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A new method to determine the refractive index of planar optical waveguides
OPTICS
Optics Communications 91 (1992) 334-336 North-HoUand
COMMUNICATIONS
A new method to determine the refractive index of planar optical waveguides S.P. Pogossian CNRS-L MMM, 92195 Meudon, France
Received 4 February 1992; revised manuscript received 30 March 1992
A new method to evaluate the surface refractive index of dielectric planar waveguides is presented, Analytic expressions are obtained for the surface refractive index of the waveguide expressed directly by the experimental mode effective index spectrum.
1. Introduction The surface refractive index (SRI) of dielectric planar waveguides is one of the most important physical properties determining the mode structure of the optical field. To evaluate the optical waveguide SRI, first the waveguide profile is approximated by some theoretical profile, which depends on processing features of waveguide fabrication, and then by the help of the wave equation or the Wentzel-Kramers-Brillouin (WKB) approximation the profile is fitted to the experimental mode spectrum. Theoretical profile parameters are adjusted to zero in the wave equation or WKB integrals, in order to achieve a good agreement between theoretical and experimental mode values. Second, the SRI value is calculated on the waveguide surface by means of the obtained theoretical refractive index profile ( R I P ) [ 1,2]. With such an indirect determination, the waveguide SRI depends on the form of the theoretical profile, whereas it is a physical property of the waveguide and its value should not depend on the theoretical form of the profile. From this point of view the determination of the waveguide SRI from the experimental mode values is rather valuable, which also facilitates evaluating the other characteristics of the RIP. We propose a new method to determine directly the SRI from the experimental mode spectrum.
2. Determination of the SRI of a waveguide with step-index RIP According to the condition of lateral resonance for a thin film optical waveguide, the total phase change on one zigzag of the beam path must be a multiple of 2n,
2qmh- 2 arctan ( rm/ qm ) - 2 arctan (pm/ qm ) = 2 n m ,
where h is the waveguide thickness, qm=k(n~ -
n2)l/2,
pm=k(nEm-n2) 1/2,
r m = k ( n 2 - n 2 ) 1/2,
k = 2rt/2, 2 is the light wavelength in vacuum, nf the refractive index (RI) of the thin film, nc the RI of the dielectric layer adjacent to the waveguide surface, ns the RI of the substrate, nm the mode effective index values, m may acquire 0, 1, 2, ... values. It must be noted, that eq. ( 1 ) is satisfied also for the value m = - l, when q_ ~= 0 (n_ l = nr) is assumed. It means that nm = nf can be formally considered as the waveguide mode effective index with the mode number m = - I. In eq. ( l ), qm is a function of the discrete variable m. It can be interpolated to extend to a continuous distribution [ 3-5 ]. As qm is an odd function of m, we can expand in the following series in terms of m around the point m = - 1, qm = a t ( m + 1 ) -1-O~3( m q- 1 )3+ ... q- O~2n+ 1(m-l- 1 )2n+ t + ....
334
(1)
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0030-4018/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 91, number 5,6
OPTICS COMMUNICATIONS
1 August 1992
where 0~2i+1 are coefficients of the expansion (2). As n~ = n ~ - q ~ / k 2 and qm is an odd function of m, we can determine n~ as an even function of m in the form of an expansion around the point m = - 1. Taking also into account that n_ ~= nf, it can be presented in the form
(ii) Formula (7) is obtained using the first three terms of expansion (3)
n2m =nf2 + b2(m + I ) 2 + b 4 ( m + I )4+...
n~=n~+(3n~-4n21)/5+(8n2-n])/35.
+b2i(m + 1 )2i+ ....
(3)
The derivative of eq. ( I ) with respect to m has the form [3]
5qm/6m=Tt/(h+ l/rm + l/pm) .
(4)
From eqs. (2), ( 3 ) and (4) the following expression can be derived for b2 = - ct~/k2,
a~=k(-b2)l/2=Tt/(h+I/r_l +l/p_l) ,
(5)
where r _ l = k ( n f2- n ~ 2) 1/2 , p _ t = k ( n 2f-n2c) I/2. Taking into account the several terms of expansion (3), we obtain analytic expressions for t/f expressed only by mode effective index values. (i) Formula (6) is obtained using the first two terms of expansion (3)
n~f=(4nE-n2)/3.
(6)
In the papers [5-7] is obtained the same formula.
n 2=n 2+(n 2-n2)/2+(n
2-n2)/10.
