• 沒有找到結果。

BEAM PROPAGATION METHOD ANALYSIS OF TRANSVERSE-ELECTRIC WAVES PROPAGATING IN A NONLINEAR TAPERED PLANAR WAVE-GUIDE

N/A
N/A
Protected

Academic year: 2021

Share "BEAM PROPAGATION METHOD ANALYSIS OF TRANSVERSE-ELECTRIC WAVES PROPAGATING IN A NONLINEAR TAPERED PLANAR WAVE-GUIDE"

Copied!
8
0
0

加載中.... (立即查看全文)

全文

(1)

2318 J. Opt. Soc. Am. B/Vol. 8, No. 11/November 1991

T.-T. Shi and S. Chi

Beam propagation method analysis of transverse-electric

waves propagating in a nonlinear tapered planar waveguide

Tian-Tsorng Shi and Sien Chi

Institute of Electro-Optical Engineering and Centerfor Telecommunications Research, National Chiao Tung University, Hsinchu, Taiwan, China

Received October 22, 1990; revised manuscript received June 26, 1991

The propagation of TE-polarized guided waves in a nonlinear tapered waveguide is studied. The incident fields are chosen from the stable branches of the nonlinear dispersion curve in its uniform section. The propagations of the field in both directions are calculated by the beam-propagation method. Wave propagation in the ta-pered waveguide is discussed in terms of the nonlinear dispersion curves of the waveguide. Depending on the intensity, the field can radiate as a spatial soliton or it can have an abrupt transition between the guided mode and the surface polariton. The wave propagation of an incident field that is composed of a dominant TEo wave and a small TE, wave is investigated. Finally, using a property of this wave propagation, we present a new device for continuous signal routing.

1. INTRODUCTION

Wave propagation in nonlinear waveguides has been stud-ied extensively,'5

and many nonlinear itegrated optical devices have been suggested.6'15 Among them, the

non-linear directional coupler has received the most attention, and its coupling characteristics depend on its input light intensity. The nonlinear X junction is one of the most promising devices for future applications; in it a strong signal beam can be routed by a weaker control beam.

However, wave propagation in the nonlinear tapered waveguide has been less extensively discussed.16

In a lin-ear multimode waveguide that is tapered to a single-mode waveguide the higher-order eigenmodes radiate out, and only the fundamental eigenmode remains in the single-mode waveguide. In a nonlinear waveguide the shape of the field and the propagation constant of the eigenmode depend on the intensity of the field. The nonlinear dis-persion relations have been calculated for several

differ-ent structures.7 8 We use the nonlinear dispersion

curves of the nonlinear tapered waveguide to explain the wave propagation in it. The waveguide consists of a lin-ear film that is bounded by a linlin-ear substrate and a non-linear self-focus cladding. The field propagating in this nonlinear tapered waveguide varies as the power of the incident field. In this paper only TE waves are consid-ered. TM wave propagation can be calculated by the finite-element method.'6

Wave propagation will be simulated by the beam propa-gation method,'9 and its behavior will be explained by the nonlinear dispersion curves of the waveguide. The propa-gation of the fundamental mode is considered first. Then the propagation of the incident wave, composed of a domi-nant TEo wave and a small TE1 wave, is investigated.

Finally, a new optical switching or scanning device is proposed.

In Section 2 we calculate the nonlinear dispersion

curves of the waveguide for several specific film thick-nesses. In Section 3 we present some numerical results for wave propagation in the tapered waveguide. A

pro-posal for optical switching and scanning is presented in Section 4. Section 5 concludes this paper.

2. NONLINEAR DISPERSION RELATIONS

Here we briefly summarize the formulas that describe the fields and nonlinear dispersion relations. The TE wave, polarized in the y direction and propagating in the planar waveguide, is written as

Ey(x, z, t) = /2Ey(x)exp[i(ko,8z - t)] + c.c., (1)

where is the effective index and the wave number

ko = co/c. The waveguide is shown in the inset of Fig. 1,

where the dielectric constants for the substrate, the film, and the cladding are e, ef, and e = Ec + alEyl 2, respec-tively. The field solutions in each layer are

