分析導波層為非線性多層光波導結構之一般方法

(1)

行政院國家科學委員會專題研究計畫成果報告

分析導波層為非線性多層光波導結構之一般方法

計畫類別：個別型計畫計畫編號： NSC92-2215-E-151-001- 執行期間： 92 年 08 月 01 日至 93 年 07 月 31 日執行單位：國立高雄應用科技大學電子工程系計畫主持人：吳曜東計畫參與人員：蕭明泉、蔡東亨報告類型：精簡報告處理方式：本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中華民國 93 年 9 月 18 日

(2)

A General Method for Analyzing the Multilayer Optical

Waveguide with All Nonlinear Guiding Films

ABSTRACT

We propose a novel method for analyzing a multilayer optical waveguide structure with all nonlinear guiding films. This method can also be used to analyze a multibranch optical waveguide structure with all nonlinear guiding branches. The numerical results show that our analyses are correct.

中文摘要

我們提出一個分析導波層均為非線性之多層光波導結構的新方法，此方法亦可用來分析每一分支波導均為非線性之多分支光波導結構。數值分析的結果證明我們的分析是正確的。

Keywords : nonlinear, Kerr effect, Multilayers, Guided waves, waveguides planar

1. INTRODUCTION

Kerr-like nonlinear waveguide structures containing one or more media, whose refractive index depends on the local intensity, have stimulated a great deal of theoretical and experimental study [1-13]. The interest in nonlinear waveguide device, which has been growing steadily in recent years, stems from their potential use for ultrafast all-optical signal processing and optical computing systems. The confinement of optical beam in the small core area and its diffractionless propagation over long distance increase the efficiency of the nonlinear interaction and permit the use of relatively weak nonlinearities. It has been shown that these nonlinear waveguides possess a variety of novel and exciting features such as power-dependent propagation constants and field profiles leading to novel feasibilities for all-optical signal processing and optical computing.

Until now, the overwhelming majority of the published papers was aimed at analyzing the optical waveguide structures where either linear films were bounded by one or two nonlinear media [14-19] or a nonlinear film was sandwiched between linear films [20-23] or two nonlinear films embedded among three linear media [24]. If the guiding dielectric films are all nonlinear, the solution fields are much more complex and consist of Jacobi elliptic functions. Within the past several years, it became evident that the multilayer systems as semiconductor multiple quantum well structures exhibit stronger nonlinearity, low threshold power, very fast response times, and suitability for integration into circuits [25-27].

(3)

In the future, the multilayer and multibranch waveguide structures are very important components in the application of integrated optics. The multilayer waveguide structures have been extensively used in the multiple quantum well waveguide structures [28-31] and the multibranch waveguide structures have also been designed to operate as all-optical waveguide devices [32-42]. To the best of our knowledge, the general method for analyzing multilayer optical waveguide with all nonlinear guiding films that is presented in this paper has not been reported before. This paper gives a detail modal analysis of the multilayer optical waveguide with all nonlinear guiding films. The analytical results are accompanied by numerical examples. This method can also be used to predict the evolution of a wave propagating along the multibranch optical waveguide structure with all nonlinear guiding branches. The numerical results show that our analyses are correct.

2. ANALYSIS PROCESS

In this section, we use the modal theory [43-44] to derive the general formulas that can be used to analyze the multilayer optical waveguide structure with all nonlinear guiding films, as shown in Fig. 1. The multilayer optical waveguide structure is composed of nonlinear guiding films ( 1

2 m−

layers), interaction layers ( 3

2 m−

layers),

cladding, and substrate. The total number of layers is m (m=3,5,7,...). The cladding

and substrate layers are assumed to extend to infinity in the +x and –x direction, respectively. The major significance of this assumption is that there are no reflections in the x direction to be concerned with, except for those occuring at interfaces.

