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Transmission performance of a 1 × 2 Ti:LiNbO3 strip waveguide directional coupler
15 November 1990
OPTICS COMMUNICATIONS
Volume 79, number 6
Transmission performance of a 1 x 2 Ti:LiNb03 strip waveguide directional coupler Mieczyslaw
Szustakowski
and Marian
Institute of Plasma Physics and Laser Microfusion,
Marciniak 00-908 Warsaw 49, Poland
Received 29 March 1990; revised manuscript received 4 July 1990
Optical power coupling in a 1 x 2 Ti:LiNbOs strip waveguide directional coupler, used as a power divider, has been investigated at I= 1.3 urn by means of a PBM simulation. The transfer lengths 1, and maximum achievable fundamental mode transfers P,,,, are given for a variety of waveguide separations.
1. Introduction Symmetric Y-junctions and directional couplers are widely used in integrated-optics devices for optical power splitting [ I]. An important requirement of these power dividers is to minimize the optical power losses. Recently, we have shown that in the case of a monomode Ti:LiNbOS strip waveguide Yjunction the minimum achievable fundamental mode power loss is of the order of 7 per cent or 0.32 dB at ;1= 1.3 urn for small junction angles (below 1 degree) [2]. Thus, it is of interest to investigate the losses of an alternate power divider - a 1 X 2 directional coupler, the geometry of which is shown in fig. 1. (The symmetry of this structure ensures equal power splitting between the two output arms.) In the present paper we give a brief review of coupled mode theory and of the supermode formalism as applied to the performance analysis of the coupler and we indicate the difficulties encountered when applying these formalisms to diffused Ti:LiNb03 waveguide couplers (section 2). Then we discuss the results of a numerical investigation of power splitting in a 1 x 2 Ti:LiNbO, waveguide directional coupler (section 3). We use a combination of the propagating-beam method (PBM) and the effective-index method as proposed by Danielsen [ 3 1. The conclusion are outlined in section 4. We analyse the structure shown in fig. 1, which is obtained after diffusion from three equidistantly 0030-4018/90/$03.50
W
Fig. 1. Geometry of the 1 x 2 Ti:LiNbOs waveguide directional coupler analysed throughout the paper. The positions of initial titanium strips of width W separated by a distance dare marked. a(z) is the input mode amplitude, b(z) are the output mode amplitudes.
spaced, identical distance between
titanium strips of width the strips is d.
W. The
2. Theory of the 1 X 2 coupler Two theoretical approaches to the problem, the coupled-mode theory and the supermode formalism, will be discussed in this section and the limits of their application will be pointed out.
0 1990 - Elsevier Science Publishers B.V. (North-Holland
)
411
Volume 79, number
6
OPTICS
COMMUNICATIONS
2.1. Coupled-mode theory Let the coupling between the outer guides of the coupler shown in fig. 1 be insignificant, hence the fundamental mode amplitudes a(z) and b(z) are governed by coupled equations [ 4 ] da/dz+$,a+%ilK(b=O,
P,=pb,
(2)
the solution to ( 1), which satisfies both the power conservation condition and the condition of exciting only the input guide with its normalized fundamental mode, is
b(z) = ( -i/G)
,
sin (K’z) exp( -i/?z)
,
(3)
where K’= lKI&’ and/3=pa=&. From (2) it follows that the optical power is periodically exchanged between the input guide and the output guides. The total transfer length, I,, satisfies K’l,=n/2.
(4)
However, complete power transfer to the outer waveguides is possible only when (2) is satisfied [ 5 1. According to (3 ), it is of great importance to know the coupling coefficient .K’ (or K) when designing the proper length of the coupling region of the power divider. Unfortunately, it is difficult to evaluate K from analytic expressions [ $61, hence the authors prefer to use approximate expressions [ 71 or to determine the coupling coefficient experimentally [ 8,9 1. In addition, the above theory is hardly applicable to the Ti:LiNb03 coupler devices because, in practical cases, the waveguide spacings are two small (a few micrometers only) and the assumption of weak coupling is not justified. Moreover, the titanium concentration in the inner guide differs from 412
those in the outer guides because of the lateral Ti diffusion, unless the waveguide separations are sufficiently wide; thus the modal fields and propagation constants may differ as well. In consequence, eq. (2 ) is not satisfied and complete power transfer to the output waveguides is impossible.