EC(x) = (+) 2 cosh[koq(x - x)] Ef (x) =

IA

A

cos[koF(x - Xf)] for 132 < ef sinh[koy(x - Xf)] for 132 > ef (2) (3) Esy(x) = B exp(kopx), (4)

where q

= (132 - e)1/2, r = (f- 132)1/2 = (2 _ Ef)1/2 p = (2 _ Ec)"/2, and A, B, xc, and Xf are constants that can be determined from the boundary conditions and the ap-propriate effective index. The power of the guided wave per unit length along the y axis is obtained by integrating the Poynting vector over the x axis'8:

P ceol3J Ey2(x)dx. (5)

Because the fields depend on the effective index, the power is a function of the effective index. Numerical re-sults are presented below.

The wavelength is 1.3 m; the thickness of the film is 5 m; the indices are 1.55 and 1.57 for the substrate and the film, respectively, i.e., e = 2.4025 and ef = 2.4649;

(2)

Vol. 8, No. 11/November 1991/J. Opt. Soc. Am. B 2319

0.6

E~ E -03 3 0 -o a3 '0 CD)

0.5

0. 4 0. 3

0.2

0.1

0.0

-1.55 1.56 1.57 1.58 1.59 Effective index

Fig. 1. TE-polarized guided-wave power versus effective index for e, = 2.4025, f = 2.4649, and ec = co + alEl2, where

en = 2.4025 and a = 3.3776 x 10-'2 m2/V2. The thickness of the film is 5 Am, and the wavelength is 1.3 /.km. In the inset the z axis is the propagating axis, the linear substrate is at the left, and the nonlinear cladding is at the right.

the index of the nonlinear cladding is 1.55 + n21, where n2 = 5.3 x 10-'0 m2/W, i.e., e = eo + alE12, where

eO = 2.4025 and a = 3.3776 x 10-12 m2/V2. The

nonlin-ear dispersion curve is depicted in Fig. 1, and the two branches represent the TE, and TEO modes. It has been shown that the positively sloped TEO branches are stable and that this is not strictly true for the TE, branch.' In this paper losses by absorption and scattering are ne-glected. If the loss is large, we believe that the soliton emission can be suppressed.

0.6

E 3 -3 0 0. ~0 CD

0.5

0.4

0.3

0.2

0. 1 0.0 L 1.55

The nonlinear dispersion curves of the TEo mode with various film thicknesses are shown in Fig. 2(a). The permittivities of all the layers are the same as in the pre-vious example, and the film thicknesses are 5,4,3,2, and 1 ,um. The maximum guided power increases as the thickness of the waveguide increases. The curves nearly overlap in the surface-polariton region, which means that the field is approximately the same for different film thicknesses in such modes. For a slowly tapered wave-guide it is expected that the wave will evolve along the horizontal line in the figure; i.e., the guided power re-mains constant and the field shape slowly changes. If the input field is too large for an eigenmode of the thinner waveguide, then it will radiate partially through the spa-tial soliton. The field evolutions for different input powers are simulated by the beam propagation method; the results and explanations are presented in Section 3.

3. RESULTS

Three different situations for wave propagation in the ta-pered waveguide are simulated. In simulation 1 the wave propagates in the forward direction, with the fundamen-tal mode of the thicker end as the input. In simulation 2 the wave propagates in the backward direction, with the fundamental mode of the thinner end as the input. In simulation 3 the wave propagates in the forward direc-tion, with two modes as the input. In the simulations the beam propagation method is employed; 4096 or 8192 grid points are used in the transverse direction for a 256- or 512-Am width, and the propagation step is 0.2 Mm to en-sure that the accumulated numerical errors are negligible.

A. Simulation 1

Forward propagation with the input being the fundamen-tal mode of the thicker end is simulated. In Fig. 2(a) the

0.2

E E A L 03 3 0 C. -D -I CD

1.56

1.57

1.58

1.59

Effective index (a)

0.18

0.16

0.14

0.12 1.56 Effective index (b)

Fig. 2. (a) TEo guided-wave power versus effective index for five different film thicknesses: 5,4, ,3, 2, and 1 um. Points 1-7 represent the selected incident fields for the tapered waveguide. The dashed line between points 1 and 8 represents the adiabatic process of the field evolving in the tapered waveguide. (b) Detail of plots of the nonlinear dispersion curves for five different film thicknesses: 2.7,2.5,2.3,2.1, and 1.9 Mm. The jump is the abrupt transition between the guided mode and the surface polariton. Path A is the forward jump, shown in Fig. 3(b), and path B is the backward jump, shown in Fig. 4. The field evolves along the solid horizontal line ab for the adiabatic process in the tapered waveguide before the jump; the dotted lines represent the jumping processes, and the dashed line represents the field evolution after the jump.

