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Guided and semileaky modes in anisotropic optical waveguides of the LiNbO3 type
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
GUIDED AND SEMILEAKY MODES IN ANISOTROPIC OPTICAL WAVEGUIDES OF THE LiNbO 3 TYPE J. CTYROKY and M. CADA Institute o f Radio Engineering and Electronics, Czechoslovak Academy of Sciences, 182 51 Prague 8, Czechoslovakia
Received 7 July 1978
Light propagation in anisotropic optical waveguides formed by an isotropic or a uniaxial layer onto a uniaxial substtate is studied theoretically. Special attention is paid to semileaky modes which are lossy due to the radiation of energy into the substrate. Both their propagation constants and loss coefficients are calculated numerically for waveguide parameters close to As-S glass layer onto LiNbO3 substrate and to LiNbO3 out-diffused and in-diffused waveguides.
I. Introduction Various kinds of anisotropic optical waveguides have been widely used in integrated optics [ 1 - 7 ] . In the present paper, light propagation in planar anisotropic waveguides is studied theoretically. Three waveguide structures are analysed here: (i) an isotropic layer onto a uniaxial substrate, (ii) a uniaxial layer which differs from the uniaxial substrate only in the extraordinary index of refraction, and (iii) a uniaxial layer with both indices of refraction different from those of the substrate. The representative example of the structure (i) is the As2Sx (x > 3) glass film onto LiNbO 3 substrate. The structures (ii) and (iii) may be regarded as simple slab models of out-diffused [9], [10] and in-diffused [11], [12] LiNbO 3 waveguides, respectively. Besides the effective indices of guided modes, special attention is paid to so called semileaky modes which are lossy due to the radiation of energy into the substrate. The angular and thickness dependences of loss coefficient are calculated for typical waveguide parameters. The results obtained are compared to those published in a very recent paper [18] on the same subject which have appeared during the preparation of this communication. The consequence of lossy character of the waveguides is briefly discussed from the point
x! 2 Y
Fig. 1. Anisotropic slab waveguide configuration. 0 - uniaxial substrate, noo, hoe (< noo). 1 - (i) isottopie layer, n I (> hoe), (ii) uniaxial layer, n 1o (= noo), n 1e (> noe), (iii) uniaxial layer, nto (> n0o), n 1e (> hoe). 2 - superstrate (air), nz. of view of device applications.
2. Theory The waveguide configuration with the coordinate system is shown in fig. 1. The optic axes of both substrate and waveguiding layer are supposed to be parallel with the coordinate z-axis. To find the exact dispersion equation of the structure, the solutions of Maxwell equations in the form of a superposition of both TE and TM, or both ordin353
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
ary and extraordinary plane waves for each region 0, 1, and 2 are to be matched at both interfaces according to the boundary conditions. Then, the dispersion equation can be expressed in the general form
y
nl, ROe
det D(N) = O,
(1)
where D(N) is the 8 × 8 matrix function of the mode effective index N [13]. For arbitrary direction of propagation, the explicit form of (1) has been found. It follows from the structure symmetry that pure TE and/or TM modes can propagate independently without loss in the direction along and perpendicularly to the optic axis. In other directions of propagation, both TE and TM, or both ordinary and extraordinary waves, contribute to the total mode field. As far as the effective index of any mode (as a real solution of (1)) is higher than both indices n0o and noe of the substrate, the mode propagates without loss. If, however, the effective index were to be lower than n0o, the total reflection at the substrate boundary would be violated. In that case, the mode effective index becomes complex reflecting so the mode field attenuation due to the radiation of the ordinary wave into the substrate. Such modes seem to have been reported for the first time in [18] where they are referred to as "surface leaky waves". The term "semileaky mode" is used here to stress the fact that only a part of the mode field - the ordinary wave in this case - is responsible for the leakage of energy. It is an anisotropic generalization of the leaky wave [14] very similar to that of the prism or grating coupler [15]. On the basis of this physical consideration, the choice of the sheet of the Riemann surface of the complex function det D(N), on which the real or complex zeroes are to be found, may be made using the same arguments as in the case of anisotropic structures [14].