(1)
where /Ia and /I,, are the modal propagation constants in the input and output guides, respectively, K is the coupling coefficient between two neighbouring guides and the multiplication by a factor of 2 in the first equation appears because the input field is coupled to two output guides. In eq. (1 ), field propagation along the z-axis of the form exp( -$z) is assumed. Assuming weak coupling and
exp( -$z)
1990
2.2 Supermode formalism
db/dz+iWb+i(Kla=O,
a(z) =cos(K’z)
15 November
This approach is based on the concept of supermodes, i.e. eigenmodes of the whole three-guide or seven-layer structure, which was previously considered in the context of a two-guide coupler [ IO,1 I]. Because of the symmetry of the coupler only the symmetric supermodes have to be considered; they are shown in fig. 2. Using the effective-index approximation [ 121, the propagation constants and field distributions of the supermodes can be determined in a manner similar to the case of an isolated inhomogeneous planar waveguide; the multilayer approximation of the inhomogenous guide [ 3 ] has been shown to be very precise [ 13 1. If the waveguide separation d is large, the incoming fundamental mode of the input guide is a sum of the two supermodes with equal amplitudes, as fig. 2 shows. Power transfer between the waveguides can now be described as an interference phenomenon with a spatial period (or beat period) & satisfying (Po-P*V,=2n.
(5)
Hence the total coupling length 1, of the coupler is equal to half the beat period between the two supet-modes [ 10 ] : ~,=&/2=~/uk-/32)~
(6)
The above procedure, simple in principle, becomes more complicated when the waveguides are placed close to each other and the modal field overlap is significant, as is usually the case [ 9 1. Now, the supermodes differ substantially from the modal fields of isolated guides. Consequently, guided and radiation supermodes have to be taken into consideration when expanding the incoming fundamental mode field in the input guide, what complicates the analysis considerably.
Volume 79. number
6
OPTICS
I5 November
COMMUNICATIONS
1990
lithium niobate indices, caused by the Ti atoms concentration c, are [ 15 ] tin, = EC )
&z,=(FC)Y,
(7)
where,!?= 1.2~ 1O-23 cm3,f=0.13x 1O-24 cm3, and ~~0.55 at A=633 nm. When the LiNb03 crystallographic axes coincide with the coordinate system (fig. 1 ), as it is in the practical cases, then solving the 2D anisotropic diffusion equation leads to the following expression for c [2,10,16]: c=co(&)
l-4,--4 (bl Fig. 2. (a) Supermodes of the coupling region, when the waveguides are designed to be monomode. The first order mode is not shown since it is of no interest in a symmetric structure. (b) The sum of the two supermodes at the beginning of the coupling region and at the total coupling length I,= n/ (/?0-~2). Note, that the power conservation requires that the mode amplitude in the central guide is greater by a factor of $ than that of any of the outer guides.
3. Numerical results In the previous section we have pointed out the difficulties of two theoretical formalisms when analyzing Ti:LiNb03 couplers. Those difficulties do not occur with the PBM, which is a totally nonmodal method [ 111. Hence, this approach is well suited to couplers with small waveguide separations d, when the above cited theoretical formalisms are hardly applicable. The PBM has been widely discussed throughout the literature [ 3,11,14], hence we restricted ourselves to give in this section a short description of the refractive-index distribution in the diffused Ti:LiNbO, coupler, and to report the results of an investigation of light propagation through the coupler, obtained by means of the PBM combined with the effective-index method [ 3 1. 3. I. Refractive-index distribution It has been found experimentally that local increases of ordinary (6n,) and extraordinary (6~)
f(YlQ)
9
s(xl&)
(8)
where td is the diffusion time, D, and Dy are the diffusion depths in the x and y directions, respectively. The constant co(&) is [ 15,161 uoph co=
=6.4x
1O22cm-3$,
ig&
(9) Y
where ao=6.022x lO*‘/mol, p is the titanium density, p is its molar mass, h is the initial Ti strip thickness. The depth profile has a gaussian shape in practical cases [lo]: KY, td) =exp( -y*lD:)
.
