Polarization effects in tapered dielectric waveguides

doi:10.1364/OE.11.001931

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Polarization effects in tapered dielectric waveguides

Ge Jin, Shouyuan Shi, Ahmed Sharkawy, and Dennis Prather »View Author Affiliations

Optics Express, Vol. 11, Issue 16, pp. 1931-1941 (2003)
http://dx.doi.org/10.1364/OE.11.001931

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▸ Article Outline
– Abstract
1. Introduction
2. Tapered slab waveguide and rectangular waveguide
3. Analysis method
4. Results and discussion
5. Conclusions
– References and links

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Abstract

The 3D finite-difference time-domain (FDTD) method is used to analyze the polarization effects in two kinds of linearly tapered optical waveguides: slab waveguides with only lateral tapers and rectangular cross section waveguides with both lateral and vertical tapers. For the slab waveguides, each guided mode of both the back reflected and output powers are determined and compared. For rectangular cross section waveguides, the output power of TE and TM modes with respect to taper length are computed and compared.

1. Introduction

Tapered waveguides have been under study for a long time due to their ability to manipulate and transform optical mode sizes to connect optical devices of different cross-sectional dimensions in optical integrated circuits (OICs) [1

Thomas Dillion, Anita Balcha, Dr.Janusz Murakowski, and Dr. Dennis Prather, “Process development and application of grayscale lithography for efficient three-dimensionally profiled fiber-to-waveguide couplers,” SPIE’s 48th annual meeting, (to be published).

]. To achieve a highly efficient power coupling, the structure must operate under radiation-loss-free and mode-conversion-free conditions. Although several studies have shown that an adiabatic taper can be used [2

G. Agrawal, Fiber-Optic Communication Systems (Weily, New York, 1992).

–6

R. Weder, “Dielectric three-dimensional electromagnetic tapers with no loss,” IEEE J. Quantum Electron. 24, 775–779 (1988). [CrossRef]

], these structures require precise control over taper shape and index profile. It is almost impossible to make such tapers using practical waveguide fabrication techniques. As a result, a number of methods have been developed to analyze the radiation loss of tapered waveguides, including intrinsic mode calculations [7

E. Marcatili, “Dielectric tapers with curved axes and no loss,” IEEE J.Quantum Electron, QE 21, 307–314 (1985). [CrossRef]

], step transition method [8

I. Lu, “Intrinsic modes in wedge-shaped taper above an anisotropic substrate,” IEEE J.Quantum Electron , 27, 2373–2377 (1991). [CrossRef]

], coupled mode calculations [9

S.El Yumin, K. Komori, S. Arai, and G. Bendelli, “Taper-shape dependence of tapered-waveguide traveling wave semiconductor laser amplifier (TTW-SLA),” IEICE Tran. Electron, E77-C 4, 624–632 (1994).

], beam propagation method [10

A. Milton and W. Burns, “Mode coupling in optical waveguide horns,” IEEE J.Quantum Electron, QE-13 10, 828–834 (1977). [CrossRef]

,11

C. Vassallo, “Analysis of tapered mode transformers for semiconductor optical amplifiers,” Opt. Quantum Electron. 26, 1025–1026 (1996).

] and finite-difference third-order simplified wave equation method [12

Z.N. Lu and R. Bansal, “A finite-difference third-order simplified wave equation method: an assessment and application,” IEEE Microwave Theory Technol. 42, 132–136 (1994). [CrossRef]

,13

Z.N. Lu, R. Bansal, and Peter K. Cheo, “Radiation losses of tapered dielectric waveguides: a finite difference analysis with ridge waveguide applications,” IEEE J. Lightwave Technol. , 12, 1373–1377 (1994). [CrossRef]

]. Those methods agree that, if a taper is long enough and has a small enough taper slope, the taper can achieve low loss. Early investigations have shown that 2D linear tapers [14

G.R. Hadley, “Design of tapered waveguides for improved output coupling,” IEEE Photon. Technol. Lett. 5, 1068–1070 (1993). [CrossRef]

