Abstract

The method of lines (MoL) has been developed to study coupling efficiency on hemispherical lens. In this paper, the physical shape of the lens is approximated by cascading a number of straight waveguide segments. The perfectly matched layer (PML) is applied as an absorber for the MoL to reduce numerical reflection in the simulation region. Analysis is done by calculating coupling efficiency at the plane of integration where the coupling efficiency is an overlap integral between laser diode field and fiber field. The result of coupling efficiency in this analysis is compared to the experiment and ABCD matrix. It is found that MoL gives good result accuracy.

© 2009 Optical Society of America

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  1. S. Gangopadhyay and S. Sarkar, "ABCD matrix for reflection and refraction of Gaussian light beams at surfaces of hyperboloid of revolution and efficiency computation for laser diode to single-mode fiber coupling by way of a hyperbolic lens on the fiber tip," Appl. Opt. 36, 8582-8586 (1997).
    [CrossRef]
  2. K. Sambanthan and F. A. Rahman, "Method to improve the coupling efficiency of a hemispherically lensed asymmetric tapered-core fiber," Opt. Commun. 254, 112-118 (2005).
    [CrossRef]
  3. W. L. Emkey and C. A. Jack, "Analysis and evaluation of graded-index fiber-lenses," J. Lightwave Technol. 12, 1156-1164 (1987).
    [CrossRef]
  4. R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
    [CrossRef]
  5. J. Yamauchi, K. Nishio and H. Nakano, "Analysis of a lensed coreless fiber by a hybrid technique combining FD-BPM and FD-TDM," J. Lightwave Technol. 16, 465-471 (1998).
    [CrossRef]
  6. Z. Wang, B. Mikkelsen, B. Pedersen, K. E. Stubkjaer and D. S. Olesen, "Coupling between angled-facet amplifiers and tapered lens-ended fibers," J. Lightwave Technol. 9, 49-55 (1991).
    [CrossRef]
  7. Y. He, S. K. Mondal and F. G. Shi, "Design optimization of wedge-shaped lensed fibers for fiber-laser coupling: Fresnel reflection and non-Gaussian mode effects," J. Lightwave Technol. 21, 2271-2275 (2003).
    [CrossRef]
  8. K. Okamakoto, Fundamental of Optical Waveguides (Academic Press, California, 2000).
  9. T. Wongcharoen, B. M. A. Rahman, M. Rajarajan and K. T. V. Grattan, "Spot-size conversion using uniform waveguide sections for efficient laser-fiber coupling," J. Lightwave Technol. 19, 708-716 (2001).
    [CrossRef]
  10. M. Rajarajan, B. M. A. Rahman and K. T. V. Grattan, "Numerical study of spot-size expanders for an efficient OEIC to SMF coupling," IEEE Photonics Technol. Lett. 10, 1082-1084 (1998).
    [CrossRef]
  11. M. N. O. Sadiku and C. N. Obiozor, "A simple introduction to the method of lines," Int. J. Electr. Eng. Educ. 37, 282-296 (2000).
  12. W. Huang and R. R. A. Syms, "Analysis of folded erbium-doped planar waveguide amplifiers by the method of lines," J. Lightwave Technol. 17, 2658-2664 (1999).
    [CrossRef]
  13. H. A. Jamid, M. Z. M. Khan and M. Ameeruddin, "A compact 90◦ three-branch beam splitter based on resonant coupling," J. Lightwave Technol. 23, 3900-3906 (2005).
    [CrossRef]
  14. A. Abdullah and M. A. Majid, "Analysis of multi-layer ARROW," J. Microwaves Optoelectron. 3, 1-8 (2003).
  15. A. Dreher and R. Pregla, "Analysis of planar waveguides with the method of lines and absorbing boundary conditions," IEEE Microwave Guided Wave Lett. 1, 138-140 (1991).
    [CrossRef]
  16. B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Comp. 31, 629-651 (1977).
    [CrossRef]
  17. G. R. Hadley, "Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron. 28, 363-370 (1992).
    [CrossRef]
  18. D. Weiping and Z. Linchang, "An improvement algorithm of Mur’s First-Order absorbing boundary condition," in IEEE 1997 International Symposium on Electromagnetic Compatibility, (Austin,USA 1997), pp. 592-595.
  19. C. Vassallo and F. Collino, "Highly efficient absorbing boundary conditions for the beam propagation method," J. Lightwave Technol. 14, 1570-1577 (1996).
    [CrossRef]
  20. H. A. Jamid, "Enhanced PML performance using higher order approximation," IEEE Trans. Microwave Theory Tech. 52, 1166-1174 (2004).
    [CrossRef]
  21. M. Z. M. Khan, "Analysis of one and two dimensional bandgap structures using automated method of lines with arbitrary longitudinal discontinuities," Master dissertation, (King Fahd University of Petroleum and Minerals, Saudi Arabia, 2004).
  22. A. A. Shittu, "Study of periodic waveguides by the finite-difference time domain method and the method of lines," PhD dissertation, (King Fahd University of Petroleum and Minerals, Saudi Arabia, 1994).
  23. J. John, T. S. M. Maclean, H. Ghafouri-Shiraz and J. Niblett, "Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength," IEE Proc.-Optoelectron. 141,178-184 (1994).
    [CrossRef]
  24. W. T. Chen and L. A. Wang, "Out-of-plane optical coupling between an elliptical Gaussian beam and an angled single-mode fiber," J. Lightwave Technol. 16, 1589-1595 (1998).
    [CrossRef]
  25. F. A. Rahman, K. Takahashi and C. H. Teik, "A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber," Opt. Commun. 208,103-110 (2002).
    [CrossRef]
  26. J. Alda, "Laser and Gaussian beam propagation and transformation," in Encyclopedia of Optical Engineering, R. G. Driggers (Marcel Dekker, New York, 2003), pp. 999-1013.
  27. T. Saitoh, T. Mukai and O. Mikami, "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," J. Lightwave Technol. LT-3, 288-293 (1985).
    [CrossRef]
  28. C. A. Edwards, H. M. Presby and L. W. Stulz, "Effective reflectivity of hyperbolic microlenses," Appl. Opt. 32, 2099-2103 (1993).
    [CrossRef] [PubMed]
  29. H. Kuwahara, Y. Onoda, M. Goto and T. Nakagami, "Reflected light in the coupling of semiconductor lasers with tapered hemispherical end fibers," Appl. Opt. 22, 2732-2738 (1983).
    [CrossRef] [PubMed]

