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... Electro-Optics and Lasers, University of Central Florida, 12424 Research Parkway, Orlando, Florida 32826 ... bar state (solid curves) and the total -throughput.
710

OPTICS LETTERS

/ Vol. 17, No. 10 / May 15, 1992

Effects of three-photon absorption on

nonlinear directional coupling C. C. Yang, A. Villeneuve, and G. I. Stegeman

Centerfor Research in Electro-Opticsand Lasers, University of Central Florida, 12424Research Parkway, Orlando,Florida 32826 J. S. Aitchison Department of Electronicsand Electrical Engineering, Universityof Glasgow,GlasgowG12 8QQ, UK Received January 8, 1992

The effects of three-photon absorption on nonlinear directional coupling in semiconductors below half the gap are studied for both cw and sech2 pulse signals. The figure of merit associated with three-photon absorption includes not only material properties but also device design parameters. Numerical results show that at high powers, three-photon absorption significantly degrades nonlinear coupling, reduces throughput, and distorts output pulses. Good agreement is found between the theoretical model and experimental data.

In the search for nonlinear materials for efficient nonlinear directional couplers (NLDC's), a large ultrafast nonlinear refractive index is usually accompanied by nonlinear absorption. This produces a limitation to nonlinear coupling. For instance, in semiconductors it has been found that two-photon absorption (2PA) is a limiting factor when a NLDC is operated near but below their fundamental band gaps.1 -4 As the wavelength approaches the half band gap, 2PA becomes progressively weaker. Alloptical switching in AlGaAs NLDC's operated just below one half its band gap, where 2PA approaches zero, has been reported. 5 However, three-photon absorption (3PA) can now become the fundamental

limitation. Indeed, significant nonlinear absorp-

tion has been observed in AlGaAs near its half band gap.6 -8 In this Letter we assess the effects of 3PA on the operation of NLDC's, taken here as an example of all-optical switching devices. It is found that 3PA is an important limiting factor to nonlinear switching for wavelengths longer than the 2PA edge. A figure of merit, which now includes not only the material properties but also the device design parameters, is defined. Also good agreement between the numerical results based on the theoretical model and experimental data is demonstrated. If we neglect group-velocity dispersion and freecarrier absorption, the equations governing the complex amplitudes in the two waveguides of NLDC, Al and A2 , are given by

. AlA 2

2ira2 n 2

az

A

/

+ 8 )A1,21

tive index, 2PA coefficient, and 3PA coefficient, respectively. a2 and a 3 represent the overlap integrals over the modal profiles in the waveguides for the

third- and fifth-order nonlinearities, respectively. By using the normalized parameters ; = z/L, and ql2 = (27Ta2n 2 L./A) 1 /2A1,2, with L, = 7T/2Krepresenting one half beat length, Eq. (1) becomes

i aqL 2+

+ i

1ql,212ql,2

+ i8

2

q1,214q1, 2 + 2 q 2, 1 = 0.

(2)

Here V = IkA,33a3 /a2 n2 , with I, being the cw critical intensity for switching given by A/(a2 n2 L,).9 Note that in Eq. (2) V is a figure of merit associated with 3PA, which is related not only to the material parameters (n2 and 63) but also to the device design. The device design includes the waveguide size, which determines a2 and a 3, and the linear coupling coefficient on which the cw critical intensity depends. When 2PA is the major nonlinear absorption, a decrease of linear coupling will lead to lower switching power but not affect the switching fraction. However, when 3PA is the principal source of nonlinear absorption, a decrease of linear coupling will not only reduce the switching power but will also improve the switching fraction. Equation (2) was numerically solved for cw in-

puts in half-beat-length NLDC's. We set T = 0 since the effects of 3PA are of prime interest here. Figure 1 shows the normalized transmission in the

2A1,2

bar state (solid curves) and the total -throughput + iI

3 3A

2

1,214A 1,2

+ KA 2,1 = 0.

(1)

Here z, A, and K are the propagation coordinate, wavelength, and linear coupling coefficient, respectively. The 2PA figure of merit T is given by T = 2,62Ak/n2.l n2 , ,32, and 83 are the nonlinear refrac0146-9592/92/100710-03$5.00/0

(dashed curves) as a function of the cw input power, normalized by the cw critical power (the normalized cw critical power is 2iv). The curves are plotted for V = 0, 0.08, 0.175, 0.35, 0.7, 1.05, and 1.58 (from the

top to the bottom, respectively). The decreasing trend of the normalized transmission owing to 3PA is similar to that owing to 2PA.' By defining the © 1992 Optical Society of America

