Determination of a dielectric waveguide propagation ... - OSA Publishing
1026
OPTICS LETTERS / Vol. 14, No. 18 / September 15, 1989
Determination of a dielectric waveguide propagation constant using a multifilament-current model H. Cory, Z. Altman, and Y. Leviatan Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000,Israel Received February 27, 1989; accepted June 11, 1989 A moment method using a multifilament-current
model is presented
to analyze the modes propagating
in a
cylindrical dielectric waveguide. In this model, analytically derivable fields of filamentary electric and magnetic currents (of yet unknown propagation constant and amplitude) 'are used to simulate the field of each mode inside and outside the guiding core. A simple point-matching procedure is subsequently used to enforce the boundary conditions at the core periphery and results in a homogeneous matrix equation. The longitudinal propagation constant of each mode and the currents that yield the field distribution of this mode are then found by solvingthis equation. As an example, a circular dielectric waveguideis analyzed and the results are presented.
A moment method using a multifilament-current model has been applied successfully to solve timeharmonic two-dimensional electromagnetic scattering problems at normal incidence.",2 In this Letter the method is applied to find the Brillouin diagrams of dielectric waveguides of arbitrary homogeneous cross section, which is a problem of great interest in optics. In the above-mentioned scattering problems, the amplitudes of the filamentary currents are assumed constant because the incidence is normal, whereas in the fl-w determination of dielectric waveguides they are assumed to have the eijz variation of the propagating modes. Other methods have also been used to find the f3-wdiagrams, such as the modal point-matching- 5 and the finite-difference methods.6 Let us consider a lossless z-directed dielectric waveguide of uniform cross section bounded by an arbitrary curve S. The core (region 1) has parameters el and gu
while its surrounding lossless cladding (region 2) has parameters
62
and Ito. We assume that el >
62.
The
waveguide propagation takes place in the hybrid mode, which is represented as the sum of TE and TM waves that together satisfy the boundary conditions at S. For circular waveguides, TE and TM modes can also propagate independently, along with hybrid modes. The time dependence of the fields and the
Fig. 2. These filaments are called internal filaments and are used to approximate the fields in region 2. The two parts of the model are connected through the boundary conditions that require the continuity of the tangential components of the electric and magnetic fields across S as follows:
A X (E'-
EO)= 0,
A X (HI
0146-9592/89/181026-03$2.00/0
HO) = 0,
where superscripts I and 0 denote that the given quantity is related to region 1 or 2, respectively. Let i3, k8, and k,8 denote the longitudinal propagation constant, the wave number, and the radial wave number, respectively, in region s (s = 1, 2). In region 1 (inside the core) k1 2 = W2 El6M - kp12 + p2, while in region 2 (outside the core) k2 2
=
cA2e2/o = kPS2
+ ,32. It
should be noted that kp, is real and kp2 is imaginary. Since, as stated above, the fields of the propagating modes vary with z according to e-jfz, the electric and magnetic currents flowing in the filaments are chosen as follows: Ii = Ioj e-ilz,
KE= KO e
2-f
In the transverse plane, the position of the ith filament
currents is taken as ejwt.
The analysis for hybrid mode propagation has two interacting parts. In the first part, we consider an infinite homogeneous space with parameters el and go in which the mathematical curve S defines the core region 1. Two sets of filaments are placed in region 2 outside S: electric filaments 11°)and magnetic filaments fK91,as shown in Fig. 1 (i = 1, 2,.. .). These filaments are called external filaments and are used to approximate the fields in region 1. In the second part, we consider an infinite homogeneous space with parameters E2and ,uoin which an identical mathematical curve S defines the cladding region 2. Two sets of filaments are placed in region 1 inside S: electric filaments {I1 and magnetic filaments JKI1,as shown in
-
unbounded homogeneous space (el, ,o)
-rAezion)
-
//
region
1
"-filamentary
currents
{I°} +{KI} Fig. 1.