0
noo n~ z
0
%0 n~ z
Fig. 2. Dispersion curves of the waveguide (i) for thinner (a) and thicker (b) layer. Solid line - guided mode, dashed line semileaky mode. o - TE mode, • - TM mode, 0 T - semileaky-to-guided mode transition angle, 0 C - semileaky "cutofF' angle.
y
h
nle ~--z'z--.. no~, - "--z-..
0
n~,~,
Fig. 3. Dispersion curves of the three-mode waveguide (ii) (symbols as in fig. 2).
n°e
3. Results
/ / OeT1/Oc / noo nlo z OTO ey2 Fig. 4. Dispersion cruves of the waveguide (iii) (symbols as in fig. 2).
The dispersion equation (1) has been solved numerically for all three structures (i), (ii), and (iii) with the waveguide parameters being chosen close to the chalcogenide As-S glass f'drn onto a Y-cut LiNbO 3 , the out-diffused Y-cut LiNbO 3 and the in-diffused Y-cut LiNbO 3 waveguides,:respectively. The typical mode dispersion curves obtained are shown schematically
(not to scale) in the normalized wave vector diagrams in figs. 2, 3, and 4. It is seen from these figures that the mode dispersion curves do not intersect each other, although the guided-mode index difference can be nonmeasurably small. It is in accordance with theoretical results in [18] and with experimental ones in [16].
354
Volume 27, n u m b e r 3
OPTICS COMMUNICATIONS
December 1978
400 120
{X
[dB~]
e(
[dB/cm]
/"xx\
T
300
\
I 100 \
20(2
8O
\
'\
60
100 \\
, ocl t
30
60 - e [degI
90 20
Fig. 5. The loss coefficient o f the semileaky m o d e versus the angle o f propagation in the structure (i); n 0 o = 2.2885, n e e = 2.2014; solid line - d = 0 . 1 1 / a n , n I = 2.60 (fig. 2a) dashed line - d = 1/~m, n l = 2.3135 (fig. 2b).
As the propagation direction changes from 0 = 0 ° to 0 = 90 ° with respect to the optic axis, some TE modes of the structures (i) and (iii) turn to TM modes and vice versa, depending on the particular mode spectra in 0 ° and 90 ° directions. The semileaky mode can either turn into the guided mode (figs. 2a and 4) as discussed in [18] or disappear at its "cut-off" (figs. 2b, 3, and 4). Although general properties of all three structures are quite similar, there are some differences worth noting. Thus, in the structure (i), at most one mode can turn from semileaky to guided mode. In the structure (ii), only semileaky modes can propagate in the general
oc 6 [dS/c~
i
0 er~eTO
30
60
9O = 8 [deg]
Fig. 7. T h e loss coefficient o f the semileaky m o d e s versus the angle o f propagation in the structure (iii) n e e = 2.2967, n0e = 2.2082, h i e = 2.3027, h i e = 2 . 2 1 3 2 , d = 5 ~ n ; dashed line - f u n d a m e n t a l (zeroth) m o d e , solid line - first mode.
direction since there is no guiding mechanism for the ordinary wave. There are different "cut-off" angles for each semileaky mode. The greater is the real part of the
[dB/cm] l-
I,ool 2OO
100 2
,
OcO
=
30 8cl
60
90 ,. e [de~3
Fig. 6. The loss coefficient o f three semfleaky m o d e s versus the angle o f propagation in the structure (fi) n o o = n 1o = 2.2967, n e e = 2.2082, h i e = 2.2140, d = 5/~m.
0
0.8
1J)
1.2
1.4
1.6
Fig. 8. T h e thickness dependence o f t h e f u n d a m e n t a l semileaky m o d e in the waveguide (i): n l = 2.3135, the substrate parameters as in fig. 5, the direction o f propagation 45 ° from the optic axis.