For simplicity, we consider the transverse electric polarized waves propagating along the z direction. The wave equation can be reduced to

2 2 2 2 2 t E c n E i yi yi ∂ ∂ = ∇ , i=1,2,…,m (1)

with solutions of the form

)] ( exp[ ) ( ) , , (x z t x j t k0z Eyi =εi ω −β (2)

where

ω

is the angular frequency, k is the wave number in the free space, and ₀ β

is the effective refractive index. For a Kerr-type nonlinear medium [45-47], the square of the refractive index of the guiding film can be expressed as

(4)

2 2 0 2 ) (x n ni = i +αiεi , i=2,4,…,m-1 (3)

where n₀_i and

α

_i are the linear refractive index and the nonlinear coefficient of the

i , ) ' ' ( ' ' ₂ ₂ 2 i i i i b a b l + = ,

where the constants a , _i a , _i' q , _i Q , _i K and _i x₀_i are shown in Appendix. For

simplicity, the modulus l is omitted in the following discussions. _i

Considering the case

β

<n_i (i=2,4,...,m−1) and matching the boundary

conditions, we can obtain the following eigenvalue equations:

for m≥7, 2 2 2 2 2 1 2 1 1 1 2 1 1 1 1 ( ) ( ) ( ) m m 1 (1 m ) ( ) m m m m m m m m m p cn A d sn A d dn A d l sn A d A A b − + + − − − − − − − − − − − −    _∆ ₊ ∆  ₋ ₋∆           2 2 2 2 2 2 1 2 1 1 2 1 2 1 1 1 1 1 [1 (1 m ) ( ) ( )] (1 m ) ( ) ( ) ( ) m m m m m m m m m m m l sn A d dn A d l sn A d cn A d dn A d b b + + + − + − − − − − − − − − − − −  _∆ _∆  = ∆ − − − ∆ −  2 2 2 2 2 2 1 1 1 4 1 1 1 1 [ ( ) ( ) ( ) ( )] m m m m m m m m m m p cn A d dn A d l sn A d cn A d A b − + − − − − − − − − − −  ∆ ∆  + −   2 2 2 2 1 4 1 1 1 (1 m ) ( ) m m m m l sn A d p b + − − − −  _∆  − −       , (9) where ∆+_m₋₂ =E_I₍_m₋₄₎ exp(−p_m₋₂w)+E_I₍_m₋₃₎ , 2 ( −4)exp( −2 ) ( −3) − − = − − ∆m EI m pm w EI m _.

(6)

Otherwise, for m=3, 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 [1 (1 )] ( ) ( ) (1 ) ( ) ( ) ( ) c s s s s E P E E E E l sn A d dn A d l sn A d cn A d dn A d A b b b b b  = − − − − −  2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 [ ( ) ( ) ( ) ( )] ( ) ( ) ( ) s s s s c c E p E E E p cn A d dn A d l sn A d cn A d cn A d sn A d dn A d b E E A b A    − _{ } + _    (10) for 5m= , 2 2 3 3 3 3 4 4 4 4 2 4 4 4 4 ( ) p ( ) ( ) 1 (1 ) ( ) cn A d sn A d dn A d l sn A d A A b − + +   _∆ ₋ ∆  ₋ ₋∆           2 2 2 3 3 3 4 2 4 4 3 4 2 4 4 4 4 4 [1 l (1 )] (sn A d dn A d) ( ) l (1 )sn A d dn A d cn A d( ) ( ) ( ) b b + + + +  _∆ _∆    = ∆ − − − ∆ − +     2 2 2 2 2 2 3 3 3 3 4 4 4 2 4 4 2 4 5 4 4 4 ( )[ ( ) ( )] 1 (1 ) ( ) p cn A d dn A d l sn A d l sn A d p b b A − + _{ } + _ ∆ ₋ ∆  ₋ ₋∆         , (11) where ∆ =₃+ E_I₁exp(−p w₃ )+E_I₂, ∆ =₃− E_I₁exp(−p w₃ )−E_I₂,

and sn and dn are Jacobi elliptic functions. These eigenvalue equations (9), (10), and

(11) can be solved numerically. When the constants β and E are determined, all _s

the other constants q , _i p , _i K , _i A , _i a , _i b , _i l , _i x₀_i, E , and _Ii E are also _c

determined (Appendix).