(7)
(iii) Formula (8) is obtained using the first four terms of expansion (3) (8)
It must be noted that formulas (6), ( 7 ), (8) are valid for both TE and TM modes. In order to demonstrate the accuracy of SRI evaluation by the formulas (5), (6), (7), the following calculations are presented. From eq. (1) the mode values are calculated with the help of given values of nf, ns and h. Then the value of nf is calculated usingi formula (5), (6), (7). In the table l different theoretical profiles are considered and the value of nf is obtained using formulas (5), (6) and (7).
3. Determination of the waveguide SRI with Fermitype RIP In order to demonstrate the validity of this method applied to waveguides with a RI distribution, different from the step index profile, Fermi type RIP is considered. In the paper [ 8] an analytic expression
Table 1 In the presented table three theoretical profilesare considered. In the left columnare giventhe values ofnf, n, and h, whereh is measured in ~tmand the laser wavelength is 1.152 ~tm.In the middle column are given the values of modes calculated by the help of eq. ( I ) with the given values ofnf, n, and h. In the right column the values of nfare calculated by the formulas (6), (7) and (8). The values of the parameters of theoretical profile
The mode values calculated byeq. (1)
The calculated values of the SRI by the formulas (6)
(7)
(8)
1. n f = 2 . 2 0 0 0 0 0 0 0 n,= 1.94500000 h= 1.50000000
no = 2 . 1 7 5 9 4 7 2 9 nl = 2 . 1 0 3 5 4 8 8 4 n2= 1.98566995
nf= 2.19955046
nf= 2.20033119
2. n f = 2 . 2 0 0 0 0 0 0 n, = 1 . 9 4 5 0 0 0 0 0 h = 2.00000000
no=2.18536977 n~ = 2 . 1 4 1 2 4 8 4 4 n2 = 2 . 0 6 7 2 7 5 7 9 n3 = 1 . 9 6 6 6 7 7 8 3
nf=2.19988017
nf= 2.0003569
nf=2.19993393
3. n f = 2 . 2 0 0 0 0 0 0 0 rts = 1 . 9 4 5 0 0 0 0 0 h = 2.50000000
no = 2 . 1 9 0 1 7 4 2 6 n~ ----2 . 1 6 0 5 5 9 0 3 n2 = 2. l 1 0 7 9 8 9 7 n3 -- 2 . 0 4 0 6 9 8 9 2 n4= 1.95414272
nfm 2 . 1 9 9 9 5 7 5 5
nf= 2.0000762
nf= 2.19999527
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OPTICS COMMUNICATIONS
of the dispersion equation (10) is obtained for Fermi-type RIP [9 ],
n2(x)=n2+(n2-n2)/{l+exp[(x-h)/u]),
(9)
where h, u, n 2 are parameters o f the RIP and nf is the SRI of the waveguide. q,, h - arctan ( rm /q,~ ) - arctan (p,,/q,~ ) - S,, = n m ,
(lO) qm=k(n~-n2) I/2, pm=k(n2-n~) '/2, rm= k ( n 2 - n ~ ) '/2, and
where
S~ = ~ { arctan(2uq,Ji) - 2 arctan[uqJ(r~u+i) ] ) .
the SRI from the experimental mode values, which facilitates the calculation o f the SRI value and the other characteristics of the waveguide.
Acknowledgements I wish to thank J. Gouzerh, A.A. Stashkevitch, E.A. Arutunian and S.Kh. Galoyan for valuable discussions and encouragement.
References ( 11 )
In eq. (10), qm is an odd function o f m and for the value m = - 1 eq. ( 10 ) is satisfied only if q_ ~ is assumed to be 0. Therefore the formulas (6), (7) and (8) are valid for them.
4. Conclusion In conclusion it must be noted, that the present method o f evaluation of the planar optical waveguide SRI, allows finding the direct dependence o f
336
1 August 1992
[ 1] P.T. Tien and R. Ulrich, J. Opt. Soe. Am. 60 (1970) 1325. [ 2 ] M.J. Adams, An introduction to optical waveguides (Wiley, New York, 1981). [ 3] E.A. Arutunyan, S.Kh. Galoyan, S.P. Pogossian and D.A.N. Armianskoi, S.S.R., Phizika 90 (1990) 16. [4] M.O. Vassell, J. Opt. Soc. Am. 65 (1975) 1019. [ 5 ] G. Svantesson,J. Magn. Soc. Jpn. ! 1; supplement S 1 ( 1987) 405. [6] G.H. Charter and P.C. Jaussand, J. Appl. Phys. 49 (1978) 917. [ 7 ] E.A.Arutunyan and S.Kh. Galoyan,Optics Comm. 56 (1986) 399. [8] E.A. Arutunyan, S.Kh. Galoyan and S.P. Pogassian, Izvestia Akademii Nauk Armianskoi S.S.R., Phizika 90 ( 1989) 215.