(10)
The lateral profile g(x) is a generalization of the results for two-guide system [ 2,10,16 ] to three-guide system: g(x, t),=0.5[erC1-erfB -erfC+erfD+erfE-erfF]
,
(11)
with a=3 Wf2+d,
b= W/2+d,
c= W/2,
A= (a-x)/D,, B= (-A-x)/D,, C= (b-x)/D,, D= (-b-x)/D,, E= (c-x)/D,, F= (-c-x)/D,, a, b and c determine the Ti strip boundaries - see fig. 1. The Ti concentration distributions obtained from eqs. (S)- ( 11) for several titanium strip separation values are shown in fig. 3. Note the difference of the concentration distribution within an isolated waveguide and in a system of well-coupled closely-spaced waveguides. When distribution of the Ti concentration is determined, the refractive-index distribution in the 413
Volume 79, number 6
OPTICS COMMUNICATIONS
0
15 November 1990
0.8
1.6
2.4
3.2
llmm
.r
Fig. 3. Distribution of the titanium concentration in the threeguide Ti:LiNbOS coupling area for several values of the waveguide separation d. The lines correspond to the concentration values: 0.1 co, 0.5 co, 0.9 c,,, increasing in the direction of the initial Ti strip (dashed area). Due to the symmetry only half of the structure is shown. The width of the Ti strip is W= 10 pm, the diffusion depths are D,=D,,= 3 pm.
coupler can be obtained
Fig. 4. The coupling characteristics of a 1 x 2 Ti:LiNbO, directional coupler at fundamental Te mode (carrying a power of unity) excitation of the input arm. X-cut Y-propagating waveguides are assumed, titanium strip width is W= 10 urn, the diffusion depths are 0,=0,=3 urn, the bulk refractive-index is n,=2.145, Anc0.01, the wavelength is 1.3 pm. P is the fundamental mode power in each of the output arms. The numbers l9 correspond to different d values given in table 1. Note, that for complete power transfer to the output arms, P would be 0.5.
with the use of expressions
(7). 3.2. Coupling characteristics An excitation of the input guide by its fundamental mode carrying a power of unity has been assumed. (Note that this mode is not an eigenmode of the three-guide or seven-layer coupler structure). Then the coupling effkiency P (defined as the fundamental mode power in each output arm) has been calculated along the propagation direction for different values of waveguide separation d. The results are shown in fig. 4 and reported in table 1. Note, that
for large separations the maximum efficiency, P,,,,,, increases (because the supermodes of the seven-layer structure become close to the mode of an isolated waveguide), but the coupling length increases considerably too.
4. Conclusions We have given analytic expressions for the refractive-index distribution in a 1 x2 Ti:LiNbO, directional coupler; the results indicate that the index distribution within the guides is perturbed by the
Table 1 The data from fig. 4: total coupling length l,, maximum coupling efficiency P,,,, and corresponding power losses for different values of waveguide separation d. The numbers l-9 correspond to those in fig. 4. No
1 2 3 4 5 6 7 8 9
414
4
km
1.oo 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
L, mm
1.4 1.5 1.7 2.1 2.4 2.75 3.2 3.55 4.0
Pmax
0.44 0.45 0.45 0.46 0.47 0.47 0.48 0.48 0.48
losses l -2P,,
dB
0.12 0.10 0.10 0.08 0.06 0.06 0.04 0.04 0.04
0.45 0.42 0.42 0.38 0.31 0.3 1 0.23 0.23 0.23
Volume 79, number 6
OPTICS COMMUNICATIONS
presence of the neighbouring guides in the practical range of waveguide separations. In this context the theoretical methods reported in the paper: coupled mode theory and supermode formalism are hardly applicable to the Ti:LiNb03 devices. As an alternative we have applied the PBM method to determine the dependence of the total coupling length 1, and of the maximum obtainable power transfer Pm, on the waveguide separation d.