For the case

β

>n_i (n_i =2,4,...,m−1), on the other hand, we just have to replace

the constants A , _i a , _i b , and _i l in Eqs.(9)–(11) with the constants _i A'_i, a'_i, b'_i,

and l' , and get these equations with similar expressions. When constants _i β and E _s

are determined, all the other constants Q _i, p _i, K _i, A'_i, a'_i, b'_i, l' , _i x₀_i, E , _Ii

(7)

3.NUMERICAL RESULTS AND DISCUSSION

In this section, we use the general formulas derived in the preceding section and in the Appendix to calculate the transverse electric field function in each layer of the multilayer optical waveguide structure with all nonlinear guiding films. Several

numerical examples are presented as follows. When m=3, we can simplify the

general formulas to analyze three-layer optical waveguide structure with nonlinear guiding film. The numerical results are shown in Figs. 2(a) and (b). Fig. 2(a) shows a

dispersion curve of TE symmetric modes of three-layer nonlinear optical waveguide ₀

structure with the constants d =3µm , 55n₁ =n₃ =1. , 57n₀₂ =1. ,

2 2 3786 .

6 µm V

α = (for MBBA Liquid Crystal), and the free sp ace wavelength is

m

µ

λ =1.3 . Fig. 2(b) shows the electric field distributions for the various input powers

with respect to points A-D as shown in Fig. 2(a). When m=7, we can simplify the

general formulas to analyze seven-layer optical waveguide structure with all nonlinear guiding films. The numerical results are shown in Figs. 3(a) and (b). Fig. 3(a) shows a

dispersion curve of TE symmetric modes of seven-layer nonlinear optical waveguide ₀

structure with the constants d =3µm , w=5µm , n₁= =n₃ n₅ =n₇ =1.55 ,

57 . 1 06 04 02 =n =n = n , 2 2 3786 . 6 µm V

α = , and λ =1.3µm. Fig. 3(b) shows the

electric field distributions for the various input powers with respect to points A-F as shown in Fig. 3(b). As shown in Figs. 2 and 3, as the guided power increases and

consequently β increases, the field distributions gradually narrow, and the optical

wave will be tightly confined in the nonlinear guiding film.

We can also use these general formulas to analyze the multibranch optical waveguide structure with all nonlinear guiding branches, as shown in Fig. 4. In the following analysis the multibranch optical waveguide structure is divided into four sections from bottom to top: the straight-line section (three-layer waveguide), the tapered section (three-layered tapered waveguide), the nonlinear multibranch section (multilayer waveguides with tapered interaction layers), and the isolated separating section (multilayer waveguides isolated from one another). For simplicity we use N to denote the nonlinear medium. Here we show an example of the three-branch optical waveguide structure with all nonlinear guiding branches, as shown in Fig. 5.

The electric field distributions of the four sections (positions Z₁, Z₂, Z , ₃ Z₄ in

Fig. 5) are shown in Figs. 6(a)-(d), respectively. In Fig. 6(a) the electric field

distribution of the straight waveguide section is plotted with the constants d_f =3µm,

1.55

c s I

n = =n n = , n_f₀ =1.57, 5662β =1. , α =6.3786µm2 V2 , and λ =1.3µm.

In Fig. 6(b) the electric field distribution of the straight waveguide section is plotted

with the constants d_f =6µm , 1.55n_c = =n_s n_I = , n_f₀ =1.57 , 5663β =1. ,

2 2 3786 .

6 µm V

(8)

m

µ

λ =1.3 . In Fig. 6(c) the electric-field distribution of the coupled

separating-waveguide section is plotted with d_f =3µm , 2d_I = µm ,

1.55

c s I

n = =n n = , n_f₀ =1.57, 5662β =1. , α =6.3786µm2 V2 , and λ =1.3µm.

And in Fig. 6(d) the electric-field distribution of the isolated separating-waveguide

section is plotted with d_f =3µm , 5d_I = µm , 1.55n_c= =n_s n_I = , n_f₀ =1.57 ,

5664 . 1 =

β , α =6.3786µm2 V2 , and λ=1.3µm.