2005

K. Sambanthan and F. A. Rahman, "Method to improve the coupling efficiency of a hemispherically lensed asymmetric tapered-core fiber," Opt. Commun. 254, 112-118 (2005).
[CrossRef]

H. A. Jamid, M. Z. M. Khan and M. Ameeruddin, "A compact 90◦ three-branch beam splitter based on resonant coupling," J. Lightwave Technol. 23, 3900-3906 (2005).
[CrossRef]

2004

H. A. Jamid, "Enhanced PML performance using higher order approximation," IEEE Trans. Microwave Theory Tech. 52, 1166-1174 (2004).
[CrossRef]

2003

2002

F. A. Rahman, K. Takahashi and C. H. Teik, "A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber," Opt. Commun. 208,103-110 (2002).
[CrossRef]

2001

2000

M. N. O. Sadiku and C. N. Obiozor, "A simple introduction to the method of lines," Int. J. Electr. Eng. Educ. 37, 282-296 (2000).

R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

1999

1998

1997

1996

C. Vassallo and F. Collino, "Highly efficient absorbing boundary conditions for the beam propagation method," J. Lightwave Technol. 14, 1570-1577 (1996).
[CrossRef]

1994

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz and J. Niblett, "Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength," IEE Proc.-Optoelectron. 141,178-184 (1994).
[CrossRef]

1993

1992

G. R. Hadley, "Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron. 28, 363-370 (1992).
[CrossRef]

1991

A. Dreher and R. Pregla, "Analysis of planar waveguides with the method of lines and absorbing boundary conditions," IEEE Microwave Guided Wave Lett. 1, 138-140 (1991).
[CrossRef]

Z. Wang, B. Mikkelsen, B. Pedersen, K. E. Stubkjaer and D. S. Olesen, "Coupling between angled-facet amplifiers and tapered lens-ended fibers," J. Lightwave Technol. 9, 49-55 (1991).
[CrossRef]

1987

W. L. Emkey and C. A. Jack, "Analysis and evaluation of graded-index fiber-lenses," J. Lightwave Technol. 12, 1156-1164 (1987).
[CrossRef]

1985

T. Saitoh, T. Mukai and O. Mikami, "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

1983

1977

B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Comp. 31, 629-651 (1977).
[CrossRef]

Abdullah, A.