References [ 1] E. Voges and A. Neyer, J. Lightwave Technology, LT-5 (1987) 1229.
[ 2 ] M. Szustakowski and M. Marciniak, Proc. SPIE, Vol. 1085 (1989). [3] P. Danielsen, IEEE J. Quantum Electron. QE-20 (1984) 1093.
IS November 1990
[4] W.K. Bums and A.F. Milton, IEEE Transactions on Microwave Theory and Techniques, MTT-30 ( 1982) 1778. [ 5] A. Yariv, IEEE Quantum Electronics, QE-9 ( 1973) 919. [6] U. Jam, A. Sharma, K. Thyagarajan and A.K. Ghatak, J. Opt. Sot. Am. 72 (1982) 1545. [ 71 R.G. Hunsperger, Integrated optics: theory and technology, Second Edition (Springer, Berlin, 1984) [ 8 ] R.C. Alfemess, R.V. Schmidt and EM. Turner, Appl. Optics (1979) 4012. [9] R.R.A. Syms and R.G. Peall, Optics Comm. 74 ( 1989) 46. [lo] J. Ctyroky, M. Hoffman, J. Janta and J. Schrofel, IEEE J. Quantum Electron. QE-20 ( 1984) 400. [ 111 Z. Weissman, A. Hardy and E. Marom, IEEE J. Quantum Electron. 25 (1989) 1200. 121 G.B. Hacker and W.K. Bums, Appl. Optics (1977) 113. 13 ] M. Marciniak, Ph.D. Thesis, Technical Academy of Warsaw, WAT, Warsaw 1989, p. 139. 141 L. Thyltn, Opt. Quantum Electron. 15 ( 1983) 433. 15 ] G.P. Bava, 1. Montrosset, W. Sohler and H. Suche, IEEE J. Quantum Electron. QE-23 (1987) 42. [ 161 M. Szustakowski and M. Marciniak, Optica Applicata, XIX (1989) 349.
415
OPTICS COMMUNICATIONS
Volume 79, number 6
Transmission performance of a 1 x 2 Ti:LiNb03 strip waveguide directional coupler Mieczyslaw
Szustakowski
and Marian
Institute of Plasma Physics and Laser Microfusion,
Marciniak 00-908 Warsaw 49, Poland
Received 29 March 1990; revised manuscript received 4 July 1990
Optical power coupling in a 1 x 2 Ti:LiNbOs strip waveguide directional coupler, used as a power divider, has been investigated at I= 1.3 urn by means of a PBM simulation. The transfer lengths 1, and maximum achievable fundamental mode transfers P,,,, are given for a variety of waveguide separations.
1. Introduction Symmetric Y-junctions and directional couplers are widely used in integrated-optics devices for optical power splitting [ I]. An important requirement of these power dividers is to minimize the optical power losses. Recently, we have shown that in the case of a monomode Ti:LiNbOS strip waveguide Yjunction the minimum achievable fundamental mode power loss is of the order of 7 per cent or 0.32 dB at ;1= 1.3 urn for small junction angles (below 1 degree) [2]. Thus, it is of interest to investigate the losses of an alternate power divider - a 1 X 2 directional coupler, the geometry of which is shown in fig. 1. (The symmetry of this structure ensures equal power splitting between the two output arms.) In the present paper we give a brief review of coupled mode theory and of the supermode formalism as applied to the performance analysis of the coupler and we indicate the difficulties encountered when applying these formalisms to diffused Ti:LiNb03 waveguide couplers (section 2). Then we discuss the results of a numerical investigation of power splitting in a 1 x 2 Ti:LiNbO, waveguide directional coupler (section 3). We use a combination of the propagating-beam method (PBM) and the effective-index method as proposed by Danielsen [ 3 1. The conclusion are outlined in section 4. We analyse the structure shown in fig. 1, which is obtained after diffusion from three equidistantly 0030-4018/90/$03.50
W
Fig. 1. Geometry of the 1 x 2 Ti:LiNbOs waveguide directional coupler analysed throughout the paper. The positions of initial titanium strips of width W separated by a distance dare marked. a(z) is the input mode amplitude, b(z) are the output mode amplitudes.
spaced, identical distance between
titanium strips of width the strips is d.