] provide 90% coupling efficiency and more complicated taper shapes such as parabolic [10

A. Milton and W. Burns, “Mode coupling in optical waveguide horns,” IEEE J.Quantum Electron, QE-13 10, 828–834 (1977). [CrossRef]

], exponential [15

R.K. Winn and J.H. Harris, “Coupling from multimode to single mode linear waveguides using horn-shaped strctures,” IEEE Microwave Theory Tech. , 23, 3012–3015 (1975). [CrossRef]

], Gaussian and hyperbolic [7

E. Marcatili, “Dielectric tapers with curved axes and no loss,” IEEE J.Quantum Electron, QE 21, 307–314 (1985). [CrossRef]

] increase this percentage by modest amounts.

However, few papers have discussed the polarization effect in the tapered waveguides — how would the polarization change in the tapered waveguides? In this paper, we present a full wave analysis of the polarization effect in both linearly tapered slab waveguide and rectangular waveguides based on 3D-FDTD methods. For the laterally tapered slab waveguide, the back reflected power and transmitted power are computed. In addition, the reflected and transmitted power of each guided mode is calculated. For rectangular waveguide, we use TE and TM single mode sources to excite the waveguide, respectively. The back reflected power, radiation loss and transmitted power versus taper length for different polarization (TE and TM) are computed and compared. In Section 2, structures of these two waveguides are described. The numerical method used is presented in Section 3. In Section 4, simulation results and discussion are presented, followed by a brief summary in Section 5.

2. Tapered slab waveguide and rectangular waveguide

Figure 1 shows the 3D tapered slab waveguide under consideration. It has the following parameters: The height is H=0.228µm, the width of the input side is W1=1.824µm, the width of the output side is W2=0.72µm, the taper length is L=40µm, the length between the taper and the input side is L1=10µm, the length between the taper and the output side is L2=5µm; the slope of the taper is Φ=0.79°, the index inside the waveguide is n=3.5, the index outside is n=1.0. The wavelength considered here is 1µm. So, the taper is actually multimode-to-multimode. Figure 2 shows the top view and the side view of this waveguide.

Fig. 1. Tapered Slab Waveguide.

Fig. 2 (a) Top view of the waveguide

Fig. 2 (b) Side view of the waveguide.

Figure 3 shows the rectangular tapered waveguide. The dimension of the input side is 2.4×2.4 µm; dimension of the output side is 0.6×0.6 µm. The taper length L changes from 3µm to 15µm, under such condition, the lateral taper slope Φ varies from 3.4° to 16.7°; the vertical taper slope θ varies from 6.8° to 30.9°. The length between taper and the input side L1=2µm, the length between the taper and the output side is L2=12µm. Similar with the slab waveguide, the refractive index inside and outside the waveguide is 3.5 and 1.0, respectively.The wavelength is also 1.0µm. Considering that the top view of this waveguide is the same with Fig. 2(a), Fig. 4 only shows the side view of this waveguide.

Fig. 3. Tapered Rectangular Waveguide.

Fig. 4. Side view of the tapered rectangular waveguide.

Fig. 5 The cross section of the 3D rectangular waveguide

According to Marcatili’s method [16

E.A.J. Marcatilli, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell System Tech. 48, 2071 (1969).

], such a waveguide supports both TE^z or

E_{pq}^{y}

and TM^z or

E_{pq}^{z}

modes. Here, we use the lowest mode of both TE^z and TM^z to excite the waveguide, respectively, in order to find the relation between the polarization change and the taper length. Next, we will introduce the analysis method in detail.