Filamentary currents producing the fields in region 1.
© 1989 Optical Society of America
September 15,1989 / Vol. 14, No. 18 / OPTICS LETTERS unbounded homogeneous
[Z]i = =
space(62-p0) -or
E0 , Ho o Pi
/0O
I
7/1-
/ I,
-
I
filameentarycurrents
,'~
{If'}+
{K[}
mathematical surface S
Fig. 2. Filamentary currents producing tthefields i region 2.
is given by pi, and the distance from any observation point p to the filament is given by R := IP - PilA
The magnetic vector potential Ai at p due to an electric filamentary current 1 iis giver l by Ai = -j A4' H( 2)(kp Rj)e JlZz
lectriveto
the while
pteta
(s = 1, 2),
(3)
'
H(2)(kpRi)eCtZ2
(s = 1, 2).
(4)
The total magnetic vector potential SN is given as the superposition of all the potentials Ai dlueto the various electric filaments situated in region 2I t=tE Ar(Io) The other total potentials, AO, FI, and F 0 , are ob tained in a similar manner. The electromagnetic fields are derived from these total potentials. The electric filaments produce the TM fielIds,the magnetic filaments produce the TE fields, ai3d the two sets jointly produce the hybrid (HYB) fie]adst We choose N electric filaments and N magnetic filathe boundary ments in both regions 1 and 2 and apy Nly conditions [Eqs. (1)] to N points on S. We require the . nnancW: continuity of all tangential field con Hz, and Ht at these N matching poi nts (the index t denotes any tangential component in tthe transverse xy plane). We obtain in this way 4.N homogeneous equations with 4N unknowns-these being the complex amplitudes of the (electric and inagnetic) external and internal currents. As an example, the continuity equation for the tangential component of the electric field at the jth matching poi.nt in the transverse plane is given by N
the values of 1 for which f(13) A det[ZI vanishes have to
be found in the range k2 >Ep and H, >>Hz and should be independent of (p. A similar reasoning applies for the TE mode. The inclusion of all the appropriate components in the definitions of A() is necessary since for some spurious one component may be continuous while another one is discontinuous. It is anticipated that the values of the A(13)'sso defined will be considerably larger for spurious solutions than for physical ones. Each of the values of the A(O)'s is to be calculated at the Mpoints on the boundary S for all the values of the
In the TM mode we do not require the continuity of the Eo and Hz components since these components do not vanish in practice, as they should, owing to nu-
all the A(1)'s that correspond to the spurious solutions, to the largest of all the A(O)'sthat correspond to the physical solutions, is 257, showing that it is not difficult to distinguish between the two kinds of solutions. The computer time required to calculate A(1) for all the solutions is 10% of the time required to calculate L(W). The 1-w diagram of a circular dielectric waveguide
is shown in Fig. 4. The intersection of the a/X = 0.5 reference line with the HE21 , TMo1, TEO1 , and HE1 1 dispersion curves corresponds to the four values of 13 found in Fig. 3. Finally, it should be noted that the values of the propagation constants obtained by the present method for circular dielectric waveguides yield a six-figure agreement with the values obtained previously by conventional methods.8 References 1. Y. Leviatan and A. Boag, IEEE Trans. Antennas Propag. AP-35, 1119 (1987). 2. Y. Leviatan, A. Boag, and A. Boag, IEEE Trans. Anten-
nas Propag. AP-36, 1026 (1988). 3. J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969). 4. J. R. James and I. N. L. Gallett, Radio Electron. Eng. 42, 103 (1972).
5. E. Yamashita, K. Atsuki, 0. Hashimoto, and K. Kamijo, IEEE Trans. Microwave Theory Tech. MTT-27, 352 (1979). 6. E. Schweig and W. B. Bridges, IEEE Trans. Microwave
Theory Tech. MTT-32, 531 (1984). 7. E. Kreyszig, Advanced Engineering Mathematics, 5th ed. (Wiley, New York, 1983).
8. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
OPTICS LETTERS / Vol. 14, No. 18 / September 15, 1989
Determination of a dielectric waveguide propagation constant using a multifilament-current model H. Cory, Z. Altman, and Y. Leviatan Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000,Israel Received February 27, 1989; accepted June 11, 1989 A moment method using a multifilament-current
model is presented
to analyze the modes propagating
in a
cylindrical dielectric waveguide. In this model, analytically derivable fields of filamentary electric and magnetic currents (of yet unknown propagation constant and amplitude) 'are used to simulate the field of each mode inside and outside the guiding core. A simple point-matching procedure is subsequently used to enforce the boundary conditions at the core periphery and results in a homogeneous matrix equation. The longitudinal propagation constant of each mode and the currents that yield the field distribution of this mode are then found by solvingthis equation. As an example, a circular dielectric waveguideis analyzed and the results are presented.
A moment method using a multifilament-current model has been applied successfully to solve timeharmonic two-dimensional electromagnetic scattering problems at normal incidence.",2 In this Letter the method is applied to find the Brillouin diagrams of dielectric waveguides of arbitrary homogeneous cross section, which is a problem of great interest in optics. In the above-mentioned scattering problems, the amplitudes of the filamentary currents are assumed constant because the incidence is normal, whereas in the fl-w determination of dielectric waveguides they are assumed to have the eijz variation of the propagating modes. Other methods have also been used to find the f3-wdiagrams, such as the modal point-matching- 5 and the finite-difference methods.6 Let us consider a lossless z-directed dielectric waveguide of uniform cross section bounded by an arbitrary curve S. The core (region 1) has parameters el and gu
while its surrounding lossless cladding (region 2) has parameters
62
and Ito. We assume that el >
62.
The
waveguide propagation takes place in the hybrid mode, which is represented as the sum of TE and TM waves that together satisfy the boundary conditions at S. For circular waveguides, TE and TM modes can also propagate independently, along with hybrid modes. The time dependence of the fields and the
Fig. 2. These filaments are called internal filaments and are used to approximate the fields in region 2. The two parts of the model are connected through the boundary conditions that require the continuity of the tangential components of the electric and magnetic fields across S as follows:
A X (E'-
EO)= 0,
A X (HI
0146-9592/89/181026-03$2.00/0
HO) = 0,
where superscripts I and 0 denote that the given quantity is related to region 1 or 2, respectively. Let i3, k8, and k,8 denote the longitudinal propagation constant, the wave number, and the radial wave number, respectively, in region s (s = 1, 2). In region 1 (inside the core) k1 2 = W2 El6M - kp12 + p2, while in region 2 (outside the core) k2 2
=
cA2e2/o = kPS2
+ ,32. It
should be noted that kp, is real and kp2 is imaginary. Since, as stated above, the fields of the propagating modes vary with z according to e-jfz, the electric and magnetic currents flowing in the filaments are chosen as follows: Ii = Ioj e-ilz,
KE= KO e
2-f
In the transverse plane, the position of the ith filament
currents is taken as ejwt.
The analysis for hybrid mode propagation has two interacting parts. In the first part, we consider an infinite homogeneous space with parameters el and go in which the mathematical curve S defines the core region 1. Two sets of filaments are placed in region 2 outside S: electric filaments 11°)and magnetic filaments fK91,as shown in Fig. 1 (i = 1, 2,.. .). These filaments are called external filaments and are used to approximate the fields in region 1. In the second part, we consider an infinite homogeneous space with parameters E2and ,uoin which an identical mathematical curve S defines the cladding region 2. Two sets of filaments are placed in region 1 inside S: electric filaments {I1 and magnetic filaments JKI1,as shown in
-
unbounded homogeneous space (el, ,o)
-rAezion)
-
//
region
1
"-filamentary
currents
{I°} +{KI} Fig. 1.
Filamentary currents producing the fields in region 1.