A. Abdullah and M. A. Majid, "Analysis of multi-layer ARROW," J. Microwaves Optoelectron. 3, 1-8 (2003).

Ameeruddin, M.

Chen, W. T.

Collino, F.

C. Vassallo and F. Collino, "Highly efficient absorbing boundary conditions for the beam propagation method," J. Lightwave Technol. 14, 1570-1577 (1996).
[CrossRef]

Dreher, A.

A. Dreher and R. Pregla, "Analysis of planar waveguides with the method of lines and absorbing boundary conditions," IEEE Microwave Guided Wave Lett. 1, 138-140 (1991).
[CrossRef]

Edwards, C. A.

Emkey, W. L.

W. L. Emkey and C. A. Jack, "Analysis and evaluation of graded-index fiber-lenses," J. Lightwave Technol. 12, 1156-1164 (1987).
[CrossRef]

Engquist, B.

B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Comp. 31, 629-651 (1977).
[CrossRef]

Gangopadhyay, S.

Ghafouri-Shiraz, H.

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz and J. Niblett, "Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength," IEE Proc.-Optoelectron. 141,178-184 (1994).
[CrossRef]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Goto, M.

Grattan, K. T. V.

T. Wongcharoen, B. M. A. Rahman, M. Rajarajan and K. T. V. Grattan, "Spot-size conversion using uniform waveguide sections for efficient laser-fiber coupling," J. Lightwave Technol. 19, 708-716 (2001).
[CrossRef]

M. Rajarajan, B. M. A. Rahman and K. T. V. Grattan, "Numerical study of spot-size expanders for an efficient OEIC to SMF coupling," IEEE Photonics Technol. Lett. 10, 1082-1084 (1998).
[CrossRef]

Hadley, G. R.

G. R. Hadley, "Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron. 28, 363-370 (1992).
[CrossRef]

He, Y.

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Huang, W.

Jack, C. A.

W. L. Emkey and C. A. Jack, "Analysis and evaluation of graded-index fiber-lenses," J. Lightwave Technol. 12, 1156-1164 (1987).
[CrossRef]

Jamid, H. A.

H. A. Jamid, M. Z. M. Khan and M. Ameeruddin, "A compact 90◦ three-branch beam splitter based on resonant coupling," J. Lightwave Technol. 23, 3900-3906 (2005).
[CrossRef]

H. A. Jamid, "Enhanced PML performance using higher order approximation," IEEE Trans. Microwave Theory Tech. 52, 1166-1174 (2004).
[CrossRef]

John, J.

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz and J. Niblett, "Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength," IEE Proc.-Optoelectron. 141,178-184 (1994).
[CrossRef]

Khan, M. Z. M.

Kuwahara, H.

Maclean, T. S. M.

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz and J. Niblett, "Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength," IEE Proc.-Optoelectron. 141,178-184 (1994).
[CrossRef]

Majda, A.

B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Comp. 31, 629-651 (1977).
[CrossRef]

Majid, M. A.

A. Abdullah and M. A. Majid, "Analysis of multi-layer ARROW," J. Microwaves Optoelectron. 3, 1-8 (2003).

Mikami, O.

T. Saitoh, T. Mukai and O. Mikami, "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

Mikkelsen, B.

Z. Wang, B. Mikkelsen, B. Pedersen, K. E. Stubkjaer and D. S. Olesen, "Coupling between angled-facet amplifiers and tapered lens-ended fibers," J. Lightwave Technol. 9, 49-55 (1991).
[CrossRef]

Mondal, S. K.

Mukai, T.

T. Saitoh, T. Mukai and O. Mikami, "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

Nakagami, T.

Nakano, H.

Niblett, J.

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz and J. Niblett, "Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength," IEE Proc.-Optoelectron. 141,178-184 (1994).
[CrossRef]

Nishio, K.

Obiozor, C. N.

M. N. O. Sadiku and C. N. Obiozor, "A simple introduction to the method of lines," Int. J. Electr. Eng. Educ. 37, 282-296 (2000).

Olesen, D. S.

Z. Wang, B. Mikkelsen, B. Pedersen, K. E. Stubkjaer and D. S. Olesen, "Coupling between angled-facet amplifiers and tapered lens-ended fibers," J. Lightwave Technol. 9, 49-55 (1991).
[CrossRef]

Onoda, Y.

Pedersen, B.