To prove the accuracy of the results shown in Figs. 6(a)-(d), we use the beam propagation method [48] to simulate the electric field propagating along this structure, from the stem to the branching waveguides. For the calculation we choose the

following numerical data: the transverse sampling points N =4096, the longitudinal

step length ∆z=0.01µm , the total propagation distance Z =6000µm, and the

branching angle θ =0.3ο. The simulation result of this structure is shown in Fig. 7. By

comparing the results shown in Fig. 6 and Fig. 7, we confirm that our analysis is correct.

4. CONCLUSIONS

In this paper, we propose a novel method for analyzing multilayer optical waveguide structure with all nonlinear guiding films. This method can also be used to predict the evolution of waves propagating along the multibranch optical waveguide structure with all nonlinear guiding branches. We give a detail modal analysis of the proposed nonlinear optical waveguide structure. The analytical results are accompanied by numerical examples. The numerical simulation results show that our analyses are correct.

5. REFERENCES

1. G. I. Stegeman, C. T. Seaton, J Chilwell, and S. D. Smith, “Nonlinear waves

guided by thin films,” Apply. Phys. Lett.,44, pp.830-832 (1984).

2. H. Vach, G. I. Stegeman, C. T. Seaton, and I. C. Khoo, “Experimental observation

of nonlinear guided waves,” Opt. Lett., 9, pp.238-240 (1984).

3. A. D. Boardman and P. Egan, “Nonlinear surface and guided polaritions of a general layered dielectric structure,” J. Phys. Colloq., C5, pp.291-303 (1984).

4. N.N. Akhmediev, “Novel class of nonlinear surface waves: Asymmetric modes in

a symmetric layered structure,” Sov. Phys. -JETP, 56, pp.299-303 (1982).

5. F. Lederer, U. Langbein, and H. E. Ponath. “Nonlinear waves guided by a dielectric slab,” Appl. Phys. B., 31, pp.187-190 (1983).

6. U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Dispersion relations for nonlinear guided waves,” J. Mol. Struct., 115, pp.493-496 (1984).

(9)

an asymmetric layered structure,” Rev. Roumaine Phys. 29, pp.365-371 (1984). 8. D. J. Robbins, “TE modes in a slab waveguide bounded by nonlinear media,”

Opt. Commun., 47, pp.309-312 (1983).

9. U. Langbein, F. Lederer, and H. E. Ponath, “A new type of nonlinear slab-guided

wave,” Opt. Commun., 46, pp.167-169 (1983).

10. F. Fedyanin and D. Mihalache, “P-poiarized nonlinear surface polaritons in layered structures,” Z. Phys. B, 47, pp.167-173 (1982).

11. F. Lederer, U. Langbein, and H. E. Ponath. “Nonlinear waves guided by a dielectric slab,” Appl. Phys. B. 31, pp.187-190 (1983).

12. D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “P-polarized nonlinear surface waves in symmetric layered structures.” Phys. Scripta, 29, pp.269-275 (1984).

13. A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys.-JETP, 45, pp.896-905 (1977). 14. C. T. Steaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and

D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron., 21, pp.774-783 (1985).

15. L. Leine, C. Wacher, U. Langbein, and F. Lederer, “Evolution of nonlinear guided optical fields down a dielectric film with nonlinear cladding,” J. Opt. Soc. Amer. B, 5, pp.547-557 (1988).

16. S. She and S. Zhang, “Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding,” Opt. Comm., 161, pp.141-148 (1999). 17. Y. D. Wu, M. H. Chen, and H. J. Tasi, “A General Method for Analyzing the

Multilayer Optical Waveguide with Nonlinear Cladding and Substrate”, SPIE Design, Fabrication, and Characterization of Photonic Dervice II, 4594, pp.323-333 (2001).

18. Y. D. Wu and M. H. Chen, “Analyzing multilayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B, 19, No.8, pp.1737-1745 (2002).

19. Y. D. Wu and M. H. Chen, “The fundamental theory of the symmetric three layer nonlinear optical waveguide structures and the numerical simulation,” J. Nat. Kao. Uni. of App. Sci., 32, pp.133-147 (2002).