(3)

2320 J. Opt. Soc. Am. B/Vol. 8, No. 11/November 1991

T.-T. Shi and S. Chi

I

[mm] 3. 0 [mm] S t E E 16 [urm] (a) Es [Jrm] (c) -16 0 16

-

3. 0 [mm] _2.5 .2.0 1 .5 1 .0 0.5 0.0 [uam] (e) 3. 0 [mm] E E Et 3. 0 [mm] Es Et _ _ 3. 0 mm] _2.5 1.5 1.0 0.5 E ,urn]

/t

~

r T~

- , -

I

. |.J I _ .___ ____ _--__

0.

0

, 16 0 16 -6 0 16 -16 . 0 16 (b m T [Jr] Tjim] (b) (d) (0

Fig. 3. Field evolution for forward propagation with various input powers. The tapered waveguide is shown at the left of each panel. The tapered section is 2 mm long, and the uniform section at each end is 0.5 mm long. The thickness of the thicker end is 5 m and that of the thinner end is 1 m. (a) Input power, 0.08 W/mm, point 1 of Fig. 2(a). (b) Input power, 0.16 W/mm, point 2 of Fig. 2(a). (c) Input power, 0.24 W/mm point 3 of Fig. 2(a). (d) Input power, 0.32 W/mm, point 4 of Fig. 2(a). (e) Input power, 0.4 W/mm, point 5 of Fig. 2(a).

(f) Input power, 0.48 W/mm, point 6 of Fig. 2(a).

six powers indicated from the positively sloped TEO branch of the nonlinear dispersion curve of the thicker end are used as the inputs for forward wave propagation in the tapered waveguide.

For a low input power, e.g., at point 1 in Fig. 2(a), P = 0.08 W/mm, the field evolves along the dashed line from point 1 to point 8, as indicated in Fig. 2(a), and the

field profile evolution is shown in Fig. 3(a). As the field propagates to the thinner waveguide, the field profile be-comes thinner and the peak of the field moves toward the film-cladding interface. For such low guided power in-put, the field evolves in an adiabatic process.

For an input power that is twice that in the above case, i.e., point 2 in Fig. 2(a), the evolution of the field is more

E f

1

EtC

(4)

Vol. 8, No. 11/November 1991/J. Opt. Soc. Am. B 2321

dramatic. Basically, the field still evolves along the hori-zontal line in Fig. 2(a), but a discontinuity occurs at the point at which the waveguide thickness equals 2.2 /,m and the maximum guided power is less than the input value. Figure 3(b) shows that the field propagates stably and is compressed by the waveguide, as in the case of low input power, at first. As the thickness of the film decreases further, the field is compressed more and more, and the

intensity at the film-cladding interface becomes high

enough to radiate energy as a spatial soliton at a propaga-tion distance of 1.9 ,tm. The field radiates and is recap-tured by the interface immediately. After correlation integral calculation, the recaptured field is shown to be a surface polariton, and the surface polariton swings as it propagates successively along the waveguide. The propa-gating phenomena described can be illustrated with refer-ence to path A of Fig. 2(b). At first the field evolves adiabatically along the horizontal line through point a to point b, shown as a solid line in the figure. As the field propagates successively, there is no corresponding guided mode for the field, and the field tends to radiate. After the field radiates a small amount of power, it becomes a surface polariton at point c. Then the field evolves con-tinuously through point d to the corresponding surface

polariton of the thinner end. We call this abrupt transi-tion between the guided mode and the surface polariton a jump. The mode considered here is entirely on the posi-tively sloped branches of the dispersion curve. It is called a guided mode when the effective index is less than the film index, and it is called a surface polariton when the effective index is larger than the film index.

