Beam 1

THREE Beam Coupling

Beam 1 Beam 1

Beam2

Figure 3.2 Beam coupling configuration. Two plane waves of arbitrary polarizations are incident on the photorefractive crystal. The beam propagation coordinate system is shown as the x, y, and z axes.

of the propagation direction.

The propagation of the two interacting waves can be written, in terms of the normal modes, as

E1 (z, t)

=

½[All) (z)ell) ^{ei(ki1) ·r-wt)}

+

Al2) (z)el2) i(kil) ·r-wt)]

+

c.c., E2(z, t)

=

½[A11) (z)e11) ei(k~1) ·r-wt)

+

A12) (z)e12) ^ei(k~l) •r-wt)]

+

c.c.,

(3.la) (3.lb) where e~m) (m, n

=

1, 2) denotes a unit complex vector representing a state of polarization at the m ^th normal mode of the beam n in the beam propagation coordinate system, kim) the corresponding wave vector, and Aim) the complex amplitude of the beam n at the m^th normal mode. Aim) is allowed to depend only on z. This corresponds to the paraxial approximation in photorefractive beam coupling. The effect of optical activity can be incorporated into the normal mode representation. We note that the orthonormality of the normal modes, i.e., e}m) · e}~)*

=

bmn (1, I', m, n

=

1, 2) can still hold in the beam propagation coordinate system. The wave equation for the complex amplitude of the total field vector Ew (

= Ei + E~)

of angular optical frequency w is given by

(3.2) where c is the speed of light in vacuum, 1 the second-rank linear dielectric tensor,

a

the second-rank conductivity tensor, and PNL the nonlinear polarization given by Eq. (2.30), all measured in the beam propagation coordinate system.

The arbitrary polarized two waves given by Eqs. (3.1) form an intensity- interference pattern in the crystal

(3.3)

and

(3.4a)

_ !

[A(2) A(2) * ( (2) (2) *)]

m2 - Io 1 2 e 1 · 8 2 • (3.4b)

This interference pattern creates the space-charge electric field in the photorefractive crystal [3.53], thereby causing PNL in Eq. (3.2) via the electrooptic effect. The wave vector and grating configuration ink space is depicted in Fig. 3.3.

In what follows we shall consider the case where the two input waves possess the same eigen polarizations which we denote

e1

^{1) and e~1). In} this case one photorefractive grating with grating vector K (l) is created initially inside the crystal. Because of the possible anisotropic Bragg condition ki²)

=

^k~1)

+

K(l)

and k~²)

=

^{k~1) -K(l)} [see Fig. 3.l(c)], two new waves having k vectors of ki²) and

k1

²⁾ are then created by the initial grating and write a new photorefractive grating with the grating vector K(²). We note that, for the most general case where the two input waves are arbitrarily polarized, at least six waves and three photorefractive gratings are involoved in the beam coupling process. These coupled-wave equations can be straightforwardly obtained by the symmetry consideration of the following result.

Substituting Eqs. (2.30), (2.32), and (3.1) into Eq. (3.2) and using the orthonormality of the normal modes and the slowly varying field approximation [3.8], i.e., ld²Ahm) /dz²¹

<

lkim)dAhm) /dzl, we obtain the following set of coupled- wave equations for the four interacting waves [3.54]:

d

dz

where

A(l)

A 1 ⁽²⁾ 1 A~l) A~2)

A ⁽¹⁾ A 1 (2)

A 1 ⁽¹⁾

A 2 (2) 2

(3.5)

z

Figure 3.3 Wave vector and grating configuration in k space. The case of a biaxial crystal such as KNbO₃ is depicted. In this example the x, y, and z axes correspond to the crystallographic c, a, and b axes, respectively. The k vectors k~¹) and k~²) correspond to the two normal modes of beam 1 inside the crystal, while k~¹⁾ and k~²⁾ correspond to those of beam 2. The grating vectors K(1) (= kp) - k~¹)) and K(2 ) (= k~²) - k~²)) correspond to the two different photorefractive gratings.

cos0(l) l

0

cos o<:a> _:a

(3.6a)

(3.6b)

(3.6d) and a: = 41rwu/c²kim) ~ a(m, n = 1, 2), where we assumed an isotropic conductivity of the crystal

a=

^{ul (I} is a unit matrix). Also in Eqs. {3.6) m1 and

m2 are given by Eqs. {3.4), and tl.k

=

{k~²) - k~¹)) · ez, tl.K

=

(K(^{2 ) -} K(^{1)) ·} ez, K!¹)

=

^{K(l) · ez, 9}₁(m)

=

^cos-1(k}m) · ez/lk}m)I) (1, m

=

1, 2). The parameters tl.n, d,

f,

"f, and g in Eqs. {3.6) are given in Table 3.1. We see from Eq. (3.5) that the submatrices [Pf] and [P~] denote the absorption and linear birefringence due to the application of the external DC electric field and the submatrices [P{']

and [P~'] den~te the photorefractive anisotropic self-diffraction [see Fig. 3.l(c)], while the submatrices

[Qi]

and

[Q

2 ] denote the photorefractive isotropic and cross- polarization beam coupling [see Figs. 3.l(a) and (b)]. We note that the coupling

'

Table 3.1 Parameters used in Eqs. (3.6).

w * ) n (l) *

i12) _ _____,,---,--::;(l)* i:(12)( (1) . ~

~.

