A Determination Method of Diffusion-Parameter Values in the Ion-Exchange Optical Waveguides in Soda-Lime glass Made by Diluted AgNO 3 with NaNO 3

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1 大阪電気通信大学研究論集 ( 自然科学編 ) 第 51 号 A Determination Method of Diffusion-Parameter Values in the Ion-Exchange Optical Waveguides in Soda-Lime glass Made by Diluted AgNO 3 with NaNO 3 Takuya IWATA and Kiyoshi KISHIOKA Abstract This paper describes a conventional determining method of optical parameter values in the optical waveguides fabricated in glass substrate by using the thermal ionexchange technique. An useful relationship between the effective indices of the guided mode and waveguide parameters is presented in a simple form, and applying it to measured values of the effective indices, the values of the substrate surface index n 0, substrate index n b and diffusion depth of the doped ion are gotten. Applied the proposed method to the Ag + -ion diffused waveguides fabricated in the soda-lime glass using AgNO 3 salt diluted with NaNO 3 as the ion source, dependences of the optical parameter values on the dilution of the ion source are performed. Values of the diffusion coefficient D of the dopant Ag + -ion are also estimated based on the guess index-profiles in the ion-diffused layer. 1. K + Ag o C NO 3 Soda-lime Pyrex [1-3] IWKB Inverse WKB Method [4] 年 5 月 9 日受理 37

2 AgNO 3 NaNO 3 Ag + AgNO 3 Ag + AgNO 3. Na n C(x, t) C x = 1 D C t D D K + (1) [1] Ag + Soda-lime D t x =0 (1) ( ) x C(x, t) = erfc () Dt [1] erfc(=1-erf) n(x) n 0 ( ) x n(x) = n 0 erfc (3) Dt (1) : n 0 = n 0 n b 38

3 n 0 n b 1 D t C(x, t)) n 0 e ( x a) (4) a 1 n 0 a D 3. [5] 3.1 β x n(x) 3 39

4 β x =0 x = x t zigzag-path β x =0 θ(0) Mode-line β n(x)k 0 x Q(x) x θ(x) x [ ( β k 0 = n p sin A + sin 1 sin θi n p )] Mode-line 3. β xt Q(x)dx ϕ c ϕ t =Mπ, M =0, 1,,... (5) 0 Q(x) =n(x)k 0 cos θ(x) = k 0n (x) β, k 0 = π λ 4 40

5 k 0 λ ϕ c ϕ t x = x t x t Q(x t )=0 n =1 ϕ c ϕ c = tan 1 β k 0 n 0k 0 β (n 0k0 β ) << (β k0) ϕ c π ϕ t π [6] (5) xt Q(x)dx = π(m + 3 ), M =0, 1,,... (6) 0 3. n 0 n b a n(x) =(n 0 n b )e ( x a) + nb (7) (6) n (x) n (x) n b + (n 0 n b)e ( x a) (8) (n 0 n b ) << n b (n 0 n b ) n b (n 0 + n b ) n b = n 0 (n 0 n b) (8) n (x) n 0 [ 1 ( 1 e x a )], = n 0 n b n 0 (9) (6) Q(x) n (x) (9) xt ( ) 1 [v e ] ( x a w 1 dx = M + 3 ) π, (M =0, 1,,...) (10) 0 a v =k 0a n 0, w = a (β k 0n b) x t v e x/a w =0 x t =a ln(v/w) (10) u = e x a u (10) ( 1 I = 4v 1 b ) 1 w du, b = b u v 5 = β k 0n b k 0(n 0 n b ) (11) 41

6 cos η = b/u u η I =4v ηt b tan ηdη =4v b(tan η t η t ) =4v [ (1 b) 1 b cos 1 b ] (1) 0 η t cos 1 b (1) (6) 1 b b cos 1 b = π ( M + 3 ), M =0, 1,,... (13) 4v 0 <b<1 φ = cos 1 b 1 b = sin φ (13) sin φ φ cos φ = π 4v ( M + 3 ), M =0, 1,,... (14) 3.3 b φ b = w v = (β k 0n b) k 0(n 0 n b ), φ = cos 1 b (15) β β b φ β k 0 n b k 0 n 0 b 0 1 π φ 0 (Mode cut-off) ( θ = 0) b Case(a):1 Case(b):0 (14) β Case(a) b 1 φ 0 sin φ φ, cos φ 1 1 φ 6 4