3. Analysis method

Using Marcatili’s method, the magnetic field of TE^z mode in the 3D optical waveguide can be expressed as:

H_{z} = {\begin{matrix} A \cos (k_{y} y - Φ) \cos (k_{z} z - Ψ) Region 1 \\ A \cos (k_{y} a - Φ) e^{- γ_{y} (y - a)} \cos (k_{z} z - Ψ) Region 2 \\ A \cos (k_{y} y - Φ) e^{- γ_{z} (z - d)} \cos (k_{z} d - Ψ) Region 3 \end{matrix}

(1)

Here, regions 1–3 are explained in Fig. 5. The important assumption of this method is that the electromagnetic field in the shaded area in Fig. 5 is small enough to be neglected since the electromagnetic field of the well-guided mode decays quite rapidly in the cladding region. Using the orthogonality of the guided modes, Eq. (1) can be interpreted as:

H_{mn}^{z} = A_{mn} B_{mn} (y, z)

(2)

Here, m and n represent the mode number; A is the amplitude constant, B stands for the mode distribution in the cross section. In order to find the power carried by each guided mode, the amplitude has to be found. From orthogonality, we have:

\iint B_{mn} (y, z) B_{\bar{mn}} (y, z) dy dz = {\begin{matrix} C_{mn} m = \bar{m}, n = \bar{n} \\ 0 else \end{matrix}

(3)

Also, recalling that the field in the optical waveguide can be written as the sum of each different mode:

H_{total}^{z} = \sum_{m = 1 n = 1}^{I} H_{mn}^{z} = \sum_{m = 1 n = 1}^{I} A_{mn} B_{mn} (y, z)

(4)

In Eq. (4), I stands for the highest mode number the waveguide can support. Therefore, combining Eqs. (3) and (4), we have

A_{mn} C_{mn} = \iint H_{total}^{z} B_{mn} (y, z) dy dz

(5)

Note that B_mn (y, z) and C_mn are known from Eqs. (1) and (3), respectively, so the amplitude constant on each guided mode is then given from Eq. (5):

A_{mn} = \frac{\iint H_{total}^{z} B_{mn} (y, z) dy dz}{C_{mn}}

(6)

Once the amplitude is determined, the power carried by each mode can be calculated. To calculate the transmitted power distribution, substitute

H_{total}^{z}

into Eq. (6) with the transmitted field

H_{out}^{z}

. To calculate the back-reflected power distribution, substitute

H_{total}^{z}

into Eq. (6) with the reflected field

H_{reflect}^{z}

We use 3D-FDTD methods to conduct the numerical simulation. The source is introduced on the connecting boundary, as shown in Fig. 6. As a result, the field before the connecting boundary, in region 1, is the reflected and back-scattered field. The field after the connecting boundary, in region 2, is the total field. The shaded region is the PML absorbing boundary.

Fig. 6. The diagram of the computation region.

The computational cost is great due to the full 3D-FDTD simulation. So we wrote the computer program based on message passing interface (MPI) in order to realize the parallel computing on our cluster with 10 computers [17

D.P. Rodohan and S.R Saunders, “Parallel implementations of the finite difference time domain (FDTD) method,” Computation in Electromagnetics, Second International Conference, 367–370 (1994).

,18

C. Guiffaut and K. Mahdjoubi, “A parallel FDTD algorithm using the MPI library,” IEEE Antennas and Propagation Magazine , 43, 94–103 (2001). [CrossRef]

4. Results and discussion

4.1. Laterally tapered slab waveguide

The waveguide is excited with the

{TE}_{11}^{z}

mode. The wavelength is set to λ=1µm. The computational step size was set to 0.02µm. Figure 7 shows the steady state field components in the middle xy plane. Because it is multimode-to-multimode, the field distribution is not perfectly regular. This is more obvious in E_z and H_y field components in case that they are the dominant components of TM^z mode. The amplitude of the dominant components of TE^z mode is much bigger than those of the TM^z mode.

Fig. 7. Steady state field in the middle xy plane.

Figure 8 shows the normalized amplitude and the power of each mode at the output port. The amplitude is normalized to the source

{TE}_{11}^{z}

mode. The power is normalized to the total output power. Our simulation shows that although 100% power of

{TE}_{11}^{z}

mode is excited, it remains 81% at the output, some power is coupled into higher order TE modes and TM modes, some is back reflected and scattered, the other is radiated and absorbed by the PML absorbing boundary. In the output power,

{TE}_{31}^{z}

accounts for 15% while in the other TE modes, the power is relatively small. Compared with TE mode, TM mode power is small and the result shows that the power is more likely to couple into higher order TM mode.

