1 + i) + i

β

1 − a 2

− ns

1 + i

−

2

i

2

2

+ · · · .

≈ (1 + i)

1 −

β − 1 − a

−

β − 1 − a

2

2

4