20. A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron., 22, No.2, pp.319-324 (1986).

21. H. Murata, M. Izutsu, and T. Sueta, “Optical bistability and all-optical switching in novel waveguide junctions with localized optical nonlinearity,” IEEE/OSA J. Lightwave Technol.,16, No.5, pp.833-840 (1998).

(10)

22. Y. D. Wu and Y. C. Jang, “Analyzing and Numerical study of Seven-Layer Optical Waveguide with Localized Nonlinear Central guiding Film,” Proceedings Electrical and Information Engineering Symposium, pp.24-28 (2003).

23. Y. D. Wu, “Analyzing multiplayer optical waveguides with a localized arbitrary nonlinear guiding film,” J. Quantum Electron., 40, No.5, pp.529-540, May (2004).

24. T. Sakakibara and N. Okamoto, “Nonlinear TE waves in a dielectric slab waveguide with two optical nonlinear layers,” IEEE J. Quantum Electron., 23, pp.2084-2088 (1987).

25. M. Cada, R. C. Gauthior, B. A. Paton, J. Chrostowski, “Nonlinear guided waves coupled nonlinearly in a planar GaAs/GaAlAs multiple-quantum-well structure,” Appl. Phys. Lett., 49, pp.755-757 (1986).

26. T. H. Wood, “Multiple-quantum-well (MQW) waveguided modulator,” J. Lightwave Technol. 6, pp.743-757 (1988).

27. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third

order nonlinear integrated optics,” J. Lightwave Technol., 6, pp.953-970 (1988).

28. S. R. Cvetkovic and A. P. Zhao, “Finite-element formalism for linear and nonlinear guided waves in multiple-quantum-well waveguides,” J. Opt. Soc. Amer. B, 10, pp.1401-1407 (1993).

29. S. Selleri and M. Zobol, “Stability analysis of nonlinear TE polarized waves in multiple-quantum-well waveguides,” IEEE J. Quantum Electron., 31, pp.1785-1789 (1995).

30. C. J. Hamiltoin, J. H. Marsh, D. C. Hutchings, J. S. Aitchison, G. T. Kennedy and W. Sibbett, “Localized Kerr-type nonlinearities in GaAs/AlGaAs multiple quantum well structure at 1.55μm,” Appl. Phys. Lett., 68, pp.3078-3080 (1996). 31. C. Rigo, L. Gastaldi, D. Campi, L. Faustini, C. Coriasso, C. Cacciatore and D.

Sholdani, “Multiple quantum well compressive strained heterostructures for low driving power all-optical waveguide switches,” J. Crystal. Growth., 188, pp.317-332 (1998).

32. M. H. Chen, and Y. D. Wu, “Numerical Study of TE waves Propagating in Multibranch Couplers,” Fiber and Integrated Optics, 11, pp.395-402 (1992). 33. Y. D. Wu, M. H. Chen, and C. H. Chu, “All-Optical Logic Device Using Bent

Nonlinear Tapered Y-Junction Waveguide Structure,” Fiber and Integrated Optics, 20, pp.517-524 (2001).

34. Y. D. Wu, M. H. Chen, and R. Z. Tasy, “A new all-optical Switching Device by using the nonlinear Mach-Zehnder interferometer with a control waveguides,” Proceedings CLEO/Pacific Rim Conference on Laser and Electro-Optics, I,

(11)

pp.292-295 (2003).

35. Y. D. Wu, M. H. Chen, and H. J. Tasi, “Novel All-optical Switching Device with Localized Nonlinearity,” Optical Society of America, Optics in Computing Devices, pp.297-299 (2002).

36. Y. D. Wu, “Nonlinear All-Optical Switching Device by Using the Spatial Soliton Collision,” Fiber and Integrated Optics, 23.4, August, to be published (2004).

37. T. Yabu,M. Geshiro,T. Kitamura,K. Nishida, and S. Sawa, “All-Optical Logic

Gates Containing a Two-Mode Nonlinear Waveguide,” IEEE J. Quantum Electron., 38, pp.37-46 (2002).