355
Volume 27, number 3
OPTICS COMMUNICATIONS
60
50
[dB~/cm]
l
30
2O 10
r
/
\ I
I
3
I
I
5
I
I
7
I
I
t
9 d [~um]
Fig• 9. The thickness dependence of the fundamental semileaky mode in the waveguide (iii). The slab and substrate parameters as in fig. 7, the direction of propagation 37° from • • o . the optic axis (sohd hne) and 60 (dashedhne). mode effective index the smaller is the corresponding "cut-off' angle 0 c. The calculated angular dependence of the loss coefficients is shown in figs. 5, 6, and 7 for three waveguide structures considered here. In figs. 8 and 9, the thickness dependences of the loss coefficient of semileaky modes in the structures (i) and (iii) are plotted, respectively. As it is seen from the figures, the loss coefficients range from practically zero to tens or even hundreds of dB/cm. In the vicinity of the semileaky "cut-off", similarly as near to the semileaky-to-guided mode transition angles, a rapid change of the loss coefficient takes place. Note the zero values of the loss coefficient at 0 = 54 ° in fig. 5, in which particular case the leaky ordinary wave contribution to the total mode field vanishes. For the structures (ii) and Off), comparatively low loss coefficients have been obtained in fairly large angular sectors (figs. 6 and 7).
4. Discussion
The characteristic feature of the anisotropic waveguides of the LiNbO 3 type which have been analysed here is the existence of lossy semileaky modes. Very good agreement of the results obtained for the structure (iii) with those in [18] (compare our figs. 4 and 7 with figs. 11 and 9 in [18]) leads to the conclusion 356
December 1978
that even the much simpler calculations performed are applicable also for the actual graded-index LiNbO 3 waveguides not only for the understanding and the qualitative description of waveguide dispersion properties but also for the estimation of the magnitude of the semileaky mode losses. The reason for this stems from the fact that the dominant loss mechanism is the extraordinary (guided) to ordinary (leaky) wave conversion at the LiNbO3-air interface, which is in fact the same for both slab and graded-index waveguides. It should be noted that the modes propagating in the direction nearly perpendicular to the optic axis in LiNbO 3 waveguide which are of the greatest importance for device applications due to their strong electrooptic and acoustooptic effects are of the semileaky type. Note also that the semileaky mode character is retained in the stripe waveguide geometry, too. To avoid undesirable loss, the light propagation direction used in these devices should not deviate too much from the direction perpendicular to the optic axis. On the other hand, large loss coefficients (30-300 dB/cm) might be used for the light coupling into or out of the waveguide through the birefringent substrate, the coupling mechanism being similar to that of [17]. The coupling length might be adjusted by changing the propagation direction or the waveguide thickness.
References
[ 1] I.P Kaminow, IEEE Trans• MicrowaveTheory Tech. MTT-23 (1975) 57. [2] R.V. Schmidt and H. Kogelnik, Appl. Phys. Lett. 28 (1976) 503. [3] E.G.H. Lean, J.M. White and C.D.W. Wilkinson, Proc• IEEE 64 (1976) 779. [4] K.W. Loh, W.S.C. Chang and R.A. Becker, Appl. Phys. Lett. 28 (1976) 109. [5] T.P. Sosnowski, Opt. Commun. 4 (1972) 408. [6] T.P. Sosnowski and H.P. Weber, Opt. Commun. 7 (1973) 42. [7] Y. Okamura, S. Yamamoto and T. Makimoto, App1. Phys. Lett. 32 (1978) 161. [8] Y. Okamura, S. Yamamoto and T. Makimoto, J. Opt. Soc. Amer. 67 (1977) 539. [9] R.V. Schmidt, I.P. Kaminow and J.R. Carruthers, Appl. Phys. Lett. 23 (1973) 417. [10] J. Noda, N. Uchida and T. Saku, Appl. Phys. Lett. 25 (1974) 131. [ 11 ] R.V. Schmidt and I.P. Kaminow, Appl. Phys. Lett. 25 (1974) 458.
Volume 27, number 3
OPTICS COMMUNICATIONS
[12] J. Noda, N. Uchida, S. Saito, T. Saku and M. Minakata, Appl. Phys. Lett. 27 (1975) 19. [13] M.S. Kharusi, J. Opt. Soc. Amer. 64 (1974) 27. [14] R.E. Collin and F.J. Zucker, Antenna Theory, Part II (McGraw-Hill, New York, 1969). [15] T. Tamir, in: Integrated optics (Springer, Berlin 1975).
December 1978
[ 16] V.V. Lemanov and B.V. Sucharev, Fizika tverdogo tela 18 (1976) 3553. [17] I.J. Kurland and H.L. Bertoni, Appl. Opt. 17 (1978) 1030. [18] K. Yamanouchi, T. Kamiya and K. Shibayama, IEEE Trans. Microwave Theory Tech. MTF-26, (1978) 298.