May 15, 1992 / Vol. 17, No. 10 / OPTICS LETTERS

711

an AlGaAs NLDC at 1.55 ,um is also shown.8 The NLDC consisted of two parallel 6-mm-long waveguides. Both were strip loaded, 1.5 tum thick and 6 ,um wide, and separated by 5 jim. The aluminum content was chosen to provide a band gap of 1.65 eV (750 nm) in the waveguide, which ensured operation below half of the band gap energy at a wavelength of 1550 nm. The short-dashed curve is plotted to

1.2

0*.0 Cl)

fit the experimental data based on Eq. (2) with

V = 0.1748 and T = 0.22. In the NLDC used for the experiments, the input waveguide is longer than the other waveguide by a leading section. The input peak intensity for the data points and the shortdashed curve is referred to that at the input facet,

0.0 0.2 0

not at the beginning of the coupler. The long-

NORMIALIZEDINPUT POWER

Fig. 1. Normalized transmission in the bar state (solid curves) and throughput (dashed curves) versus the normalized cw input power in a half-beat-length NLDC. For both solid and dashed curves, V increases from top to as V

bottom

=

0, 0.08, 0.175, 0.35, 0.7, 1.05, and 1.58,

respectively. 1.0

z

° 0.8

Gc

U) U) 5-

N

L'i

0 0~

0.6

U0

0

.F

C)

LLI

0N10.4

U)

z 0.2

0.0

0.8 V

Fig. 2. Normalized transmission in the bar state (dashed curve, left scale) and switching power (solid curve, right scale) versus V for cw inputs in a half-beat-length NLDC.

switching power as the input power leading to the

first maximum normalized transmission in the bar

state, this transmission maximum (dashed curve, left scale) and the switching power (solid curve, right scale) versus V are plotted in Fig. 2. It can be

seen that for the normalizedtransmission in the bar

state to be larger than 0.5, V (=I, 2 /33a3 L,) must be smaller than 0.68. In other words, the 3PA must be sufficiently weak for the total nonlinear absorption

dashed curve is drawn for the same results as the short-dashed curve versus the peak intensity at the beginning of the coupler. With T = 0, three solid curves are arranged from top to bottom for V = 0, 0.1784, and 0.8, respectively, for comparison with the long-dashed curve. Note that even for zero nonlinear absorption, complete nonlinear switching is not possible because of pulse breakup unless square pulses or solitons are used as signals. The comparison between the long-dashed curve and the solid curve closest to it shows the small effect of 2PA for the NLDC at the wavelength considered. Since the NLDC used for Fig. 3 is not exactly one half beat length, the output power in the bar state does not start from zero at the low-power limit. Pulse breakup for a half-beat-length NLDC is demonstrated in Figs. 4 and 5. Figure 4 shows the pulse shapes in the bar state for V = 0, 0.08, 0.24, and 0.8, respectively (from the top to the bottom), and Fig. 5 shows those in the cross state for V = 0, 0.24, and 0.8. The input peak power is three times the cw critical power. When V = 0 (/l33= 0), the sech2 pulse shape is basically preserved in the bar 1-0 -....

EXP. DATA

V=0, 0.1784, 0.8; T=0

T=0.22 - - V=0.1784;

z 00.8 U)

0.6 H-

N 0.4

at the cw critical intensity over the half beat length

~0.2

criterion for nonlinear switching associated with

0.0

to be smaller than 0.68. This can be regarded as a 3PA. It is equivalent to requiring that Ic2 /33 a3 L,
3.2ir (required nonlinear phase

shift at the cw critical intensity). The criterion for V is similar to the one associated with 2PA, i.e., Ica2,l32L,< 0.5 (T < 1). Next we consider the sech2 pulse inputs. Figure 3

shows the normalized transmission in the bar state

of a NLDC of 0.84 half beat length long as a function of the input peak power, normalized to the cw critical power. A set of experimental data from

0)

l

2

3

NORMALIZED INPUT PEAK POWER

4

Fig. 3. Normalized transmission in the bar state versus the input peak intensity, normalized by the cw critical intensity, of a sech2 input pulse in a NLDC of 0.84 half beat length. The solid curves (T = 0; from top to bottom V = 0, 0.1784, and 0.8) and the long-dashed curve (V = 0.1784 and T = 0.22) are drawn versus the normalized peak intensity at the beginning of the coupler. The shortdashed curve (V = 0.1784 and T = 0.22) fits the experimental data well with the horizontal scale representing the input intensity in the leading section.

712

OPTICS LETTERS

/ Vol. 17, No. 10 / May 15, 1992

16

ures). Also from Figs. 4 and 5, an almost square pulse and a weak multihump pulse are expected in

the bar and cross states, respectively. Free-carrier

12