{TM}_{42}^{z}

mode accounts for 0.76%,

{TM}_{41}^{z}

mode accounts for 0.48%,

{TM}_{22}^{z}

mode accounts for 0.30%,

{TM}_{21}^{z}

mode accounts for 0.07%.

Fig. 8. (a) Normalized field amplitude of the output field. (b) Power distribution of output field

Figure 9 shows the normalized amplitude and the power of each mode of the back-reflected field. We should point out that the back reflected power is around 5.4% of the source and our figure only included the relative dominant value so that some lower modes such as

{TE}_{42}^{z}

{TE}_{52}^{z}

{TM}_{11}^{z}

{TM}_{12}^{z}

are not plotted. Our results show that the

{TE}_{11}^{z}

mode is still the major contributor to the back-reflected field, it accounts for more than 85% of the total back-reflected power. For TM mode, the value of most of the modes is small except

{TM}_{21}^{z}

and

{TM}_{42}^{z}

, which take 0.72% and 0.31% respectively.

Fig. 9. (a) Normalized field amplitude of back reflected field. (b) Power distribution of back-reflected field

4.2. Tapered rectangular waveguide

In order to find the relation between polarization change and the taper length, we repeated the numerical experiment several times with different taper length, from 3µm to 15µm. We used

{TE}_{11}^{z}

and

{TM}_{11}^{z}

modes as sources, respectively and found that TM modes have less radiation loss and better polarization stability. Figure 10 shows the radiation loss comparison and back-reflected loss comparison. As you can see, the radiation loss of TE mode is almost twice the radiation loss of TM mode, for taper length longer than 6µm. They reach the same level only when the taper length is equal to 3µm. As for back-reflected loss, both of them remain at thevery low level except at 3µm, the TM mode outreaches TE mode by 7% with respect to the source power.

Fig. 10. (a) Radiation loss comparison. (b) Back reflection loss comparison.

Figure 11 shows the percentage of output power with respect to source power versus taper length, Fig. 11(a) corresponds to the TM source and Fig. 11(b) corresponds to the TE source. Our simulation shows that for the same taper length, TM mode has less polarization change compared with TE mode. In other words, TM mode has better polarization stability. For example, when the taper length is 9µm, if the source is TM, then 81.6% power remains in TM mode, 11.8% power changes into TE mode, but if the source is TE, only 43% power remains in TE mode, 28.5% power changes into TM mode. What’s more, our simulation shows that the polarization remains well with long taper length and tends to change when the taper length becomes short. This is reasonable since in the short taper, the mode profile changes so fast that the EM field no longer satisfies the adiabatic boundary conditions. As a result the polarization will change in order to adapt to the new boundary condition. As is well know, an infinitely long taper is perfectly adiabatic in which case the polarization should remain unchanged, since it is in fact a straight waveguide. Figure 11 also shows that both the TM mode output power and TE mode output power decrease rapidly due to the increasing radiation loss and back-reflection loss, when the taper length becomes shorter than 9µm.

Fig. 11. (a) Output power percentage versus taper length, TM source. (b) Output power percentage versus taper length, TE source.

5. Conclusions

Polarization effects in linearly tapered waveguide are studied with full wave 3D-FDTD methods. Two kinds of tapered structures are presented and studied. For the tapered slab waveguide that we studied, more than 81% output power remains in the source mode:

{TE}_{11}^{z}

mode, 15% power couples into

{TE}_{31}^{z}

. Although TM output mode power is less than 2%, most of them are coupled into higher order modes.