Z. Wang, B. Mikkelsen, B. Pedersen, K. E. Stubkjaer and D. S. Olesen, "Coupling between angled-facet amplifiers and tapered lens-ended fibers," J. Lightwave Technol. 9, 49-55 (1991).
[CrossRef]

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

A. Dreher and R. Pregla, "Analysis of planar waveguides with the method of lines and absorbing boundary conditions," IEEE Microwave Guided Wave Lett. 1, 138-140 (1991).
[CrossRef]

Presby, H. M.

Rahman, B. M. A.

T. Wongcharoen, B. M. A. Rahman, M. Rajarajan and K. T. V. Grattan, "Spot-size conversion using uniform waveguide sections for efficient laser-fiber coupling," J. Lightwave Technol. 19, 708-716 (2001).
[CrossRef]

M. Rajarajan, B. M. A. Rahman and K. T. V. Grattan, "Numerical study of spot-size expanders for an efficient OEIC to SMF coupling," IEEE Photonics Technol. Lett. 10, 1082-1084 (1998).
[CrossRef]

Rahman, F. A.

K. Sambanthan and F. A. Rahman, "Method to improve the coupling efficiency of a hemispherically lensed asymmetric tapered-core fiber," Opt. Commun. 254, 112-118 (2005).
[CrossRef]

F. A. Rahman, K. Takahashi and C. H. Teik, "A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber," Opt. Commun. 208,103-110 (2002).
[CrossRef]

Rajarajan, M.

T. Wongcharoen, B. M. A. Rahman, M. Rajarajan and K. T. V. Grattan, "Spot-size conversion using uniform waveguide sections for efficient laser-fiber coupling," J. Lightwave Technol. 19, 708-716 (2001).
[CrossRef]

M. Rajarajan, B. M. A. Rahman and K. T. V. Grattan, "Numerical study of spot-size expanders for an efficient OEIC to SMF coupling," IEEE Photonics Technol. Lett. 10, 1082-1084 (1998).
[CrossRef]

Sadiku, M. N. O.

M. N. O. Sadiku and C. N. Obiozor, "A simple introduction to the method of lines," Int. J. Electr. Eng. Educ. 37, 282-296 (2000).

Saitoh, T.

T. Saitoh, T. Mukai and O. Mikami, "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

Sambanthan, K.

K. Sambanthan and F. A. Rahman, "Method to improve the coupling efficiency of a hemispherically lensed asymmetric tapered-core fiber," Opt. Commun. 254, 112-118 (2005).
[CrossRef]

Sarkar, S.

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Shi, F. G.

Stubkjaer, K. E.

Z. Wang, B. Mikkelsen, B. Pedersen, K. E. Stubkjaer and D. S. Olesen, "Coupling between angled-facet amplifiers and tapered lens-ended fibers," J. Lightwave Technol. 9, 49-55 (1991).
[CrossRef]

Stulz, L. W.

Syms, R. R. A.

Takahashi, K.

F. A. Rahman, K. Takahashi and C. H. Teik, "A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber," Opt. Commun. 208,103-110 (2002).
[CrossRef]

Teik, C. H.

F. A. Rahman, K. Takahashi and C. H. Teik, "A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber," Opt. Commun. 208,103-110 (2002).
[CrossRef]

Vassallo, C.

C. Vassallo and F. Collino, "Highly efficient absorbing boundary conditions for the beam propagation method," J. Lightwave Technol. 14, 1570-1577 (1996).
[CrossRef]

Wang, L. A.

Wang, Z.

Z. Wang, B. Mikkelsen, B. Pedersen, K. E. Stubkjaer and D. S. Olesen, "Coupling between angled-facet amplifiers and tapered lens-ended fibers," J. Lightwave Technol. 9, 49-55 (1991).
[CrossRef]

Wongcharoen, T.

Yamauchi, J.

Appl. Opt.

IEEE J. Quantum Electron.

G. R. Hadley, "Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron. 28, 363-370 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

IEEE Microwave Guided Wave Lett.

A. Dreher and R. Pregla, "Analysis of planar waveguides with the method of lines and absorbing boundary conditions," IEEE Microwave Guided Wave Lett. 1, 138-140 (1991).
[CrossRef]

IEEE Photonics Technol. Lett.

M. Rajarajan, B. M. A. Rahman and K. T. V. Grattan, "Numerical study of spot-size expanders for an efficient OEIC to SMF coupling," IEEE Photonics Technol. Lett. 10, 1082-1084 (1998).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

H. A. Jamid, "Enhanced PML performance using higher order approximation," IEEE Trans. Microwave Theory Tech. 52, 1166-1174 (2004).
[CrossRef]

Int. J. Electr. Eng. Educ.