357
OPTICS COMMUNICATIONS
December 1978
GUIDED AND SEMILEAKY MODES IN ANISOTROPIC OPTICAL WAVEGUIDES OF THE LiNbO 3 TYPE J. CTYROKY and M. CADA Institute o f Radio Engineering and Electronics, Czechoslovak Academy of Sciences, 182 51 Prague 8, Czechoslovakia
Received 7 July 1978
Light propagation in anisotropic optical waveguides formed by an isotropic or a uniaxial layer onto a uniaxial substtate is studied theoretically. Special attention is paid to semileaky modes which are lossy due to the radiation of energy into the substrate. Both their propagation constants and loss coefficients are calculated numerically for waveguide parameters close to As-S glass layer onto LiNbO3 substrate and to LiNbO3 out-diffused and in-diffused waveguides.
I. Introduction Various kinds of anisotropic optical waveguides have been widely used in integrated optics [ 1 - 7 ] . In the present paper, light propagation in planar anisotropic waveguides is studied theoretically. Three waveguide structures are analysed here: (i) an isotropic layer onto a uniaxial substrate, (ii) a uniaxial layer which differs from the uniaxial substrate only in the extraordinary index of refraction, and (iii) a uniaxial layer with both indices of refraction different from those of the substrate. The representative example of the structure (i) is the As2Sx (x > 3) glass film onto LiNbO 3 substrate. The structures (ii) and (iii) may be regarded as simple slab models of out-diffused [9], [10] and in-diffused [11], [12] LiNbO 3 waveguides, respectively. Besides the effective indices of guided modes, special attention is paid to so called semileaky modes which are lossy due to the radiation of energy into the substrate. The angular and thickness dependences of loss coefficient are calculated for typical waveguide parameters. The results obtained are compared to those published in a very recent paper [18] on the same subject which have appeared during the preparation of this communication. The consequence of lossy character of the waveguides is briefly discussed from the point
x! 2 Y
Fig. 1. Anisotropic slab waveguide configuration. 0 - uniaxial substrate, noo, hoe (< noo). 1 - (i) isottopie layer, n I (> hoe), (ii) uniaxial layer, n 1o (= noo), n 1e (> noe), (iii) uniaxial layer, nto (> n0o), n 1e (> hoe). 2 - superstrate (air), nz. of view of device applications.
2. Theory The waveguide configuration with the coordinate system is shown in fig. 1. The optic axes of both substrate and waveguiding layer are supposed to be parallel with the coordinate z-axis. To find the exact dispersion equation of the structure, the solutions of Maxwell equations in the form of a superposition of both TE and TM, or both ordin353
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
ary and extraordinary plane waves for each region 0, 1, and 2 are to be matched at both interfaces according to the boundary conditions. Then, the dispersion equation can be expressed in the general form
y
nl, ROe
det D(N) = O,
(1)
where D(N) is the 8 × 8 matrix function of the mode effective index N [13]. For arbitrary direction of propagation, the explicit form of (1) has been found. It follows from the structure symmetry that pure TE and/or TM modes can propagate independently without loss in the direction along and perpendicularly to the optic axis. In other directions of propagation, both TE and TM, or both ordinary and extraordinary waves, contribute to the total mode field. As far as the effective index of any mode (as a real solution of (1)) is higher than both indices n0o and noe of the substrate, the mode propagates without loss. If, however, the effective index were to be lower than n0o, the total reflection at the substrate boundary would be violated. In that case, the mode effective index becomes complex reflecting so the mode field attenuation due to the radiation of the ordinary wave into the substrate. Such modes seem to have been reported for the first time in [18] where they are referred to as "surface leaky waves". The term "semileaky mode" is used here to stress the fact that only a part of the mode field - the ordinary wave in this case - is responsible for the leakage of energy. It is an anisotropic generalization of the leaky wave [14] very similar to that of the prism or grating coupler [15]. On the basis of this physical consideration, the choice of the sheet of the Riemann surface of the complex function det D(N), on which the real or complex zeroes are to be found, may be made using the same arguments as in the case of anisotropic structures [14].