© 1989 Optical Society of America
September 15,1989 / Vol. 14, No. 18 / OPTICS LETTERS unbounded homogeneous
[Z]i = =
space(62-p0) -or
E0 , Ho o Pi
/0O
I
7/1-
/ I,
-
I
filameentarycurrents
,'~
{If'}+
{K[}
mathematical surface S
Fig. 2. Filamentary currents producing tthefields i region 2.
is given by pi, and the distance from any observation point p to the filament is given by R := IP - PilA
The magnetic vector potential Ai at p due to an electric filamentary current 1 iis giver l by Ai = -j A4' H( 2)(kp Rj)e JlZz
lectriveto
the while
pteta
(s = 1, 2),
(3)
'
H(2)(kpRi)eCtZ2
(s = 1, 2).
(4)
The total magnetic vector potential SN is given as the superposition of all the potentials Ai dlueto the various electric filaments situated in region 2I t=tE Ar(Io) The other total potentials, AO, FI, and F 0 , are ob tained in a similar manner. The electromagnetic fields are derived from these total potentials. The electric filaments produce the TM fielIds,the magnetic filaments produce the TE fields, ai3d the two sets jointly produce the hybrid (HYB) fie]adst We choose N electric filaments and N magnetic filathe boundary ments in both regions 1 and 2 and apy Nly conditions [Eqs. (1)] to N points on S. We require the . nnancW: continuity of all tangential field con Hz, and Ht at these N matching poi nts (the index t denotes any tangential component in tthe transverse xy plane). We obtain in this way 4.N homogeneous equations with 4N unknowns-these being the complex amplitudes of the (electric and inagnetic) external and internal currents. As an example, the continuity equation for the tangential component of the electric field at the jth matching poi.nt in the transverse plane is given by N
the values of 1 for which f(13) A det[ZI vanishes have to
be found in the range k2 >Ep and H, >>Hz and should be independent of (p. A similar reasoning applies for the TE mode. The inclusion of all the appropriate components in the definitions of A() is necessary since for some spurious one component may be continuous while another one is discontinuous. It is anticipated that the values of the A(13)'sso defined will be considerably larger for spurious solutions than for physical ones. Each of the values of the A(O)'s is to be calculated at the Mpoints on the boundary S for all the values of the
In the TM mode we do not require the continuity of the Eo and Hz components since these components do not vanish in practice, as they should, owing to nu-
all the A(1)'s that correspond to the spurious solutions, to the largest of all the A(O)'sthat correspond to the physical solutions, is 257, showing that it is not difficult to distinguish between the two kinds of solutions. The computer time required to calculate A(1) for all the solutions is 10% of the time required to calculate L(W). The 1-w diagram of a circular dielectric waveguide
is shown in Fig. 4. The intersection of the a/X = 0.5 reference line with the HE21 , TMo1, TEO1 , and HE1 1 dispersion curves corresponds to the four values of 13 found in Fig. 3. Finally, it should be noted that the values of the propagation constants obtained by the present method for circular dielectric waveguides yield a six-figure agreement with the values obtained previously by conventional methods.8 References 1. Y. Leviatan and A. Boag, IEEE Trans. Antennas Propag. AP-35, 1119 (1987). 2. Y. Leviatan, A. Boag, and A. Boag, IEEE Trans. Anten-
nas Propag. AP-36, 1026 (1988). 3. J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969). 4. J. R. James and I. N. L. Gallett, Radio Electron. Eng. 42, 103 (1972).
5. E. Yamashita, K. Atsuki, 0. Hashimoto, and K. Kamijo, IEEE Trans. Microwave Theory Tech. MTT-27, 352 (1979). 6. E. Schweig and W. B. Bridges, IEEE Trans. Microwave
Theory Tech. MTT-32, 531 (1984). 7. E. Kreyszig, Advanced Engineering Mathematics, 5th ed. (Wiley, New York, 1983).
8. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