Optics Communications 91 (1992) 334-336 North-HoUand
COMMUNICATIONS
A new method to determine the refractive index of planar optical waveguides S.P. Pogossian CNRS-L MMM, 92195 Meudon, France
Received 4 February 1992; revised manuscript received 30 March 1992
A new method to evaluate the surface refractive index of dielectric planar waveguides is presented, Analytic expressions are obtained for the surface refractive index of the waveguide expressed directly by the experimental mode effective index spectrum.
1. Introduction The surface refractive index (SRI) of dielectric planar waveguides is one of the most important physical properties determining the mode structure of the optical field. To evaluate the optical waveguide SRI, first the waveguide profile is approximated by some theoretical profile, which depends on processing features of waveguide fabrication, and then by the help of the wave equation or the Wentzel-Kramers-Brillouin (WKB) approximation the profile is fitted to the experimental mode spectrum. Theoretical profile parameters are adjusted to zero in the wave equation or WKB integrals, in order to achieve a good agreement between theoretical and experimental mode values. Second, the SRI value is calculated on the waveguide surface by means of the obtained theoretical refractive index profile ( R I P ) [ 1,2]. With such an indirect determination, the waveguide SRI depends on the form of the theoretical profile, whereas it is a physical property of the waveguide and its value should not depend on the theoretical form of the profile. From this point of view the determination of the waveguide SRI from the experimental mode values is rather valuable, which also facilitates evaluating the other characteristics of the RIP. We propose a new method to determine directly the SRI from the experimental mode spectrum.
2. Determination of the SRI of a waveguide with step-index RIP According to the condition of lateral resonance for a thin film optical waveguide, the total phase change on one zigzag of the beam path must be a multiple of 2n,
2qmh- 2 arctan ( rm/ qm ) - 2 arctan (pm/ qm ) = 2 n m ,
where h is the waveguide thickness, qm=k(n~ -
n2)l/2,
pm=k(nEm-n2) 1/2,
r m = k ( n 2 - n 2 ) 1/2,
k = 2rt/2, 2 is the light wavelength in vacuum, nf the refractive index (RI) of the thin film, nc the RI of the dielectric layer adjacent to the waveguide surface, ns the RI of the substrate, nm the mode effective index values, m may acquire 0, 1, 2, ... values. It must be noted, that eq. ( 1 ) is satisfied also for the value m = - l, when q_ ~= 0 (n_ l = nr) is assumed. It means that nm = nf can be formally considered as the waveguide mode effective index with the mode number m = - I. In eq. ( l ), qm is a function of the discrete variable m. It can be interpolated to extend to a continuous distribution [ 3-5 ]. As qm is an odd function of m, we can expand in the following series in terms of m around the point m = - 1, qm = a t ( m + 1 ) -1-O~3( m q- 1 )3+ ... q- O~2n+ 1(m-l- 1 )2n+ t + ....
334
(1)
(2)
0030-4018/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 91, number 5,6
OPTICS COMMUNICATIONS
1 August 1992
where 0~2i+1 are coefficients of the expansion (2). As n~ = n ~ - q ~ / k 2 and qm is an odd function of m, we can determine n~ as an even function of m in the form of an expansion around the point m = - 1. Taking also into account that n_ ~= nf, it can be presented in the form
(ii) Formula (7) is obtained using the first three terms of expansion (3)
n2m =nf2 + b2(m + I ) 2 + b 4 ( m + I )4+...
n~=n~+(3n~-4n21)/5+(8n2-n])/35.
+b2i(m + 1 )2i+ ....
(3)
The derivative of eq. ( I ) with respect to m has the form [3]
5qm/6m=Tt/(h+ l/rm + l/pm) .
(4)
From eqs. (2), ( 3 ) and (4) the following expression can be derived for b2 = - ct~/k2,
a~=k(-b2)l/2=Tt/(h+I/r_l +l/p_l) ,
(5)
where r _ l = k ( n f2- n ~ 2) 1/2 , p _ t = k ( n 2f-n2c) I/2. Taking into account the several terms of expansion (3), we obtain analytic expressions for t/f expressed only by mode effective index values. (i) Formula (6) is obtained using the first two terms of expansion (3)
n~f=(4nE-n2)/3.
(6)
In the papers [5-7] is obtained the same formula.
n 2=n 2+(n 2-n2)/2+(n
2-n2)/10.
