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Semicircular glass waveguide dependence on exchange-time and window width
Optics Communications 247 (2005) 313–317 www.elsevier.com/locate/optcom
Semicircular glass waveguide dependence on exchange-time and window width Zigang Zhou *, Desen Liu School of Physics, Southwest China Normal University, Chongqing 400715, China Received 26 April 2004; received in revised form 14 October 2004; accepted 17 November 2004
Abstract A detailed theoretical and experimental study of the semicircular waveguide dependence of surface Tl+–Na+ ionexchange waveguide on exchange-time and window width is reported. Modeling, which includes the effect of ionexchange time t and window width W, agrees well with our experiments showing that a surface semicircular waveguide pffiffiffiffiffiffiffiffiffi depends on Deff t W =2. The result may be used in the proper design of arrayed microlens and integrated optical circuits, such as diffractive waveguide gratings, and so on. 2004 Elsevier B.V. All rights reserved. PACS: 42.79.Gn; 42.79.Ry; 42.82.Et; 82.39.Wj Keywords: Tl+–Na+ion-exchange; Exchange-time; Window width; Semicircular waveguide
1. Introduction Researchers have used ion-exchange techniques for fabrication of integrated devices such as arrayed microlens, waveguide lasers, gratings, sensors, couplers, and phase and amplitude modulators for a number of years [1]. Glass waveguides are used extensively in integrated optical devices for optical communications because they *
Corresponding author. Tel: +86 23 68254607; fax: +86 23 68254608. E-mail address: [email protected] (Z. Zhou).
feature very low loss and are well matched with fibers [2]. In many cases, it is necessary to use surface semicircular waveguides in different sections of an integrated optical circuit, such as arrayed waveguide gratings and passive devices based on multimode waveguide. Because the elliptical spot of semiconductor laser become the small spot through surface semicircular waveguides. And different radii of semicircular waveguides are formed refractive lens arrays [3]. In this paper, surface channel waveguides are fabricated by performing a thermal Tl+ exchange through a window on the glass surface. Tl+ ions are
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.11.074
314
Z. Zhou, D. Liu / Optics Communications 247 (2005) 313–317
exchanged for sodium ions, and the refractive index is locally increased in glass. We report on a detailed study of the dependence of the initial window widths and exchange-time in the case of surface Tl+-ion-exchanged waveguides in glass.
2. Semicircular waveguide theory We first present our theoretical results that we developed to accurately calculate the radii of semicircular waveguides. This behavior is explained by the different mobilities of Tl+ and Na+ ions in the glass used in the experiments [4]. A binary ion exchange, such as Tl+–Na+ exchange, involves a system with two kinds of monovalent ions: indiffusing ions (Tl+) and outdiffusing ions (Na+) in the glass substrate. There are two types of force that drive the exchange of ions. One is due to the concentration gradient of ions and the other is due to the drift of ions resulting from the electric field in the glass during the exchange. The electric field is an internal field built by the local charge imbalance caused by the difference in radii of ions. The development of concentration distribution in time can be given by [5] " # 2 oC Tlþ DTlþ aðrC Tlþ Þ 2 : ð1Þ ¼ r C Tlþ þ ot 1 aC Tlþ 1 aC Tlþ The two terms result from the concentration gradient of ions and the internal field due to the distribution of ions with different radii, respectively. Here C Tlþ ¼ cTlþ =c0 is the normalized concentration of Tl+ ions, and cTlþ is the concentration of Tl+ ions, and c0 is the total ionic concentration. DTlþ and DNaþ are the selfdiffusion coefficients of Tl+ ions and Na+ ions, respectively, and a ¼ 1 DTlþ =DNaþ is a measure of the difference in ion mobilities. When M (the ratio between the self-diffusion coefficient of two ions, DTlþ =DNaþ ) is a small value, that is a 1, we get " # 2 oC Tlþ DTlþ ðrC Tlþ Þ 2 ¼ r C Tlþ þ : ð2Þ ot 1 C Tlþ 1 C Tlþ
A complete solution of (2) requires knowledge of the initial and boundary conditions. Therefore, we construct a theoretical model of the Tl+ ion-exchange waveguide. Fig. 1 shows the surface waveguide used for the diffusion process. We get 8 C Tlþ ðx,y,0Þ ¼ 0 for jxj > W =2, y > 0, > > > < C þ ðx,1,tÞ ¼ 0 for t P 0, Tl > C for t P 0, > Tlþ ð1,y,tÞ ¼ 0 > : C Tlþ ðx,0,tÞ ¼ 1 for t P 0 and jxj 6 W =2: To model the channel waveguide fabrication process by the ion-exchange process where Tl+ ions are introduced in glass, we calculated a change of concentration with a finite-difference algorithm to solve Eq. (2) under the above conditions. It is well known that there is a close correlation between the index profile and the glass waveguides chemical composition. The change in refractive index is taken to be proportional to the concentration of the Tl+ dopant ion introduced in the glass. The refractive index distribution function n(x,y) for channel waveguides, with two-dimensional diffusion from a long narrow strip of source material on the surface of the substrate, is given by W W x þx 1 2 2 ffiffiffiffiffiffiffiffiffi þ erf p ffiffiffiffiffiffiffiffiffi nðx,yÞ ¼ ns þ ðDnÞ erf p 2 2 Deff t 2 Deff t y erfc , ð3Þ dy where ns is the substrate index, Dn is the maximum surface index change, and dy is the diffusion depth of the channel waveguide. Here, we
Fig. 1. Schematic representation of Tl+ ion-exchange process.