(2))* i21 _ _ l_i12) l - (1) ""'sc r,.l el q el ' l - (2) l

2n₁ c n₁

/ (12) = W ::;(l) (12)( (l)*.

~ ~.

(2)) /(21) = n~l) /(12)*

l (1) ""'sc ~l e2 q e2 ' l (2) l

2n₂ c n₂

,(ij)

=

_w_::;(l)a(ij) (e(i)* . ~q~. e(j))

l 2 ⁽ⁱ⁾ ""'sc l 1 2

n₁ ^C

(ij) ^W ...,(l)* (ji)*( (i)* A~ (j)) n~i) (ji)*

91

=

----ny-~sc al e2 • u.q · el

=

"'l'I)'l

2n₂ c n₂

~nJij)

=

2

Efi) (a?j)ou

+

a[ij)t521)(eJi)* ·

~q·

^{e}j)), (i,j,l}

=

1,2) n,

.

A~ ~ (Z ) ~ u.q

=

^{l : r : e8c : £}

d12)

= ! / l

ei[(~k-KC'>)-(:z:e.,+11e11)ldx dy

~(12)

=

^..!:..

1· ^{r i[}

(.6.k-.6.K+KC'))·(:z:e.,+11e11)ldx dy

l

V ls .

(11) _ (22) _ l

al - a2 -

a~12)

=

^[a~21)]*

= ~ J l

i[(.6.k-.6.K)·(xe.,+ye 11 )ldx dy a~12)

=

^[a~21)]*

= ~ J l

ei[.6.k•(xe.,+ye 11 )ldx dy a~ll)

=

^[a~22)]*

= ~ J l

ei(.6.K-(:i:e.,+11e11)ldx dy

V

= 1• {

^dxdy

}(grating area)S

In the above formulae E₀ and

aiQ

are related to the space-charge electric field as

follows:

into other possible waves due to the linear birefringence is neglected for simplicity.

The inclusion of these waves also requires the inclusion of other possible photorefractive gratings. The present treatment is, however, complete for the case of anisotropic crystals without DC electric field and the two special cases described in the following section.

Some of the elements in the above matrices involve the modulation indices m₁ and m2 • The set of coupled-wave equations is thus nonlinear and general solutions to Eq. (3.5) are difficult to solve exactly. We therefore look for the exact and/or approximate analytic solutions for two special cases which are described in the following two sections.

3.3 Scalar two-beam coupling 3.3.1 Coupled-wave equa~ions

Scalar two-beam coupling is the simplest case since

ll.q

[= ll.x/(-Esc/41r), see Table 2.1] and therefore the submatrices [P1], [P2], [Q1], and [Q2] are all diagonal. This occurs, for example, when the space-charge field is parallel to the crystallographic

<

0()1

>

axis and

a) for 2mm, 4mm and 3m crystal symmetry classes the input beams are linearly polarized along one of the principal axes, or

b) for cubic 43m crystal symmetry class the input beams are linearly polarized along the crystallographic

<

110 > axis or

< 110

> axis [see Eqs. (2.33)]. In these cases Eq. (3.5) can be reduced to the following simple form:

, dA 1 a , ^I

1

2 . ( w ) ( 11)

· cos81 -d

=

--A1 - -

1 A1 A2 - i - ll.n 1 A1,

Z 2 O C

(3.7a) (3.7b) where we set A 1

=

Ai1

), A2

=

A~1

) and Ai2

)

=

A~2

)

=

0 for the two input waves of the same eigen polarizations (e₁

=

e~¹) and e₂

=

e~¹

>), 8L

2 is measured in a

crystal, and

(3.8) in which n~ is ne or n0 depending on whether the mixing beams are of extraordinary or ordinary polarization, and Bsc(=

a!J))

is given, for example, by

1 Eqe(En-iEo)

1

+

/3enw/seio Eqe

+

En - iEo. (3.9) for one species and one type of carrier transport [see Eq. (2.14)].

Equations (3.7a) and (3.7b) can be rewritten, in terms of the intensity and the phase of each beam A,-= .jf;ei,t,,(j = 1,2), as

, dI1 I1I2 - cos8i- = - f - - - al1

dz Io ' (3.10a)

1 dl2 I1h