For point 3 in Fig. 2(a) the input power 3PO is associated with no corresponding point in the dispersion curve of the waveguide when -its film thickness is less than 3.2 /m; therefore the power is no longer guided, and the field is expected to radiate a spatial soliton in order to catry away the excess power in the waveguide, which is similar to the soliton emission from the uniform nonlinear wave-guide.3 Because the power radiated is much larger than that of a surface polariton, the field will not be recaptured by the film-cladding interface. The radiated field carries away the most power from the waveguide near a propaga-tion distance of 1.55 mm. In Fig. 3(c) the radiated field is identified as a spatial soliton after correlation integral calculation; the remaining power in the waveguide is small, and only the fundamental mode exists in the wave-guide. In the successive tapered process the field evolves adiabatically, as in the case of a low input power.

The guided power is 4PO at point 4 in Fig. 2(a), and the spatial soliton radiation is obvious; the remaining field in the waveguide excites mainly the fundamental mode. The propagating behavior of the remaining field in the waveguide is much like that of two-mode propagation, which is discussed below. The field profile evolution is

shown in Fig. 3(d).

The field profile evolution of an input power 5Po is shown in Fig. 3(e). The characteristics of spatial-soliton radiation and the remaining field trapped by the wave-guide are similar to those of Fig. 3(d), but the power of the remaining field is higher, and it jumps to the surface polariton in the successive tapered waveguide, as in the case of the 2Po input in Fig. 3(b).

In the incident field at point 6 in Fig. 2(a) two spatial

soliton radiations are observed. The remaining field is small after the two successive radiations and evolves adia-batically as shown in Fig. 3(f).

B. Simulation 2

Backward propagation with the input being the funda-mental mode of the thinner end is simulated. The al-lowed guided power of the guided mode of a thinner

waveguide is less than that of a thicker waveguide; the allowed powers are the same for the surface polariton. Thus the thicker waveguide is always able to capture the guided mode that is incident from the thinner waveguide, and spatial soliton radiation is impossible for a guided wave propagating in such a backward-tapered waveguide. Both the guided mode and the surface polariton propagat-ing in the tapered waveguide evolve adiabatically. How-ever, in some specific cases, such as at point 7 in Fig. 2(a), the field evolves with a jump, as shown in Fig. 4. The field jumps from the surface polariton of the thinner waveguide into the guided mode of the thicker end, which is illustrated by path B in Fig. 2(b). The excited guided wave in the successive propagation is similar to the wave propagation of the incident field that is composed of a dominant TEo wave and a small TE, wave.

C. Simulation 3

Forward propagation with the input being a dominant TEO wave and a small TE, wave is simulated. Wave propaga-tion is considered in the forward-tapered direcpropaga-tion, be-cause the thinner waveguide supports only a TEo wave. A low guided power of the TEO wave, which will not radiate solitons in the propagation associated with a small TE, wave, is launched from the thicker end of the tapered waveguide. We would expect the TE, wave to radiate

ES E f E,

. 5 1.0

/-16 0 16

Fig. 4. Profile evolution for backward propagation with an input power of 0.16 W/mm, point 7 in Fig. 2(a). The tapered waveguide is shown at the left, and its dimension is the same as that in Fig. 3; the linear substrate is to the left and the nonlinear clad-ding is to the right of the waveguide.

(5)

2322 J. Opt. Soc. Am. B/Vol. 8, No. 11/November 1991

T.-T. Shi and S. Chi

E Et 3.0 (mm] .2.5 .2.0 .5 .0 5 EC [(Jm] (a a 0. 0 3.0[mm] 2.0 1.5 1.0 .0.5

E

£s Ef -,= ono -16 0 16 (i m] (b)

Fig. 5. (a) Wave propagation in the nonlinear tapered waveguide of the incident field composed of a dominant TEo wave and a weak TE, wave. (b) Same as (a) but for the linear case.

beyond cutoff and leave only the TEo wave in the wave-guide. However, on the contrary, the numerical results show that after propagation in the tapered waveguide and in the 20-mm-long uniform single-mode waveguide the swing of the wave is retained. This result means that the wave propagation of the incident field that is composed of two modes will not approach the fundamental mode that was calculated in Section 2. The field profile evolution in the nonlinear tapered waveguide is shown in Fig. 5(a), and the linear case is shown in Fig. 5(b) for comparison,

where the input powers of the TEo and TE, modes are 0.08

and 3.2 mW/mm, respectively.