7 (14) (16) 1 φ3 π ( M + 3 ), M =0, 1,,... (16) 4v φ = cos 1 b π ( ) b ( ) 1 π 3 (1 ) 3 1 b = π ( M + 3 ), M =0, 1,,... (17) 4v (1 b 1 ) b = 1 1 b 1 1 (1 b) (17) ( 1 )5 ( ) ( π 3 n 0 n eff n 0 n b ) 3 = ( π M + 3 ), M =0, 1,,... (18) 4k 0 a(n 0 n b) 1 β/k 0 = n eff n eff n eff = n 0 4 ( ) π 4 3 ( n 0 n b k 0 a ) 3 ( M + 3 ) 3, M =0, 1,,... (19) n eff Case(b) b 0 φ π sin φ cos φ φ π sin φ 1 1 (14) ( φ ı ) π, cos φ φ ( π ) = φ φ = cos 1 b π ( ) 1 1 b (14) 1 1 ( 8 π b 1 π = M + 3 ) 4v 7 43

8 Case(a) n eff n eff = n b + 4 ( ) [ ( 1 4k π 4 k0a 0 a n 0 n b π M + 3, M =0, 1,,... (0) )] n eff 4. n eff (= β/k 0 ) n eff n eff Case (a) (19) M (M + 3) 3 Case (b) (0) π(m + 3 ) 3 n eff (M + 3) 3 4 π(m + 3) n eff (M + 3) 3 π(m + 3 ) 3 n eff (M + 3) 3 ( 4 n eff π(m + 3) (a) n 0 n eff (19) n eff (M + 3) 3 1 (M + 3) 3 = 0 (19) n eff = n 0 3 n eff n 0 n eff 8 44

9 (b) n b n eff (0) π(m + 3) =4k 0a n 0 n b (= x min) n eff n b (0) n eff n b (c) a 3 γ = 4 ( ) π 4 3 ( n 0 n b k 0 a γ (b) n b a ) 3 a = 8 (n 0 n b) k 0 γ 3 π (1) 4. Soda-lime Ag + NaNO 3 AgNO 3 Ag C 1 1 [mol %] NaNO 3 AgNO 3 [NaNO 3 ] [AgNO 3 ] mol% = [AgNO 3 ] [AgNO 3 ] + [NaNO 3 ] 100 % 9 45

10 5. n 0 a n b β µm He-Ne 5.1 n 0 n 0 n 0 [mol %] n a a 3 3 a [1/µ m] mol%

11 5.3 n b n b = mol % mol % 7.0 mol % 11 47

12 6. D Ag + t n(x, t) =(n 0 n b )erfc ( x Dt ) + n b () D D erfc- exp- D 8 D D 8 erfc exp D D 4 D 4 D mol % D [µm /min]

13 9 11 D mol % mol % 11.0 mol %

14 ( ) x + C = 1 y D C (3) t C(0,y)=1, ( y < W/) C(0,y)=0, ( y > W/) Ditechret x <0 C(x, y, t) = 1 e {erf ξ π x Dt [ ( ξ y + W x ) ] erf [ ( ξ y W x ) ]} dξ (4) [1] 13 D Ag + 1 (a) t=10 min. (b) t=40 min. (c) t=80 min. 13 mol %) 14 50

15 8. n eff (1) n 0 3 n eff n 0 n 0 4 ()n b 4 4 n b 14 n b ( 0.4 mol %) 3 n eff 14 n eff n b 14 n b 9. Soda-lime Ag

16 Ag + [1] K.Kishioka, Determination of Diffusion-Parameter Values in K + -Ion Exchange Waveguides Made by Diluted KNO 3 in Soda-Lime Glass, IEICE Trans., Electron. Vol-E78-C, No.10, pp , [] G.L.Yip, P. C. Noutsios, and K.Kishioka, Characteristics of optical waveguides made by elctric-fieldn-assisted K + -ion exchange, Opt., Lett., Vol.15, no.14, pp , [3] BK-7 NO 3 C Vol.13, No., pp , 003. [4] J. M. White and P. F. Heidrich, Optical waveguide refractive index profiles determined from measurement of mode indices: A simple analysis, Appl. Opt., Vol.15, No.1, pp , Jan., [5],, RS15-14 (016 ). [6] C. Vassallo; Optical Waveguide Concepts, Elsevier, pp ,

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