{TM}_{42}^{z}

mode accounts for 0.76%,

{TM}_{41}^{z}

mode accounts for 0.48%. For rectangular waveguide, our simulation showed that TM mode has better polarization stability and less radiation loss. Under the same condition, the radiation loss of TE mode is almost twice that of the TM mode. In addition, less power is changed into other polarization if the taper is excited with TM mode than TE mode.

Even though two examples of linearly tapered waveguides were discussed in details in the paper, the analysis can be generalized to include any taper shapes such as parabolic, exponential, Gaussian and hyperbolic tapers, which will be studied in the future.

References and Links

1.	Thomas Dillion, Anita Balcha, Dr.Janusz Murakowski, and Dr. Dennis Prather, “Process development and application of grayscale lithography for efficient three-dimensionally profiled fiber-to-waveguide couplers,” SPIE’s 48th annual meeting, (to be published).
2.	G. Agrawal, Fiber-Optic Communication Systems (Weily, New York, 1992).
3.	M. Wu, P. Fan, and C. Lee, “Completely adiabatic s-shaped bent tapers in optical waveguides,” IEEE Photon. Tech. Lett. 9, 212–214 (1997). [CrossRef]
4.	C. Lee, M. Wu, L. Sheu, P. Fan, and J. Hsu, “Design and analysis of completely adiabatic tapered waveguides by conformal mapping,” IEEE J. Lightwave Technol. 15, 403–410 (1993).
5.	J. Sakai and E. Marcatili, “Lossless dielectric tapers with three-dimensional geometry,” IEEE J. Lightwave Technol. 9, 386–393 (1991). [CrossRef]
6.	R. Weder, “Dielectric three-dimensional electromagnetic tapers with no loss,” IEEE J. Quantum Electron. 24, 775–779 (1988). [CrossRef]
7.	E. Marcatili, “Dielectric tapers with curved axes and no loss,” IEEE J.Quantum Electron, QE 21, 307–314 (1985). [CrossRef]
8.	I. Lu, “Intrinsic modes in wedge-shaped taper above an anisotropic substrate,” IEEE J.Quantum Electron , 27, 2373–2377 (1991). [CrossRef]
9.	S.El Yumin, K. Komori, S. Arai, and G. Bendelli, “Taper-shape dependence of tapered-waveguide traveling wave semiconductor laser amplifier (TTW-SLA),” IEICE Tran. Electron, E77-C 4, 624–632 (1994).
10.	A. Milton and W. Burns, “Mode coupling in optical waveguide horns,” IEEE J.Quantum Electron, QE-13 10, 828–834 (1977). [CrossRef]
11.	C. Vassallo, “Analysis of tapered mode transformers for semiconductor optical amplifiers,” Opt. Quantum Electron. 26, 1025–1026 (1996).
12.	Z.N. Lu and R. Bansal, “A finite-difference third-order simplified wave equation method: an assessment and application,” IEEE Microwave Theory Technol. 42, 132–136 (1994). [CrossRef]
13.	Z.N. Lu, R. Bansal, and Peter K. Cheo, “Radiation losses of tapered dielectric waveguides: a finite difference analysis with ridge waveguide applications,” IEEE J. Lightwave Technol. , 12, 1373–1377 (1994). [CrossRef]
14.	G.R. Hadley, “Design of tapered waveguides for improved output coupling,” IEEE Photon. Technol. Lett. 5, 1068–1070 (1993). [CrossRef]
15.	R.K. Winn and J.H. Harris, “Coupling from multimode to single mode linear waveguides using horn-shaped strctures,” IEEE Microwave Theory Tech. , 23, 3012–3015 (1975). [CrossRef]
16.	E.A.J. Marcatilli, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell System Tech. 48, 2071 (1969).
17.	D.P. Rodohan and S.R Saunders, “Parallel implementations of the finite difference time domain (FDTD) method,” Computation in Electromagnetics, Second International Conference, 367–370 (1994).
18.	C. Guiffaut and K. Mahdjoubi, “A parallel FDTD algorithm using the MPI library,” IEEE Antennas and Propagation Magazine , 43, 94–103 (2001). [CrossRef]