Z. Zhou, D. Liu / Optics Communications 247 (2005) 313–317
have defined an effective diffusion constant as Deff ¼ 2DTlþ =ðM þ 1Þ, and t is the total ionexchange time. The model usually adopted to study a narrow rectangle considers that the depth is unaffected by the mark window width W [6]. The Tl+ diffusion depth dy is given by pffiffiffiffiffiffiffiffiffi d y d eff ¼ 2 Deff t: ð4Þ From Eq. (3), the index profile resulting from the simulations can be approximated with reasonable accuracy as the product of two functions describing the depth and the breadth, respectively. With the values we chose, we can approximate the depth distribution with an erfc function and the transversal one with two error functions. When the model is adopted to study a narrow rectangle, from [7], we get the normalized diffused functions in the x-direction and the y-direction as 8 W x W þx > < gðx=d x Þ ¼ 12 erf p2 ffiffiffiffiffiffiffi þ erf p2 ffiffiffiffiffiffiffi , 2 Deff t 2 Deff t > : f ðy=d Þ ¼ erfc y , y dy ð5Þ where dx is the diffusion breadth of the channel waveguide. From [8], if the ion-exchange time is longer and the window width is narrower, the diffused channel guides have assumed to have the same index profiles in the x-direction and the y-direction as
gðx=d x Þ ¼ f ðy=d y Þ:
315
ð6Þ
When x = dx and y = dy, From Eq. (6), we get W W dx þ dx 2 2 erf pffiffiffiffiffiffiffiffiffi þ erf pffiffiffiffiffiffiffiffiffi ¼ 2erfcð1Þ: ð7Þ 2 Deff t 2 Deff t In Eq. (7), we can get the breadth diffusion length dx. pffiffiffiffiffiffiffiffiffi From Eqs. (4) and (7), if Deff t W =2, that is the ion-exchange time is longer and longer or window width is narrower, we get pffiffiffiffiffiffiffiffiffi ð8Þ d y ¼ d x ¼ 2 Deff t: Therefore, a semicircular (no semielliptical) index profile is formed in glass waveguides. To study the surface channel waveguide dependence on exchange-time and window widths, our theoretical values used in work were the following: Tl+ diffusion coefficient, M = 0.01, D = 2.4 · 1016 m2/s; ion-exchange temperature, T = 530 C; window width, W = 6 lm; ns = 1.5125 (at k = 632.8 nm); Dn = 0.08. Fig. 2 gives the theoretical index profile by FDM. When W = 6 lm and t1 = 10 h, we get surface semicircular waveguides, dx = dy = 57.12 lm.
3. Experiment results and analysis To experimentally investigate surface radii dependence on different exchange times and window widths, we fabricated surface waveguides in
Fig. 2. Graded index-modulation profile by thermal ion-exchange process in glass with W = 6 lm and t = 10 h.
Z. Zhou, D. Liu / Optics Communications 247 (2005) 313–317
glass and used parameters similar to those used in theoretical modeling. The thermal ion exchange was performed in a Tl2SO4 melt at T = 530 C. Fig. 3 gives semicircular waveguides with 10 h. We developed a simple procedure for measuring the surface radii. When the window width is W = 6 lm, the surface radii were dx = 57.5 lm, dy = 56.86 lm with the exchange time t = 10 h. The same index profile is formed a curve in the interference fringes below the surface, so we get a semicircular index profile with a semicircular curve. Fig. 4 gives semicircular interference fringes below the surface waveguides with t = 10 h and W = 6 lm, and Fig. 5 gives the normalized diffused distributions in the x-direction and the y-direction of waveguide on the same axes. If the exchange-time is longer and the surface window width is narrower, the x value is equal to the y value from x,y P 4 lm. So the curve of the equal refractive index profile is semicircular from x,y P 4 lm. And, the semielliptical waveguides are formed in the range 0 < x, y < 4 lm.