0
noo n~ z
0
%0 n~ z
Fig. 2. Dispersion curves of the waveguide (i) for thinner (a) and thicker (b) layer. Solid line - guided mode, dashed line semileaky mode. o - TE mode, • - TM mode, 0 T - semileaky-to-guided mode transition angle, 0 C - semileaky "cutofF' angle.
y
h
nle ~--z'z--.. no~, - "--z-..
0
n~,~,
Fig. 3. Dispersion curves of the three-mode waveguide (ii) (symbols as in fig. 2).
n°e
3. Results
/ / OeT1/Oc / noo nlo z OTO ey2 Fig. 4. Dispersion cruves of the waveguide (iii) (symbols as in fig. 2).
The dispersion equation (1) has been solved numerically for all three structures (i), (ii), and (iii) with the waveguide parameters being chosen close to the chalcogenide As-S glass f'drn onto a Y-cut LiNbO 3 , the out-diffused Y-cut LiNbO 3 and the in-diffused Y-cut LiNbO 3 waveguides,:respectively. The typical mode dispersion curves obtained are shown schematically
(not to scale) in the normalized wave vector diagrams in figs. 2, 3, and 4. It is seen from these figures that the mode dispersion curves do not intersect each other, although the guided-mode index difference can be nonmeasurably small. It is in accordance with theoretical results in [18] and with experimental ones in [16].
354
Volume 27, n u m b e r 3
OPTICS COMMUNICATIONS
December 1978
400 120
{X
[dB~]
e(
[dB/cm]
/"xx\
T
300
\
I 100 \
20(2
8O
\
'\
60
100 \\
, ocl t
30
60 - e [degI
90 20
Fig. 5. The loss coefficient o f the semileaky m o d e versus the angle o f propagation in the structure (i); n 0 o = 2.2885, n e e = 2.2014; solid line - d = 0 . 1 1 / a n , n I = 2.60 (fig. 2a) dashed line - d = 1/~m, n l = 2.3135 (fig. 2b).
As the propagation direction changes from 0 = 0 ° to 0 = 90 ° with respect to the optic axis, some TE modes of the structures (i) and (iii) turn to TM modes and vice versa, depending on the particular mode spectra in 0 ° and 90 ° directions. The semileaky mode can either turn into the guided mode (figs. 2a and 4) as discussed in [18] or disappear at its "cut-off" (figs. 2b, 3, and 4). Although general properties of all three structures are quite similar, there are some differences worth noting. Thus, in the structure (i), at most one mode can turn from semileaky to guided mode. In the structure (ii), only semileaky modes can propagate in the general
oc 6 [dS/c~
i
0 er~eTO
30
60
9O = 8 [deg]
Fig. 7. T h e loss coefficient o f the semileaky m o d e s versus the angle o f propagation in the structure (iii) n e e = 2.2967, n0e = 2.2082, h i e = 2.3027, h i e = 2 . 2 1 3 2 , d = 5 ~ n ; dashed line - f u n d a m e n t a l (zeroth) m o d e , solid line - first mode.
direction since there is no guiding mechanism for the ordinary wave. There are different "cut-off" angles for each semileaky mode. The greater is the real part of the
[dB/cm] l-
I,ool 2OO
100 2
,
OcO
=
30 8cl
60
90 ,. e [de~3
Fig. 6. The loss coefficient o f three semfleaky m o d e s versus the angle o f propagation in the structure (fi) n o o = n 1o = 2.2967, n e e = 2.2082, h i e = 2.2140, d = 5/~m.
0
0.8
1J)
1.2
1.4
1.6
Fig. 8. T h e thickness dependence o f t h e f u n d a m e n t a l semileaky m o d e in the waveguide (i): n l = 2.3135, the substrate parameters as in fig. 5, the direction o f propagation 45 ° from the optic axis.