M. N. O. Sadiku and C. N. Obiozor, "A simple introduction to the method of lines," Int. J. Electr. Eng. Educ. 37, 282-296 (2000).

J. Lightwave Technol.

C. Vassallo and F. Collino, "Highly efficient absorbing boundary conditions for the beam propagation method," J. Lightwave Technol. 14, 1570-1577 (1996).
[CrossRef]

Z. Wang, B. Mikkelsen, B. Pedersen, K. E. Stubkjaer and D. S. Olesen, "Coupling between angled-facet amplifiers and tapered lens-ended fibers," J. Lightwave Technol. 9, 49-55 (1991).
[CrossRef]

W. L. Emkey and C. A. Jack, "Analysis and evaluation of graded-index fiber-lenses," J. Lightwave Technol. 12, 1156-1164 (1987).
[CrossRef]

T. Wongcharoen, B. M. A. Rahman, M. Rajarajan and K. T. V. Grattan, "Spot-size conversion using uniform waveguide sections for efficient laser-fiber coupling," J. Lightwave Technol. 19, 708-716 (2001).
[CrossRef]

W. Huang and R. R. A. Syms, "Analysis of folded erbium-doped planar waveguide amplifiers by the method of lines," J. Lightwave Technol. 17, 2658-2664 (1999).
[CrossRef]

J. Yamauchi, K. Nishio and H. Nakano, "Analysis of a lensed coreless fiber by a hybrid technique combining FD-BPM and FD-TDM," J. Lightwave Technol. 16, 465-471 (1998).
[CrossRef]

W. T. Chen and L. A. Wang, "Out-of-plane optical coupling between an elliptical Gaussian beam and an angled single-mode fiber," J. Lightwave Technol. 16, 1589-1595 (1998).
[CrossRef]

Y. He, S. K. Mondal and F. G. Shi, "Design optimization of wedge-shaped lensed fibers for fiber-laser coupling: Fresnel reflection and non-Gaussian mode effects," J. Lightwave Technol. 21, 2271-2275 (2003).
[CrossRef]

H. A. Jamid, M. Z. M. Khan and M. Ameeruddin, "A compact 90◦ three-branch beam splitter based on resonant coupling," J. Lightwave Technol. 23, 3900-3906 (2005).
[CrossRef]

T. Saitoh, T. Mukai and O. Mikami, "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

J. Microwaves Optoelectron.

A. Abdullah and M. A. Majid, "Analysis of multi-layer ARROW," J. Microwaves Optoelectron. 3, 1-8 (2003).

Math. Comp.

B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Comp. 31, 629-651 (1977).
[CrossRef]

Opt. Commun.

K. Sambanthan and F. A. Rahman, "Method to improve the coupling efficiency of a hemispherically lensed asymmetric tapered-core fiber," Opt. Commun. 254, 112-118 (2005).
[CrossRef]

F. A. Rahman, K. Takahashi and C. H. Teik, "A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber," Opt. Commun. 208,103-110 (2002).
[CrossRef]

Optoelectron.

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz and J. Niblett, "Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength," IEE Proc.-Optoelectron. 141,178-184 (1994).
[CrossRef]

Other

J. Alda, "Laser and Gaussian beam propagation and transformation," in Encyclopedia of Optical Engineering, R. G. Driggers (Marcel Dekker, New York, 2003), pp. 999-1013.

M. Z. M. Khan, "Analysis of one and two dimensional bandgap structures using automated method of lines with arbitrary longitudinal discontinuities," Master dissertation, (King Fahd University of Petroleum and Minerals, Saudi Arabia, 2004).

A. A. Shittu, "Study of periodic waveguides by the finite-difference time domain method and the method of lines," PhD dissertation, (King Fahd University of Petroleum and Minerals, Saudi Arabia, 1994).

K. Okamakoto, Fundamental of Optical Waveguides (Academic Press, California, 2000).

D. Weiping and Z. Linchang, "An improvement algorithm of Mur’s First-Order absorbing boundary condition," in IEEE 1997 International Symposium on Electromagnetic Compatibility, (Austin,USA 1997), pp. 592-595.

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Figures (10)

Fig. 1.
Fig. 1.

Discretization of simulation region.

Fig. 2.
Fig. 2.

Discretization of PML region [20].

Fig. 3.
Fig. 3.