W. The
2. Theory of the 1 X 2 coupler Two theoretical approaches to the problem, the coupled-mode theory and the supermode formalism, will be discussed in this section and the limits of their application will be pointed out.
0 1990 - Elsevier Science Publishers B.V. (North-Holland
)
411
Volume 79, number
6
OPTICS
COMMUNICATIONS
2.1. Coupled-mode theory Let the coupling between the outer guides of the coupler shown in fig. 1 be insignificant, hence the fundamental mode amplitudes a(z) and b(z) are governed by coupled equations [ 4 ] da/dz+$,a+%ilK(b=O,
P,=pb,
(2)
the solution to ( 1), which satisfies both the power conservation condition and the condition of exciting only the input guide with its normalized fundamental mode, is
b(z) = ( -i/G)
,
sin (K’z) exp( -i/?z)
,
(3)
where K’= lKI&’ and/3=pa=&. From (2) it follows that the optical power is periodically exchanged between the input guide and the output guides. The total transfer length, I,, satisfies K’l,=n/2.
(4)
However, complete power transfer to the outer waveguides is possible only when (2) is satisfied [ 5 1. According to (3 ), it is of great importance to know the coupling coefficient .K’ (or K) when designing the proper length of the coupling region of the power divider. Unfortunately, it is difficult to evaluate K from analytic expressions [ $61, hence the authors prefer to use approximate expressions [ 71 or to determine the coupling coefficient experimentally [ 8,9 1. In addition, the above theory is hardly applicable to the Ti:LiNb03 coupler devices because, in practical cases, the waveguide spacings are two small (a few micrometers only) and the assumption of weak coupling is not justified. Moreover, the titanium concentration in the inner guide differs from 412
those in the outer guides because of the lateral Ti diffusion, unless the waveguide separations are sufficiently wide; thus the modal fields and propagation constants may differ as well. In consequence, eq. (2 ) is not satisfied and complete power transfer to the output waveguides is impossible.
(1)
where /Ia and /I,, are the modal propagation constants in the input and output guides, respectively, K is the coupling coefficient between two neighbouring guides and the multiplication by a factor of 2 in the first equation appears because the input field is coupled to two output guides. In eq. (1 ), field propagation along the z-axis of the form exp( -$z) is assumed. Assuming weak coupling and
exp( -$z)
1990
2.2 Supermode formalism
db/dz+iWb+i(Kla=O,
a(z) =cos(K’z)
15 November
This approach is based on the concept of supermodes, i.e. eigenmodes of the whole three-guide or seven-layer structure, which was previously considered in the context of a two-guide coupler [ IO,1 I]. Because of the symmetry of the coupler only the symmetric supermodes have to be considered; they are shown in fig. 2. Using the effective-index approximation [ 121, the propagation constants and field distributions of the supermodes can be determined in a manner similar to the case of an isolated inhomogeneous planar waveguide; the multilayer approximation of the inhomogenous guide [ 3 ] has been shown to be very precise [ 13 1. If the waveguide separation d is large, the incoming fundamental mode of the input guide is a sum of the two supermodes with equal amplitudes, as fig. 2 shows. Power transfer between the waveguides can now be described as an interference phenomenon with a spatial period (or beat period) & satisfying (Po-P*V,=2n.
(5)
Hence the total coupling length 1, of the coupler is equal to half the beat period between the two supet-modes [ 10 ] : ~,=&/2=~/uk-/32)~
(6)
The above procedure, simple in principle, becomes more complicated when the waveguides are placed close to each other and the modal field overlap is significant, as is usually the case [ 9 1. Now, the supermodes differ substantially from the modal fields of isolated guides. Consequently, guided and radiation supermodes have to be taken into consideration when expanding the incoming fundamental mode field in the input guide, what complicates the analysis considerably.