OCIS Codes
(000.0000) General : General
(260.5430) Physical optics : Polarization

ToC Category:
Research Papers

History
Original Manuscript: July 24, 2003
Revised Manuscript: August 4, 2003
Published: August 11, 2003

Citation
Ge Jin, Shouyuan Shi, Ahmed Sharkawy, and Dennis Prather, "Polarization effects in tapered dielectric waveguides," Opt. Express 11, 1931-1941 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-16-1931

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References

Thomas Dillion, Anita Balcha, Dr.Janusz Murakowski and Dr. Dennis Prather, �??Process development and application of grayscale lithography for efficient three-dimensionally profiled fiber-to-waveguide couplers,�?? SPIE�??s 48th annual meeting, (to be published).
G.Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1992).
M.Wu, P.Fan and C.Lee, �??Completely adiabatic s-shaped bent tapers in optical waveguides,�?? IEEE Photon. Tech. Lett. 9, 212-214 (1997). [CrossRef]
C.Lee, M.Wu, L.Sheu, P.Fan and J.Hsu, �??Design and analysis of completely adiabatic tapered waveguides by conformal mapping,�?? IEEE J. Lightwave Technol. 15, 403-410 (1993).
J.Sakai and E.Marcatili, �??Lossless dielectric tapers with three-dimensional geometry,�?? IEEE J. Lightwave Technol. 9, 386-393 (1991). [CrossRef]
R.Weder, �??Dielectric three-dimensional electromagnetic tapers with no loss,�?? IEEE J. Quantum Electron. 24, 775-779 (1988). [CrossRef]
E.Marcatili, �??Dielectric tapers with curved axes and no loss,�?? IEEE J.Quantum Electron., QE 21, 307-314 (1985). [CrossRef]
I.Lu, �??Intrinsic modes in wedge-shaped taper above an anisotropic substrate,�?? IEEE J.Quantum Electron., 27, 2373-2377 (1991). [CrossRef]
S. El Yumin, K. Komori, S. Arai and G. Bendelli, �??Taper-shape dependence of tapered-waveguide traveling wave semiconductor laser amplifier (TTW-SLA),�?? IEICE Tran. Electron., E77-C 4, 624-632 (1994).
A. Milton and W. Burns, �??Mode coupling in optical waveguide horns,�?? IEEE J. Quantum Electron., QE-13 10, 828-834 (1977). [CrossRef]
C.Vassallo, �??Analysis of tapered mode transformers for semiconductor optical amplifiers,�?? Opt. Quantum Electron. 26, 1025-1026 (1996).
Z.N.Lu and R.Bansal, �??A finite-difference third-order simplified wave equation method: an assessment and application,�?? IEEE Microwave Theory Technol. 42, 132-136 (1994). [CrossRef]
Z.N.Lu, R.Bansal and Peter K.Cheo, �??Radiation losses of tapered dielectric waveguides: a finite difference analysis with ridge waveguide applications,�?? IEEE J. Lightwave Technol., 12, 1373-1377 (1994). [CrossRef]
G. R. Hadley, �??Design of tapered waveguides for improved output coupling,�?? IEEE Photon. Technol. Lett. 5, 1068-1070 (1993). [CrossRef]
R. K. Winn and J. H. Harris, �??Coupling from multimode to single mode linear waveguides using horn-shaped strctures,�?? IEEE Microwave Theory Tech., 23, 3012-3015 (1975). [CrossRef]
E. A. J. Marcatilli, �??Dielectric rectangular waveguide and directional coupler for integrated optics,�?? Bell System Tech. 48, 2071 (1969).
D. P. Rodohan and S. R. Saunders, �??Parallel implementations of the finite difference time domain (FDTD) method,�?? Computation in Electromagnetics, Second International Conference, 367-370 (1994).
C. Guiffaut and K. Mahdjoubi, �??A parallel FDTD algorithm using the MPI library,�?? IEEE Antennas and Propagation Magazine, 43, 94-103 (2001). [CrossRef]

Bell System Tech.