Fig. 3. Semicircular waveguides at near-field with W = 6 lm and t = 10 h.
60
x 50
y
40
µm
316
30 20 10 0 0
0.1 0. 2 0. 3 0. 4 0.5 0. 6 0. 7 0. 8 0.9
1
Cd /C0 Fig. 5. The normalized diffused distributions in the x-direction (solid line) and the y-direction (broken line) of semicircular waveguide with W = 6 lm and t = 10 h.
4. Conclusion We studied the variations in radii of modes in ion-exchanged glass waveguides for different exchange-times and window widths. We found the semicircular waveguides are formed when the exchange-time is longer and the surface window width is narrow. Otherwise, the semielliptical waveguides are formed. The experimental results were found to agree well with the modeling. We get a anticipative theoretical Eqs. (3) and (8) by which a surface semicircular waveguide is experimentally decided, Also, as seen from the modeling results that the surface semicircular waveguide is formed with increasing exchange-time and decreasing window width. Acknowledgements This work was supported by the Chongqing Science Foundation. Authors are very thankful to the referees for the fruitful comments about our paper. References
Fig. 4. Interference pattern of semicircular waveguide with W = 6 lm and t = 10 h.
[1] P. Madasamy, B.R. West, M.M. Morrell et al., Opt. Lett. 28 (2003) 1132. [2] G. Chartier, P. Collier, A. Guez et al., Appl. Opt. 19 (1980) 1092.
Z. Zhou, D. Liu / Optics Communications 247 (2005) 313–317 [3] R.P. Salmino, J. Saarinen, J. Tervonen et al., Appl. Phys. Lett. 66 (1995) 917. [4] Z.G. Zhou, D.S. Liu, Chin. Opt. Lett. 1 (2003) 406. [5] D. Salazar, H. Porte, H. Marquez, Appl. Opt. 36 (1997) 8987.
317
[6] M. Fukuma, J. Noda, H. Iwasaki, J. Appl. Phys. 49 (1978) 3693. [7] D. Chadha, Opt. Commun. 140 (1997) 207. [8] M. Oikawa, K. Iga, M. Morinaga et al., Appl. Opt. 23 (1984) 1787.
Semicircular glass waveguide dependence on exchange-time and window width Zigang Zhou *, Desen Liu School of Physics, Southwest China Normal University, Chongqing 400715, China Received 26 April 2004; received in revised form 14 October 2004; accepted 17 November 2004
Abstract A detailed theoretical and experimental study of the semicircular waveguide dependence of surface Tl+–Na+ ionexchange waveguide on exchange-time and window width is reported. Modeling, which includes the effect of ionexchange time t and window width W, agrees well with our experiments showing that a surface semicircular waveguide pffiffiffiffiffiffiffiffiffi depends on Deff t W =2. The result may be used in the proper design of arrayed microlens and integrated optical circuits, such as diffractive waveguide gratings, and so on. 2004 Elsevier B.V. All rights reserved. PACS: 42.79.Gn; 42.79.Ry; 42.82.Et; 82.39.Wj Keywords: Tl+–Na+ion-exchange; Exchange-time; Window width; Semicircular waveguide
1. Introduction Researchers have used ion-exchange techniques for fabrication of integrated devices such as arrayed microlens, waveguide lasers, gratings, sensors, couplers, and phase and amplitude modulators for a number of years [1]. Glass waveguides are used extensively in integrated optical devices for optical communications because they *
Corresponding author. Tel: +86 23 68254607; fax: +86 23 68254608. E-mail address: [email protected] (Z. Zhou).
feature very low loss and are well matched with fibers [2]. In many cases, it is necessary to use surface semicircular waveguides in different sections of an integrated optical circuit, such as arrayed waveguide gratings and passive devices based on multimode waveguide. Because the elliptical spot of semiconductor laser become the small spot through surface semicircular waveguides. And different radii of semicircular waveguides are formed refractive lens arrays [3]. In this paper, surface channel waveguides are fabricated by performing a thermal Tl+ exchange through a window on the glass surface. Tl+ ions are
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.11.074
314
Z. Zhou, D. Liu / Optics Communications 247 (2005) 313–317
exchanged for sodium ions, and the refractive index is locally increased in glass. We report on a detailed study of the dependence of the initial window widths and exchange-time in the case of surface Tl+-ion-exchanged waveguides in glass.