Volume 79. number
6
OPTICS
I5 November
COMMUNICATIONS
1990
lithium niobate indices, caused by the Ti atoms concentration c, are [ 15 ] tin, = EC )
&z,=(FC)Y,
(7)
where,!?= 1.2~ 1O-23 cm3,f=0.13x 1O-24 cm3, and ~~0.55 at A=633 nm. When the LiNb03 crystallographic axes coincide with the coordinate system (fig. 1 ), as it is in the practical cases, then solving the 2D anisotropic diffusion equation leads to the following expression for c [2,10,16]: c=co(&)
l-4,--4 (bl Fig. 2. (a) Supermodes of the coupling region, when the waveguides are designed to be monomode. The first order mode is not shown since it is of no interest in a symmetric structure. (b) The sum of the two supermodes at the beginning of the coupling region and at the total coupling length I,= n/ (/?0-~2). Note, that the power conservation requires that the mode amplitude in the central guide is greater by a factor of $ than that of any of the outer guides.
3. Numerical results In the previous section we have pointed out the difficulties of two theoretical formalisms when analyzing Ti:LiNb03 couplers. Those difficulties do not occur with the PBM, which is a totally nonmodal method [ 111. Hence, this approach is well suited to couplers with small waveguide separations d, when the above cited theoretical formalisms are hardly applicable. The PBM has been widely discussed throughout the literature [ 3,11,14], hence we restricted ourselves to give in this section a short description of the refractive-index distribution in the diffused Ti:LiNbO, coupler, and to report the results of an investigation of light propagation through the coupler, obtained by means of the PBM combined with the effective-index method [ 3 1. 3. I. Refractive-index distribution It has been found experimentally that local increases of ordinary (6n,) and extraordinary (6~)
f(YlQ)
9
s(xl&)
(8)
where td is the diffusion time, D, and Dy are the diffusion depths in the x and y directions, respectively. The constant co(&) is [ 15,161 uoph co=
=6.4x
1O22cm-3$,
ig&
(9) Y
where ao=6.022x lO*‘/mol, p is the titanium density, p is its molar mass, h is the initial Ti strip thickness. The depth profile has a gaussian shape in practical cases [lo]: KY, td) =exp( -y*lD:)
.
(10)
The lateral profile g(x) is a generalization of the results for two-guide system [ 2,10,16 ] to three-guide system: g(x, t),=0.5[erC1-erfB -erfC+erfD+erfE-erfF]
,
(11)
with a=3 Wf2+d,
b= W/2+d,
c= W/2,
A= (a-x)/D,, B= (-A-x)/D,, C= (b-x)/D,, D= (-b-x)/D,, E= (c-x)/D,, F= (-c-x)/D,, a, b and c determine the Ti strip boundaries - see fig. 1. The Ti concentration distributions obtained from eqs. (S)- ( 11) for several titanium strip separation values are shown in fig. 3. Note the difference of the concentration distribution within an isolated waveguide and in a system of well-coupled closely-spaced waveguides. When distribution of the Ti concentration is determined, the refractive-index distribution in the 413
Volume 79, number 6
OPTICS COMMUNICATIONS
0
15 November 1990
0.8
1.6
2.4
3.2
llmm
.r
Fig. 3. Distribution of the titanium concentration in the threeguide Ti:LiNbOS coupling area for several values of the waveguide separation d. The lines correspond to the concentration values: 0.1 co, 0.5 co, 0.9 c,,, increasing in the direction of the initial Ti strip (dashed area). Due to the symmetry only half of the structure is shown. The width of the Ti strip is W= 10 pm, the diffusion depths are D,=D,,= 3 pm.
coupler can be obtained
Fig. 4. The coupling characteristics of a 1 x 2 Ti:LiNbO, directional coupler at fundamental Te mode (carrying a power of unity) excitation of the input arm. X-cut Y-propagating waveguides are assumed, titanium strip width is W= 10 urn, the diffusion depths are 0,=0,=3 urn, the bulk refractive-index is n,=2.145, Anc0.01, the wavelength is 1.3 pm. P is the fundamental mode power in each of the output arms. The numbers l9 correspond to different d values given in table 1. Note, that for complete power transfer to the output arms, P would be 0.5.