E. A. J. Marcatilli, �??Dielectric rectangular waveguide and directional coupler for integrated optics,�?? Bell System Tech. 48, 2071 (1969).

IEEE Antennas and Propagation Magazine

C. Guiffaut and K. Mahdjoubi, �??A parallel FDTD algorithm using the MPI library,�?? IEEE Antennas and Propagation Magazine, 43, 94-103 (2001). [CrossRef]

IEEE J. Lightwave Technol.

Z.N.Lu, R.Bansal and Peter K.Cheo, �??Radiation losses of tapered dielectric waveguides: a finite difference analysis with ridge waveguide applications,�?? IEEE J. Lightwave Technol., 12, 1373-1377 (1994). [CrossRef]
C.Lee, M.Wu, L.Sheu, P.Fan and J.Hsu, �??Design and analysis of completely adiabatic tapered waveguides by conformal mapping,�?? IEEE J. Lightwave Technol. 15, 403-410 (1993).
J.Sakai and E.Marcatili, �??Lossless dielectric tapers with three-dimensional geometry,�?? IEEE J. Lightwave Technol. 9, 386-393 (1991). [CrossRef]

IEEE J. Quantum Electron.

R.Weder, �??Dielectric three-dimensional electromagnetic tapers with no loss,�?? IEEE J. Quantum Electron. 24, 775-779 (1988). [CrossRef]
A. Milton and W. Burns, �??Mode coupling in optical waveguide horns,�?? IEEE J. Quantum Electron., QE-13 10, 828-834 (1977). [CrossRef]

IEEE J.Quantum Electron.

E.Marcatili, �??Dielectric tapers with curved axes and no loss,�?? IEEE J.Quantum Electron., QE 21, 307-314 (1985). [CrossRef]
I.Lu, �??Intrinsic modes in wedge-shaped taper above an anisotropic substrate,�?? IEEE J.Quantum Electron., 27, 2373-2377 (1991). [CrossRef]

IEEE Microwave Theory Tech.

R. K. Winn and J. H. Harris, �??Coupling from multimode to single mode linear waveguides using horn-shaped strctures,�?? IEEE Microwave Theory Tech., 23, 3012-3015 (1975). [CrossRef]

IEEE Microwave Theory Technol.

Z.N.Lu and R.Bansal, �??A finite-difference third-order simplified wave equation method: an assessment and application,�?? IEEE Microwave Theory Technol. 42, 132-136 (1994). [CrossRef]

IEEE Photon. Tech. Lett.

M.Wu, P.Fan and C.Lee, �??Completely adiabatic s-shaped bent tapers in optical waveguides,�?? IEEE Photon. Tech. Lett. 9, 212-214 (1997). [CrossRef]

IEEE Photon. Technol. Lett.

G. R. Hadley, �??Design of tapered waveguides for improved output coupling,�?? IEEE Photon. Technol. Lett. 5, 1068-1070 (1993). [CrossRef]

IEICE Tran. Electron.

S. El Yumin, K. Komori, S. Arai and G. Bendelli, �??Taper-shape dependence of tapered-waveguide traveling wave semiconductor laser amplifier (TTW-SLA),�?? IEICE Tran. Electron., E77-C 4, 624-632 (1994).

Opt. Quantum Electron.

C.Vassallo, �??Analysis of tapered mode transformers for semiconductor optical amplifiers,�?? Opt. Quantum Electron. 26, 1025-1026 (1996).

Other

Thomas Dillion, Anita Balcha, Dr.Janusz Murakowski and Dr. Dennis Prather, �??Process development and application of grayscale lithography for efficient three-dimensionally profiled fiber-to-waveguide couplers,�?? SPIE�??s 48th annual meeting, (to be published).
G.Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1992).
D. P. Rodohan and S. R. Saunders, �??Parallel implementations of the finite difference time domain (FDTD) method,�?? Computation in Electromagnetics, Second International Conference, 367-370 (1994).

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