38. F. Garzia, and M. Bertolotti, “All-optical security coded key,” Optical Quantum Electronics, 33, pp.527-540 (2001).

39. Y. H. Pramono, and Endarko, “Nonlinear Waveguides for Optical Logic and

Computation,” Journal of Nonlinear Optical Physics & Materials, 10,

pp.209-222 (2001).

40. Y. H. Pramono, M. Geshiro, T. Kitamura, and S. Sawa, “Optical Logic OR-AND-NOT and NOR Gates in Waveguides Consisting of Nonlinear Material,” IEICE Trans. Electron., E83-C, pp.1755-1761 (2000).

41. H.-B. Lin, J. Y. Su, R. S. Cheng and, W. S. Wang, “Novel Optical Single-Mode Asymmetric Y-Branches for Variable Power Splitting,” IEEE Journal of Quantum Electronics, 35, pp.1092-1097 (1999).

42. F. Garzia, C. Sibilia, M. Bertolotti, ”All-Optical Solution Based Router,” Optics Cmmunications, 168, pp.277-285 (1999).

43. W. K. Burns, and A. F. Milton, “Mode conversion in planar dielectric separating waveguide,” IEEE J. Quantum Electron., 11, pp.32-39 (1975).

44. D. Marcuse, “Radiation losses of tapered dielectric slab wave-guides,” Bell Syst; Tech. J., 49, pp.273-290 (1970).

45. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron., 21, pp.774-783 (1985).

46. C. T. Seaton, X. Mai, G. I. Stegeman, N. G. Winful, “Nonlinear guided wave applications,” Opt. Eng., 24, pp.593-599 (1985).

47. H. Vach, G. I. Stegeman, C. T. Seaton, and I. C. Khoo, “Experimental observation of nonlinear guided waves,” Opt. Lett., 9, pp.238-240 (1984).

48. H. F. Chou, C. F. Lin, and G. C. Wang, “An Interative Finite Difference Beam Propagation Method for Modeling Second-Order Nonlinear Effects in Optical Waveguides,” J. of Lightwave Technol., 16, No.9, pp.1686-1693 (1998).

Substrate

0 = x d x= w d x= + w n d n x 2) 2 1 ( ) 1 2 1 ( − − + − − = w n d n x 1) 2 1 ( ) 1 2 1 ( − − + − − = w n d n x 1) 2 1 ( 2 1 − − + − =

Cladding

Nonlinear Guiding Film

Interaction Layer

(13)

(a) 1.56 1.58 1.6 1.62 1.64 0 0.4 0.8 1.2 1.6 Power density ( µ W /mm)

Effective refractive index A

B C

D

Fig.2. (a)Dispersion curve of the three-layer optical waveguide structure with the nonlinear central guiding film. (b)The electric field distributions with

respect to points A-D as shown in (a).

-10 -5 0 5 10 15 0 0.2 0.4 0.6 0.8 1 E*4 E*7 E*10 E*40 Transverse distance (µm) El ect ri c fi el d (V/ µ m) (b) A B C D

(14)

0 0.4 0.8 1.2 1.6 2.0 1.5 1.57 1.58 1.59 1.6 1.61 1.62 1.63 1.64 1.65 D B A C E F

Effective Refractive Index

Power density ( µ W /mm) Transverse Distance (µm) El ect ri c F iel d (V/ µ m) E D B C F -5 0 5 10 15 20 25 0 0.05 0.1 0.15 0.2 0.25 A (a) (b)

Fig.3 (a)Dispersion curve of the seven-layer optical waveguide structure with all nonlinear guiding films. (b)The electric field distributions with respect to

(15)

Fig.4.The multibranch optical waveguide structure with all nonlinear guiding

branches (

N

denoted the nonlinear medium).

…

s

n

c

n

(16)

Fig.5 The three-branch optical waveguide structure with all nonlinear guiding

branches (

N

denoted the nonlinear medium).