with the use of expressions
(7). 3.2. Coupling characteristics An excitation of the input guide by its fundamental mode carrying a power of unity has been assumed. (Note that this mode is not an eigenmode of the three-guide or seven-layer coupler structure). Then the coupling effkiency P (defined as the fundamental mode power in each output arm) has been calculated along the propagation direction for different values of waveguide separation d. The results are shown in fig. 4 and reported in table 1. Note, that
for large separations the maximum efficiency, P,,,,,, increases (because the supermodes of the seven-layer structure become close to the mode of an isolated waveguide), but the coupling length increases considerably too.
4. Conclusions We have given analytic expressions for the refractive-index distribution in a 1 x2 Ti:LiNbO, directional coupler; the results indicate that the index distribution within the guides is perturbed by the
Table 1 The data from fig. 4: total coupling length l,, maximum coupling efficiency P,,,, and corresponding power losses for different values of waveguide separation d. The numbers l-9 correspond to those in fig. 4. No
1 2 3 4 5 6 7 8 9
414
4
km
1.oo 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
L, mm
1.4 1.5 1.7 2.1 2.4 2.75 3.2 3.55 4.0
Pmax
0.44 0.45 0.45 0.46 0.47 0.47 0.48 0.48 0.48
losses l -2P,,
dB
0.12 0.10 0.10 0.08 0.06 0.06 0.04 0.04 0.04
0.45 0.42 0.42 0.38 0.31 0.3 1 0.23 0.23 0.23
Volume 79, number 6
OPTICS COMMUNICATIONS
presence of the neighbouring guides in the practical range of waveguide separations. In this context the theoretical methods reported in the paper: coupled mode theory and supermode formalism are hardly applicable to the Ti:LiNb03 devices. As an alternative we have applied the PBM method to determine the dependence of the total coupling length 1, and of the maximum obtainable power transfer Pm, on the waveguide separation d.
References [ 1] E. Voges and A. Neyer, J. Lightwave Technology, LT-5 (1987) 1229.
[ 2 ] M. Szustakowski and M. Marciniak, Proc. SPIE, Vol. 1085 (1989). [3] P. Danielsen, IEEE J. Quantum Electron. QE-20 (1984) 1093.
IS November 1990
[4] W.K. Bums and A.F. Milton, IEEE Transactions on Microwave Theory and Techniques, MTT-30 ( 1982) 1778. [ 5] A. Yariv, IEEE Quantum Electronics, QE-9 ( 1973) 919. [6] U. Jam, A. Sharma, K. Thyagarajan and A.K. Ghatak, J. Opt. Sot. Am. 72 (1982) 1545. [ 71 R.G. Hunsperger, Integrated optics: theory and technology, Second Edition (Springer, Berlin, 1984) [ 8 ] R.C. Alfemess, R.V. Schmidt and EM. Turner, Appl. Optics (1979) 4012. [9] R.R.A. Syms and R.G. Peall, Optics Comm. 74 ( 1989) 46. [lo] J. Ctyroky, M. Hoffman, J. Janta and J. Schrofel, IEEE J. Quantum Electron. QE-20 ( 1984) 400. [ 111 Z. Weissman, A. Hardy and E. Marom, IEEE J. Quantum Electron. 25 (1989) 1200. 121 G.B. Hacker and W.K. Bums, Appl. Optics (1977) 113. 13 ] M. Marciniak, Ph.D. Thesis, Technical Academy of Warsaw, WAT, Warsaw 1989, p. 139. 141 L. Thyltn, Opt. Quantum Electron. 15 ( 1983) 433. 15 ] G.P. Bava, 1. Montrosset, W. Sohler and H. Suche, IEEE J. Quantum Electron. QE-23 (1987) 42. [ 161 M. Szustakowski and M. Marciniak, Optica Applicata, XIX (1989) 349.
415