Transformation of lens from a) continuous form to b) discretized form.

Fig. 4.
Fig. 4.

Discretization of the lens in z axis with waveguide discontinuities [22]

Fig. 5.
Fig. 5.

Physical structure of hemispherical lens.

Fig. 6.
Fig. 6.

Coupling efficiency as a function of working distance.

Fig. 7.
Fig. 7.

Spotsize and radius of curvature in x-axis at plane P1.

Fig. 8.
Fig. 8.

Phase image and spotsize of the laser field at the working distance, d = 7 μm

Fig. 9.
Fig. 9.

Coupling efficiency for different radius of lens.

Fig. 10.
Fig. 10.

Effective reflectivity as a function of working distance.

Tables (2)

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Table 1. Simulation parameter for MoL and PML.

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Table 2. Parameter for LD and SMF [23].

Equations (21)

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2 E x z z 2 + 2 E x z x 2 + k 0 2 n ( x ) 2 E x z = 0
2 E z 2 + Q 2 E = 0
E = ( E 1 E 2 E 3 . . E u . . E m ) t u = 1,2 m
Q 2 = 1 Δ x 2 ( 2 1 0 . . . . 0 1 2 1 0 . . 0 0 1 2 1 . . 0 . . . . . . . . . . . . 0 . . 0 1 2 1 0 0 . . 0 1 2 ) + k 0 2 ( n 1 2 0 0 . . . . 0 0 n 2 2 0 . . . . 0 . . . . . . . . . . . . 0 0 . . n u 2 . . 0 . . . . . . . . . . . . 0 0 0 0 . . n m 2 )
x ¯ i = i ( π 2 ) ( p + 1 )
h i = h 0 + j ( ε p ) f ( x ¯ i )
Q 2 = ( a 33 p a 32 p a 31 p . . . . 0 a 34 p 1 a 33 p 1 a 32 p 1 a 31 p 1 . . 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 a 31 p 1 a 32 p 1 a 33 p 1 a 34 p 1 0 0 0 a 31 p a 32 p a 33 p ) + k 0 2 ( n 1 2 0 0 . . . . 0 0 n 2 2 0 . . . . 0 . . . . . . . . . . . . 0 0 . . n u 2 . . 0 . . . . . . . . . . . . 0 0 0 0 . . n m 2 )
( . . . . . . . . . . . . . . . . . . . . a 31 i a 32 i a 33 i a 34 i a 35 i . . . . . . . . . . . . . . . . . . . . ) = ( . . . . q i 1 p i . . . . . . . . q i . . . . 1 0 0 0 0 . . . . q i + 1 + . . . . . . . . q i + 2 + p i + 1 + . . . . ) 1
q i ± = ( 1 ± h i h i 2 2 ! ± h i 3 3 ! h i 4 4 ! )
p i ± = ( 1 ± h i h i 2 2 ! ± h i 3 3 ! h i 4 4 ! 0 1 ± h i h i 2 2 ! ± h i 3 3 ! 0 0 1 ± h i h i 2 2 ! 0 0 0 1 ± h i 0 0 0 0 1 )
Γ i = [ ( I U i 1 U i + 1 ) + ( I + U i 1 U i + 1 ) D i + 1 Γ i + 1 D i + 1 ] ×
[ ( I + U i 1 U i + 1 ) + ( I U i 1 U i + 1 ) D i + 1 Γ i + 1 D i + 1 ] 1
A i + 1 = 0.5 [ ( I + U i + 1 1 U i ) D i A i + ( I U i + 1 1 U i ) B i ]
E l 0 x y = ( 2 π ω ox ω oy ) 1 2 exp ( x 2 ω ox 2 y 2 ω oy 2 )
η = E l x y E f 0 x y * d x d y 2 E l x y 2 d x d y E f 0 x y 2 d x d y
η = η x η y
ω x , y ( z ) = ω ox , oy [ 1 + ( λz π ω ox , oy 2 ) 2 ] 0.5
R x , y ( z ) = z [ 1 + ( λz π ω ox , oy 2 ) 2 ]
ω x = 2 E lx 2 x 2 dx E lx 2 dx [ E lx 2 xdx E lx 2 dx ] 2
1 R x = π ω x 2 E lx 2 dx × [ E lx x E lx * E lx E lx * x ] × [ x E lx 2 xdx E lx 2 dx ] dx
R e = 10 log 10 [ E inc x y E ref x y dxdy 2 E inc x y 2 dxdy 2 ]

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