355
Volume 27, number 3
OPTICS COMMUNICATIONS
60
50
[dB~/cm]
l
30
2O 10
r
/
\ I
I
3
I
I
5
I
I
7
I
I
t
9 d [~um]
Fig• 9. The thickness dependence of the fundamental semileaky mode in the waveguide (iii). The slab and substrate parameters as in fig. 7, the direction of propagation 37° from • • o . the optic axis (sohd hne) and 60 (dashedhne). mode effective index the smaller is the corresponding "cut-off' angle 0 c. The calculated angular dependence of the loss coefficients is shown in figs. 5, 6, and 7 for three waveguide structures considered here. In figs. 8 and 9, the thickness dependences of the loss coefficient of semileaky modes in the structures (i) and (iii) are plotted, respectively. As it is seen from the figures, the loss coefficients range from practically zero to tens or even hundreds of dB/cm. In the vicinity of the semileaky "cut-off", similarly as near to the semileaky-to-guided mode transition angles, a rapid change of the loss coefficient takes place. Note the zero values of the loss coefficient at 0 = 54 ° in fig. 5, in which particular case the leaky ordinary wave contribution to the total mode field vanishes. For the structures (ii) and Off), comparatively low loss coefficients have been obtained in fairly large angular sectors (figs. 6 and 7).
4. Discussion
The characteristic feature of the anisotropic waveguides of the LiNbO 3 type which have been analysed here is the existence of lossy semileaky modes. Very good agreement of the results obtained for the structure (iii) with those in [18] (compare our figs. 4 and 7 with figs. 11 and 9 in [18]) leads to the conclusion 356
December 1978
that even the much simpler calculations performed are applicable also for the actual graded-index LiNbO 3 waveguides not only for the understanding and the qualitative description of waveguide dispersion properties but also for the estimation of the magnitude of the semileaky mode losses. The reason for this stems from the fact that the dominant loss mechanism is the extraordinary (guided) to ordinary (leaky) wave conversion at the LiNbO3-air interface, which is in fact the same for both slab and graded-index waveguides. It should be noted that the modes propagating in the direction nearly perpendicular to the optic axis in LiNbO 3 waveguide which are of the greatest importance for device applications due to their strong electrooptic and acoustooptic effects are of the semileaky type. Note also that the semileaky mode character is retained in the stripe waveguide geometry, too. To avoid undesirable loss, the light propagation direction used in these devices should not deviate too much from the direction perpendicular to the optic axis. On the other hand, large loss coefficients (30-300 dB/cm) might be used for the light coupling into or out of the waveguide through the birefringent substrate, the coupling mechanism being similar to that of [17]. The coupling length might be adjusted by changing the propagation direction or the waveguide thickness.
References
[ 1] I.P Kaminow, IEEE Trans• MicrowaveTheory Tech. MTT-23 (1975) 57. [2] R.V. Schmidt and H. Kogelnik, Appl. Phys. Lett. 28 (1976) 503. [3] E.G.H. Lean, J.M. White and C.D.W. Wilkinson, Proc• IEEE 64 (1976) 779. [4] K.W. Loh, W.S.C. Chang and R.A. Becker, Appl. Phys. Lett. 28 (1976) 109. [5] T.P. Sosnowski, Opt. Commun. 4 (1972) 408. [6] T.P. Sosnowski and H.P. Weber, Opt. Commun. 7 (1973) 42. [7] Y. Okamura, S. Yamamoto and T. Makimoto, App1. Phys. Lett. 32 (1978) 161. [8] Y. Okamura, S. Yamamoto and T. Makimoto, J. Opt. Soc. Amer. 67 (1977) 539. [9] R.V. Schmidt, I.P. Kaminow and J.R. Carruthers, Appl. Phys. Lett. 23 (1973) 417. [10] J. Noda, N. Uchida and T. Saku, Appl. Phys. Lett. 25 (1974) 131. [ 11 ] R.V. Schmidt and I.P. Kaminow, Appl. Phys. Lett. 25 (1974) 458.
Volume 27, number 3
OPTICS COMMUNICATIONS
[12] J. Noda, N. Uchida, S. Saito, T. Saku and M. Minakata, Appl. Phys. Lett. 27 (1975) 19. [13] M.S. Kharusi, J. Opt. Soc. Amer. 64 (1974) 27. [14] R.E. Collin and F.J. Zucker, Antenna Theory, Part II (McGraw-Hill, New York, 1969). [15] T. Tamir, in: Integrated optics (Springer, Berlin 1975).
December 1978
[ 16] V.V. Lemanov and B.V. Sucharev, Fizika tverdogo tela 18 (1976) 3553. [17] I.J. Kurland and H.L. Bertoni, Appl. Opt. 17 (1978) 1030. [18] K. Yamanouchi, T. Kamiya and K. Shibayama, IEEE Trans. Microwave Theory Tech. MTF-26, (1978) 298.
357