2. Semicircular waveguide theory We first present our theoretical results that we developed to accurately calculate the radii of semicircular waveguides. This behavior is explained by the different mobilities of Tl+ and Na+ ions in the glass used in the experiments [4]. A binary ion exchange, such as Tl+–Na+ exchange, involves a system with two kinds of monovalent ions: indiffusing ions (Tl+) and outdiffusing ions (Na+) in the glass substrate. There are two types of force that drive the exchange of ions. One is due to the concentration gradient of ions and the other is due to the drift of ions resulting from the electric field in the glass during the exchange. The electric field is an internal field built by the local charge imbalance caused by the difference in radii of ions. The development of concentration distribution in time can be given by [5] " # 2 oC Tlþ DTlþ aðrC Tlþ Þ 2 : ð1Þ ¼ r C Tlþ þ ot 1 aC Tlþ 1 aC Tlþ The two terms result from the concentration gradient of ions and the internal field due to the distribution of ions with different radii, respectively. Here C Tlþ ¼ cTlþ =c0 is the normalized concentration of Tl+ ions, and cTlþ is the concentration of Tl+ ions, and c0 is the total ionic concentration. DTlþ and DNaþ are the selfdiffusion coefficients of Tl+ ions and Na+ ions, respectively, and a ¼ 1 DTlþ =DNaþ is a measure of the difference in ion mobilities. When M (the ratio between the self-diffusion coefficient of two ions, DTlþ =DNaþ ) is a small value, that is a 1, we get " # 2 oC Tlþ DTlþ ðrC Tlþ Þ 2 ¼ r C Tlþ þ : ð2Þ ot 1 C Tlþ 1 C Tlþ
A complete solution of (2) requires knowledge of the initial and boundary conditions. Therefore, we construct a theoretical model of the Tl+ ion-exchange waveguide. Fig. 1 shows the surface waveguide used for the diffusion process. We get 8 C Tlþ ðx,y,0Þ ¼ 0 for jxj > W =2, y > 0, > > > < C þ ðx,1,tÞ ¼ 0 for t P 0, Tl > C for t P 0, > Tlþ ð1,y,tÞ ¼ 0 > : C Tlþ ðx,0,tÞ ¼ 1 for t P 0 and jxj 6 W =2: To model the channel waveguide fabrication process by the ion-exchange process where Tl+ ions are introduced in glass, we calculated a change of concentration with a finite-difference algorithm to solve Eq. (2) under the above conditions. It is well known that there is a close correlation between the index profile and the glass waveguides chemical composition. The change in refractive index is taken to be proportional to the concentration of the Tl+ dopant ion introduced in the glass. The refractive index distribution function n(x,y) for channel waveguides, with two-dimensional diffusion from a long narrow strip of source material on the surface of the substrate, is given by W W x þx 1 2 2 ffiffiffiffiffiffiffiffiffi þ erf p ffiffiffiffiffiffiffiffiffi nðx,yÞ ¼ ns þ ðDnÞ erf p 2 2 Deff t 2 Deff t y erfc , ð3Þ dy where ns is the substrate index, Dn is the maximum surface index change, and dy is the diffusion depth of the channel waveguide. Here, we
Fig. 1. Schematic representation of Tl+ ion-exchange process.
Z. Zhou, D. Liu / Optics Communications 247 (2005) 313–317
have defined an effective diffusion constant as Deff ¼ 2DTlþ =ðM þ 1Þ, and t is the total ionexchange time. The model usually adopted to study a narrow rectangle considers that the depth is unaffected by the mark window width W [6]. The Tl+ diffusion depth dy is given by pffiffiffiffiffiffiffiffiffi d y d eff ¼ 2 Deff t: ð4Þ From Eq. (3), the index profile resulting from the simulations can be approximated with reasonable accuracy as the product of two functions describing the depth and the breadth, respectively. With the values we chose, we can approximate the depth distribution with an erfc function and the transversal one with two error functions. When the model is adopted to study a narrow rectangle, from [7], we get the normalized diffused functions in the x-direction and the y-direction as 8 W x W þx > < gðx=d x Þ ¼ 12 erf p2 ffiffiffiffiffiffiffi þ erf p2 ffiffiffiffiffiffiffi , 2 Deff t 2 Deff t > : f ðy=d Þ ¼ erfc y , y dy ð5Þ where dx is the diffusion breadth of the channel waveguide. From [8], if the ion-exchange time is longer and the window width is narrower, the diffused channel guides have assumed to have the same index profiles in the x-direction and the y-direction as
gðx=d x Þ ¼ f ðy=d y Þ:
315
ð6Þ
When x = dx and y = dy, From Eq. (6), we get W W dx þ dx 2 2 erf pffiffiffiffiffiffiffiffiffi þ erf pffiffiffiffiffiffiffiffiffi ¼ 2erfcð1Þ: ð7Þ 2 Deff t 2 Deff t In Eq. (7), we can get the breadth diffusion length dx. pffiffiffiffiffiffiffiffiffi From Eqs. (4) and (7), if Deff t W =2, that is the ion-exchange time is longer and longer or window width is narrower, we get pffiffiffiffiffiffiffiffiffi ð8Þ d y ¼ d x ¼ 2 Deff t: Therefore, a semicircular (no semielliptical) index profile is formed in glass waveguides. To study the surface channel waveguide dependence on exchange-time and window widths, our theoretical values used in work were the following: Tl+ diffusion coefficient, M = 0.01, D = 2.4 · 1016 m2/s; ion-exchange temperature, T = 530 C; window width, W = 6 lm; ns = 1.5125 (at k = 632.8 nm); Dn = 0.08. Fig. 2 gives the theoretical index profile by FDM. When W = 6 lm and t1 = 10 h, we get surface semicircular waveguides, dx = dy = 57.12 lm.