(N)

-10 -5 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Transverse Distance (µm) El ect ri c F iel d (V/ µ m ) (× 10 -2 ) Fig. 6(c) -5 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Transverse Distance (µm) El ect ri c F iel d (V/ µ m ) (× 10 -2 ) Fig. 6(d)

Fig.6 Electric-field distributions of the three-branch optical waveguide structure with all

nonlinear guiding branches at positions (a)Z1 (df =3µm, 5662β =1. ), (b)Z2

(19)

Fig.7 Evolution of a wave propagating along a three-branch optical waveguide structure with all nonlinear guiding branches.

Transverse Distance (µm) Propag ation Dis tance ( µ m) Z3 Z4 Z2 Z1

− , for β >ni (ni =2,4,...,m−1),     +       − − + = [ ( ) ( ) ( )] 1 ( )(1 ) 2 1 2 2 2 2 2 2 2 2 2 1 2 1 b E d A sn l d A dn d A sn A E p d A cn E E s s s I −           − − − (1 )]- (1 ) ( ) 1 [ ) ( ) ( 2 2 2 2 2 2 2 2 2 2 2 2 2 cn A d b E l b E l d A dn d A sn E A s s s

(21)

2 2 2 2 2 2 1 2 2 2 2 2 3 2 2 2 2 2 2 ( )[ ( ) ( )] [1 ( )(1 )] s s s p E E E cn A d dn A d l sn A d p l sn A d A b b    −  − −    , 2 2 2 2 2 2 1 1 1 1 2 1 1 1 1 [ ( ) ] ( ) ( ) 1 (1 ) ( )] 2 i i i Ii i i i i i i i i p E cn A d sn A d dn A d l sn A d A b − + + − − − − + + + + + + +  _∆  _∆   =  ∆ +  − − +       2 2 2 1 2 1 2 1 1 1 1 1 [1 (1 i )][1 ( )] ( ) ( ) i i i i i i i i A l l cn A d sn A d dn A d b + + − + − + + + + + +  _∆ _∆ ₋ ₋ ₋ ₋   2 2 2 2 2 2 2 1 4 1 1 1 1 1 1 1 (1 i ) ( ) ( ) ( ) i i ( )[ ( ) i i i i i i i i i p l sn A d cn A d dn A d cn A d dn A d b A + − + − − − − + + + + + + + + ∆ ∆ ∆ − − − 2 2 2 2 2 2 2 1 2 1 1 1 4 1 1 ( )] 1 ( )(1 ) i i i i i i i i i l sn A d l sn A d p b b + + − − + + + + + +     ∆  ₋ ₋∆             _ , i=3,5...,m−2, where ∆ =_i+₋₂ E_Ii₋₂+E_Ii₋₁exp(−p w_i₋₂ ), ∆ =_i−₋₂ E_Ii₋₂−E_Ii₋₁exp(−p w_i₋₂ ). For 4, 6,...,i= m−3, 1 0 02 2 E x cn b −   = _ _  , ( 3) ( 1) ( 2) 1 0 exp( ) I i i I i i i E p w E x cn b − − − −  − +  = _ _  , 2 2 2 2 2 2 0 2 0 0( 2 1 ) 2 E K =k E n − +n α , 2 ( 1) 2 2 2 2 1 0 ( 1 ) 2 m m m m m m E K ₋ =k E n ₋ −n +α − , 2 2 ( 1) 2 2 2 2 2 2 2 2 0 ( 1) 1 0 ( 1) 1 ( 1) 2 2 2 2 ( 1) 0 ( 1) 1 ( 1) ( 1) 2 ( 1) 0 ( 1) ( 1 ( ) exp( 2 )[ exp( 2 )] 2 2 3 2 exp( )[ 2 exp( )] 2 2 exp( )[ i I i i Ii i I i i i Ii i i i i I i Ii i i i i I i Ii i i I i Ii i I i E E K k E n n k E p w n n p w E E k p w n n E E p w E E k p w E α α β α α − − + + + + − + + − + − + − = − + + − − + − + − − + + − + − 2 2 ) +EIiexp( 2− p(i+1)w)].

分析導波層為非線性多層光波導結構之一般方法

行政院國家科學委員會專題研究計畫 成果報告