4. APPLICATIONS

A device with continuous routing of a light beam for pho-tonic switching or spatial scanning is proposed in this sec-tion. At the left in Fig. 6(a) the structure of the proposed device is shown schematically. The tapered waveguide, which is the same as that discussed above, terminates with a nonlinear medium. The thicknesses of the thicker and thinner ends of the tapered waveguide are 5 and 1 ,um, respectively. The nonlinear medium at the thinner end is chosen in the same way as the nonlinear cladding of the waveguide; its length here is 0.5 mm.

The characteristics of wave propagation in the tapered waveguide were obtained in Section 3. For a low guided power input the TEo wave evolves adiabatically, and the power is confined in the waveguide. When a small com-ponent of a TE, wave is present, the field swings even if the thickness of the waveguide is reduced to the single-mode region. We use the field at the end of the tapered waveguide to excite the spatial soliton directly, and the propagation direction of the spatial soliton depends on the phase of the field at the interface between the nonlinear medium and the thinner end of the waveguide. The phase at the interface is determined by the input intensity of the TE, wave or its phase relative to the TEo wave. Hence, by changing the input condition, we can control the field propagations in the device. The desired incident TEO and TE, waves can be obtained by the converging asymmetric Y junction.8

The input power of the TEO mode is fixed at 0.12 W/mm, which is near the maximum allowed power in the tapered waveguide for the adiabatic evolution. For this input power the field evolves adiabatically from the guided mode in the thicker end to the surface-polariton-like mode in the thinner waveguide. Because the shape of the field now is like that of a surface polariton but the effective index is less than the film index, we call the field the surface-polariton-like mode. The surface-polariton-like field in the thinner end excites the spatial soliton with an extremely low radiation loss. The power of the TE wave employed here is much less than that of the TEO wave; thus the spatial soliton is approximately the same in the nonlinear medium for different routes.

A. Power-Controlled Routing

With reference to Figs. 6(a)-6(c), the routes of the spatial soliton are shown for three input powers of the TE, wave,

which are 0, 3, and 17.4 mW/mm, corresponding to 0, 2.5%,

and 14.5% of the power of the TEO wave, respectively. For the case of no TE1 wave, the field evolves from the guided

mode to the surface-polariton-like mode along the tapered waveguide, and the peak of the field moves toward the film-cladding interface. The phase front of the beam is not parallel to the interface, and the angle of the propaga-tion direcpropaga-tion of the spatial soliton is 0.11° with respect to the z axis. The angle depends on the tapered angle of the waveguide; it decreases as the tapered angle de-creases. The angle of the spatial soliton propagation with respect to the z axis versus the incident power of the TE, wave is plotted in Fig. 7. The total variation of the angle

(6)

Vol. 8, No. 11/November 1991/J. Opt. Soc. Am. B 2323 I -3. 0 [mm] _2.5

-

1.5 _1.0 _ 0.5 E E E, T E5 E f EC T-_ 0.0 16 -16 0 [m] (a) 3. 0 [mm] .3. 0 [mm] .2.5 .2.0 .1.5 _ 1.0 _0.5 _0.0 [Jm] (c) *3. 0 [mm] .2.5 .2.0 5 0 E Ef EC E EfE F= 0

'

-16 0 16 -16 0 15 T e[Im] T [eim] (b) (d)

Fig. 6. (a) TEO wave evolving in the proposed device, which is shown schematically at the left. (b) Same as (a) but with a TE, wave input of 3 mW/mm (i.e., 2.5% of the power of the TEo wave). (c) Same as (a) but with a TE, wave input of 17.4 mW/mm (i.e., 14.5% of the power of the TEo wave). (d) Same as (b) but with a relative phase difference of 180° between the TEo and TE1waves.

is 1.60; for the purpose of switching, the number of allowed channels is proportional to the length of the

non-linear medium. For a beam width of 5 ,um and a

length of 0.5 mm for the nonlinear medium, the number of allowed channels is 3.