(7)
(iii) Formula (8) is obtained using the first four terms of expansion (3) (8)
It must be noted that formulas (6), ( 7 ), (8) are valid for both TE and TM modes. In order to demonstrate the accuracy of SRI evaluation by the formulas (5), (6), (7), the following calculations are presented. From eq. (1) the mode values are calculated with the help of given values of nf, ns and h. Then the value of nf is calculated usingi formula (5), (6), (7). In the table l different theoretical profiles are considered and the value of nf is obtained using formulas (5), (6) and (7).
3. Determination of the waveguide SRI with Fermitype RIP In order to demonstrate the validity of this method applied to waveguides with a RI distribution, different from the step index profile, Fermi type RIP is considered. In the paper [ 8] an analytic expression
Table 1 In the presented table three theoretical profilesare considered. In the left columnare giventhe values ofnf, n, and h, whereh is measured in ~tmand the laser wavelength is 1.152 ~tm.In the middle column are given the values of modes calculated by the help of eq. ( I ) with the given values ofnf, n, and h. In the right column the values of nfare calculated by the formulas (6), (7) and (8). The values of the parameters of theoretical profile
The mode values calculated byeq. (1)
The calculated values of the SRI by the formulas (6)
(7)
(8)
1. n f = 2 . 2 0 0 0 0 0 0 0 n,= 1.94500000 h= 1.50000000
no = 2 . 1 7 5 9 4 7 2 9 nl = 2 . 1 0 3 5 4 8 8 4 n2= 1.98566995
nf= 2.19955046
nf= 2.20033119
2. n f = 2 . 2 0 0 0 0 0 0 n, = 1 . 9 4 5 0 0 0 0 0 h = 2.00000000
no=2.18536977 n~ = 2 . 1 4 1 2 4 8 4 4 n2 = 2 . 0 6 7 2 7 5 7 9 n3 = 1 . 9 6 6 6 7 7 8 3
nf=2.19988017
nf= 2.0003569
nf=2.19993393
3. n f = 2 . 2 0 0 0 0 0 0 0 rts = 1 . 9 4 5 0 0 0 0 0 h = 2.50000000
no = 2 . 1 9 0 1 7 4 2 6 n~ ----2 . 1 6 0 5 5 9 0 3 n2 = 2. l 1 0 7 9 8 9 7 n3 -- 2 . 0 4 0 6 9 8 9 2 n4= 1.95414272
nfm 2 . 1 9 9 9 5 7 5 5
nf= 2.0000762
nf= 2.19999527
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OPTICS COMMUNICATIONS
of the dispersion equation (10) is obtained for Fermi-type RIP [9 ],
n2(x)=n2+(n2-n2)/{l+exp[(x-h)/u]),
(9)
where h, u, n 2 are parameters o f the RIP and nf is the SRI of the waveguide. q,, h - arctan ( rm /q,~ ) - arctan (p,,/q,~ ) - S,, = n m ,
(lO) qm=k(n~-n2) I/2, pm=k(n2-n~) '/2, rm= k ( n 2 - n ~ ) '/2, and
where
S~ = ~ { arctan(2uq,Ji) - 2 arctan[uqJ(r~u+i) ] ) .
the SRI from the experimental mode values, which facilitates the calculation o f the SRI value and the other characteristics of the waveguide.
Acknowledgements I wish to thank J. Gouzerh, A.A. Stashkevitch, E.A. Arutunian and S.Kh. Galoyan for valuable discussions and encouragement.
References ( 11 )
In eq. (10), qm is an odd function o f m and for the value m = - 1 eq. ( 10 ) is satisfied only if q_ ~ is assumed to be 0. Therefore the formulas (6), (7) and (8) are valid for them.
4. Conclusion In conclusion it must be noted, that the present method o f evaluation of the planar optical waveguide SRI, allows finding the direct dependence o f
336
1 August 1992
[ 1] P.T. Tien and R. Ulrich, J. Opt. Soe. Am. 60 (1970) 1325. [ 2 ] M.J. Adams, An introduction to optical waveguides (Wiley, New York, 1981). [ 3] E.A. Arutunyan, S.Kh. Galoyan, S.P. Pogossian and D.A.N. Armianskoi, S.S.R., Phizika 90 (1990) 16. [4] M.O. Vassell, J. Opt. Soc. Am. 65 (1975) 1019. [ 5 ] G. Svantesson,J. Magn. Soc. Jpn. ! 1; supplement S 1 ( 1987) 405. [6] G.H. Charter and P.C. Jaussand, J. Appl. Phys. 49 (1978) 917. [ 7 ] E.A.Arutunyan and S.Kh. Galoyan,Optics Comm. 56 (1986) 399. [8] E.A. Arutunyan, S.Kh. Galoyan and S.P. Pogassian, Izvestia Akademii Nauk Armianskoi S.S.R., Phizika 90 ( 1989) 215.