3. Experiment results and analysis To experimentally investigate surface radii dependence on different exchange times and window widths, we fabricated surface waveguides in
Fig. 2. Graded index-modulation profile by thermal ion-exchange process in glass with W = 6 lm and t = 10 h.
Z. Zhou, D. Liu / Optics Communications 247 (2005) 313–317
glass and used parameters similar to those used in theoretical modeling. The thermal ion exchange was performed in a Tl2SO4 melt at T = 530 C. Fig. 3 gives semicircular waveguides with 10 h. We developed a simple procedure for measuring the surface radii. When the window width is W = 6 lm, the surface radii were dx = 57.5 lm, dy = 56.86 lm with the exchange time t = 10 h. The same index profile is formed a curve in the interference fringes below the surface, so we get a semicircular index profile with a semicircular curve. Fig. 4 gives semicircular interference fringes below the surface waveguides with t = 10 h and W = 6 lm, and Fig. 5 gives the normalized diffused distributions in the x-direction and the y-direction of waveguide on the same axes. If the exchange-time is longer and the surface window width is narrower, the x value is equal to the y value from x,y P 4 lm. So the curve of the equal refractive index profile is semicircular from x,y P 4 lm. And, the semielliptical waveguides are formed in the range 0 < x, y < 4 lm.
Fig. 3. Semicircular waveguides at near-field with W = 6 lm and t = 10 h.
60
x 50
y
40
µm
316
30 20 10 0 0
0.1 0. 2 0. 3 0. 4 0.5 0. 6 0. 7 0. 8 0.9
1
Cd /C0 Fig. 5. The normalized diffused distributions in the x-direction (solid line) and the y-direction (broken line) of semicircular waveguide with W = 6 lm and t = 10 h.
4. Conclusion We studied the variations in radii of modes in ion-exchanged glass waveguides for different exchange-times and window widths. We found the semicircular waveguides are formed when the exchange-time is longer and the surface window width is narrow. Otherwise, the semielliptical waveguides are formed. The experimental results were found to agree well with the modeling. We get a anticipative theoretical Eqs. (3) and (8) by which a surface semicircular waveguide is experimentally decided, Also, as seen from the modeling results that the surface semicircular waveguide is formed with increasing exchange-time and decreasing window width. Acknowledgements This work was supported by the Chongqing Science Foundation. Authors are very thankful to the referees for the fruitful comments about our paper. References
Fig. 4. Interference pattern of semicircular waveguide with W = 6 lm and t = 10 h.
[1] P. Madasamy, B.R. West, M.M. Morrell et al., Opt. Lett. 28 (2003) 1132. [2] G. Chartier, P. Collier, A. Guez et al., Appl. Opt. 19 (1980) 1092.
Z. Zhou, D. Liu / Optics Communications 247 (2005) 313–317 [3] R.P. Salmino, J. Saarinen, J. Tervonen et al., Appl. Phys. Lett. 66 (1995) 917. [4] Z.G. Zhou, D.S. Liu, Chin. Opt. Lett. 1 (2003) 406. [5] D. Salazar, H. Porte, H. Marquez, Appl. Opt. 36 (1997) 8987.
317
[6] M. Fukuma, J. Noda, H. Iwasaki, J. Appl. Phys. 49 (1978) 3693. [7] D. Chadha, Opt. Commun. 140 (1997) 207. [8] M. Oikawa, K. Iga, M. Morinaga et al., Appl. Opt. 23 (1984) 1787.