B. Phase-Controlled Routing

Routing of the spatial soliton can also be achieved by con-trolling the phase difference between the TEo and TE,

waves. The power of the TE, wave is chosen to be 3 mW/mm, -2.5% of the power of the TEo wave, and the power of the TEO wave is still 0.12 W/mm. The field pro-file evolution is shown in Fig. 6(b), where TEO and TE, waves are in phase. When the relative phase between two waves is 180°, the field profile evolution is as shown in Fig. 6(d). Figure 8 shows the dependence of the angle of the spatial soliton propagation on the phase difference be-tween the two waves.

(7)

2324 J. Opt. Soc. Am. B/Vol. 8, No. 11/November 1991

T.-T. Shi and S. Chi 1 03 03 L 01 03 c 0n (U 0) ., 4J 0 Er 0.6 0.2

-0.2

-0.6 1L Fig. 7. Depend the TE, wave.

03 03 L 0) 03 013 '-4 0) C (0 0) C 4-' j) 0 Mr 0.6 0.2

-0.2

-0.6

-1.0

0 4 8 12 16 20

Power ratio of TEI/TEO (%)

ence of the routing angle on the. input power of

Ir

0 90 180 270 360

Phase difference (degree)

Fig. 8. Dependence of the routing angle on the relative phase difference between the TEO and TE, waves.

5. CONCLUSIONS

Field evolution in a tapered nonlinear waveguide has been discussed from the point of view of nonlinear dispersion curves. Since the tapered structure is used in the

nonlin-ear waveguide circuit, the input field of the tapered

waveguide is chosen from the stable nonlinear modes. In most cases the available field in the waveguide circuit is the fundamental mode, so the evolution of the TEO mode is our main interest here.

Both propagation directions of the field in the tapered waveguide are considered. The main difference in the two propagation directions is caused by the different al-lowable guided powers for different thicknesses of the film. The allowable guided power in the thicker wave-guide is larger than that of the thinner wavewave-guide. The

propagation's behavior depends on the input powers.

When the field is incident in the thicker waveguide with a

higher power, the field propagating to the thinner wave-guide will radiate the excess power as a spatial soliton.

After the radiation, if the remaining part is small it

excites the fundamental mode of the waveguide; if the remaining part is large it swings in the propagation. On the other hand, the spatial soliton does not appear for a field propagating from the opposite direction as long as the input field is the guided mode of the thinner wave-guide. Jump behavior, which is the abrupt transition be-tween the guided mode and the surface polariton, is observed in both propagation directions. After the jump the excited field is not exactly the eigenmode of the wave-guide, and the field swings. For an input power that is less than that required in a jump, the transition between the guided modes and the surface polariton is gradual.

Wave propagation of an incident field that is composed of a strong TEO wave and a weak TE, wave in a nonlinear tapered waveguide has been investigated. The swinging effect is observed and lasts for a long distance after the tapered section. We propose a new device for continu-ously routing the signal by using this property. The spa-tial soliton is directly excited by the guided field at the end of the tapered waveguide; its propagation direction is determined by the phase of the field in the interface, and the phase depends on the input condition of the two waves. Controlling the input power of the TE, wave or the phase difference between the TEO wave and the TE, wave will allow the signal to be routed in the desired direction.

ACKNOWLEDGMENT

This research was supported by the National Science

Council of the Republic of China under contract

NSC-79-0417-E009-01.

REFERENCES

1. L. Leine, C. Wchter, U. Langbein, and F. Lederer, "Evolu-tion of nonlinear guided optical fields down a dielectric film with a nonlinear cladding," J. Opt. Soc. Am. B 5, 547-558 (1988).

2. K. Hayata, A. Misawa, and M. Koshiba, "Split-step finite-element method applied to nonlinear integrated optics," J. Opt. Soc. Am. B 7, 1772-1784 (1990).

3. M. A. Gubbels, E. M. Wright, G. I. Stegeman, C. T, Seaton, and J. V Moloney, "Numerical study of soliton emission from a nonlinear waveguide," J. Opt. Soc. Am. B 4, 1837-1842 (1987).

4. M. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V Moloney, "Effects of absorption on TEO nonlinear guided waves," Opt. Commun. 61, 357-362 (1987).

5. L. Leine, C. Wichter, U. Langbein, and F. Lederer, "Propaga-tion phenomena of nonlinear film guided waves in a configu-ration with material losses: a numerical analysis," Opt. Lett. 12, 747-749 (1987).

6. G. I. Stegeman and E. M. Wright, 'All-optical waveguide switching," Opt. Quantum Electron. 22, 95-122 (1990). 7. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and

C. T. Seaton, "Third order nonlinear integrated optics," IEEE J. Lightwave Technol. 6, 953-970 (1988).

8. G. I. Stegeman and R. H. Stolen, "Waveguides and fibers for nonlinear optics," J. Opt. Soc. Am. B 6, 652-662 (1989). 9. S. M. Jensen, "The nonlinear coherent coupler," IEEE

J. Quantum Electron. QE-18, 1580-1583 (1982).

10. A. Ankiewicz, "Novel effects in nonlinear coupling," Opt. Quantum Electron. 20, 329-337 (1988).

11. L. Thylen, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V Moloney, "Beam-propagation method analysis of a non-linear directional coupler," Opt. Lett. 11, 739-741 (1986).

(8)

T.-T. Shi and S. Chi Vol. 8, No. 11/November 1991/J. Opt. Soc. Am. B 2325 12. D. R. Heatley, E. M. Wright, and G. I. Stegeman, "Soliton

cou-pler," Appl. Phys. Lett. 53, 172-174 (1988).

13. Y Silberberg and B. G. Sfez, 'All-optical phase- and power-controlled switching in nonlinear waveguide junctions," Opt. Lett. 13, 1132-1134 (1988).

14. J. P. Sabini, N. Finlayson, and G. I. Stegeman, 'All-optical switching in nonlinear Xjunctions," Appl. Phys. Lett. 55, 1176-1178 (1989).

15. H. Fouckhardt and Y Silberberg, 'All-optical switching in waveguide X-junctions," J. Opt. Soc. Am. B 7, 803-809 (1990).

16. K. Hayata, A. Misawa, and M. Koshiba, "Nonlinear beam

propagation in tapered waveguides," Electron. Lett. 25, 661-662 (1989).

17. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. QE-21, 774-783 (1985). 18. S.-Y. Shin, E. M. Wright, and G. I. Stegeman, "Nonlinear TE

waves of coupled waveguides bounded by nonlinear media," IEEE J. Lightwave Technol. 6, 977-983 (1988).

19. M. D. Feit and J. A. Fleck, Jr., "Computation of mode proper-ties in optical fiber waveguides by a propagating beam method," Appl. Opt. 19, 1154-1164 (1980).

數據

Fig.  1.  TE-polarized  guided-wave  power  versus  effective  index for  e,  =  2.4025,  f  =  2.4649,  and  ec  =  co  +  alEl 2 ,  where
Fig. 3.  Field  evolution for  forward propagation with  various  input  powers.  The tapered  waveguide is shown at  the  left of each panel
Fig. 4.  Profile evolution for backward propagation with an input power  of 0.16  W/mm,  point  7  in  Fig
Fig. 5.  (a) Wave propagation in the  nonlinear tapered  waveguide of  the  incident  field  composed  of a  dominant  TEo wave and  a weak TE,  wave
+3

參考文獻

相關文件

The Vertical Line Test A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.. The reason for the truth of

different spectral indices for large and small structures Several scintil- lation theories including the Phase Screen, Rytov, and Parabolic Equa- tion Method

Work Flow Analysis: Since the compound appears in only 2% of the texts and the combination of two glyphs is less than half of 1% of the times when the single glyphs occur, it

More precisely, it is the problem of partitioning a positive integer m into n positive integers such that any of the numbers is less than the sum of the remaining n − 1

6 《中論·觀因緣品》,《佛藏要籍選刊》第 9 冊,上海古籍出版社 1994 年版,第 1

In the algorithm, the cell averages in the resulting slightly non-uniform grid is updated by employing a finite volume method based on a wave- propagation formulation, which is very

In Section 3, the shift and scale argument from [2] is applied to show how each quantitative Landis theorem follows from the corresponding order-of-vanishing estimate.. A number

• One technique for determining empirical formulas in the laboratory is combustion analysis, commonly used for compounds containing principally carbon and