Abstract

Graded-Index (GRIN) lenses with a diameter of 125 µm are presented. This diameter enables the assembly of the GRIN lenses onto an optical micro-system using the same passive alignment grooves as used for the light carrying optical fibers. In contrast to refractive lenses, GRIN lenses have flat endfaces and the focal distance of a GRIN lens is defined by its length. Therefore, GRIN lenses can be diced from a selected multimode optical fiber with a regular wafer dicing machine. The effects of the resulting surface roughness are reduced by immersing the optical parts into index matching oil, which can not be applied for refractive lenses. This has a further advantage since an anti-reflective coating becomes dispensable. The coupling efficiency of a collimator set-up using the GRIN lenses is studied using paraxial ray calculations. The calculated minimal coupling losses of less than 0.3 dB are in excellent agreement with the measured results. Losses smaller than 2 dB over a coupling length of 2 mm have been measured.

© 2006 Optical Society of America

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References

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  1. W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
    [Crossref]
  2. H. Toshiyoshi and H. Fujita, “Electrostatic Micro Torsion Mirrors for an Optical Switch Matrix,” IEEE Journal of Microelectromechanical Systems 5, 1996, 231–7.
    [Crossref]
  3. Corning SMF-28 product information.
  4. C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.
  5. Ph. Nussbaum, R. Völkel, H.P. Herzig, M. Eisner, and S. Haselback, “Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystem,” Pure Appl. Opt. 6, 1997, 617–636.
    [Crossref]
  6. W. R. Cox, C. Guan, and D. J. Hayes, “Microjet Printing of Micro-optical Interconnects and Sensors,” SPIE Phot. West Proc., V.3852, 2000, 400.
    [Crossref]
  7. J. Bähr, U. Krackhardt, and K.-H. Brenner, “Fabrication and Testing of planar Micro Lens Arrays by Ion Exchange Technique in Glass,” SPIE-Int. Soc. Opt. Eng. Proceedings of Spie 4455, 2001, 281–92.
  8. F. Zhu, Y. Sato, Y. Sasaki, and K. Hamanaka, “Fiber Collimator Arrays for Optical Devices/Systems,” Proc. SPIE 4564, 2001, 175–182.
    [Crossref]
  9. Corning Product Information: http://www.corning.com
  10. L Emkey William and A Jack Curtis, “Multimode fiber-lens optical coupler” 20.10.1987.
  11. Webpage: http://www.oki.com/en/press/2002/z02062e.html
  12. S. Yuan and A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt.38, No. 15, 1999.
  13. B Saleh and M Teich, Foundametals of photonics, “New York, Wiley, 1991”.
  14. International Patent Application: W. Noell, M. Zickar, C. Marxer, and N. de Rooij, “Optical Coupling Element and Method of Manufacturing the Optical Coupling Element,” PCT/IB2004/001611.
  15. M. Zickar, W. Noell, A. Oudalov, C. Marxer, and N. de Rooij, “4×4 & 8×8 Optical Matrix Switch with 250 µm Mirror Pitch,” OMEMS ‘04, Takamatsu, Japan, 162–163.

2002 (1)

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

2001 (2)

J. Bähr, U. Krackhardt, and K.-H. Brenner, “Fabrication and Testing of planar Micro Lens Arrays by Ion Exchange Technique in Glass,” SPIE-Int. Soc. Opt. Eng. Proceedings of Spie 4455, 2001, 281–92.

F. Zhu, Y. Sato, Y. Sasaki, and K. Hamanaka, “Fiber Collimator Arrays for Optical Devices/Systems,” Proc. SPIE 4564, 2001, 175–182.
[Crossref]

1997 (2)

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

Ph. Nussbaum, R. Völkel, H.P. Herzig, M. Eisner, and S. Haselback, “Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystem,” Pure Appl. Opt. 6, 1997, 617–636.
[Crossref]

1996 (1)

H. Toshiyoshi and H. Fujita, “Electrostatic Micro Torsion Mirrors for an Optical Switch Matrix,” IEEE Journal of Microelectromechanical Systems 5, 1996, 231–7.
[Crossref]

Anthamatten, O.

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

Bähr, J.

J. Bähr, U. Krackhardt, and K.-H. Brenner, “Fabrication and Testing of planar Micro Lens Arrays by Ion Exchange Technique in Glass,” SPIE-Int. Soc. Opt. Eng. Proceedings of Spie 4455, 2001, 281–92.

Bättig, R.

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

Brenner, K.-H.

J. Bähr, U. Krackhardt, and K.-H. Brenner, “Fabrication and Testing of planar Micro Lens Arrays by Ion Exchange Technique in Glass,” SPIE-Int. Soc. Opt. Eng. Proceedings of Spie 4455, 2001, 281–92.

Clerc, P-A.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

Cox, W. R.

W. R. Cox, C. Guan, and D. J. Hayes, “Microjet Printing of Micro-optical Interconnects and Sensors,” SPIE Phot. West Proc., V.3852, 2000, 400.
[Crossref]

Dändliker, R.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

de Rooij, N.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

International Patent Application: W. Noell, M. Zickar, C. Marxer, and N. de Rooij, “Optical Coupling Element and Method of Manufacturing the Optical Coupling Element,” PCT/IB2004/001611.

M. Zickar, W. Noell, A. Oudalov, C. Marxer, and N. de Rooij, “4×4 & 8×8 Optical Matrix Switch with 250 µm Mirror Pitch,” OMEMS ‘04, Takamatsu, Japan, 162–163.

Dellmann, L.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

Eisner, M.

Ph. Nussbaum, R. Völkel, H.P. Herzig, M. Eisner, and S. Haselback, “Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystem,” Pure Appl. Opt. 6, 1997, 617–636.
[Crossref]

Emkey William, L

L Emkey William and A Jack Curtis, “Multimode fiber-lens optical coupler” 20.10.1987.

Fujita, H.

H. Toshiyoshi and H. Fujita, “Electrostatic Micro Torsion Mirrors for an Optical Switch Matrix,” IEEE Journal of Microelectromechanical Systems 5, 1996, 231–7.
[Crossref]

Gretillat, M-A.

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

Guan, C.

W. R. Cox, C. Guan, and D. J. Hayes, “Microjet Printing of Micro-optical Interconnects and Sensors,” SPIE Phot. West Proc., V.3852, 2000, 400.
[Crossref]

Guldimann, B.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

Hamanaka, K.

F. Zhu, Y. Sato, Y. Sasaki, and K. Hamanaka, “Fiber Collimator Arrays for Optical Devices/Systems,” Proc. SPIE 4564, 2001, 175–182.
[Crossref]

Haselback, S.

Ph. Nussbaum, R. Völkel, H.P. Herzig, M. Eisner, and S. Haselback, “Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystem,” Pure Appl. Opt. 6, 1997, 617–636.
[Crossref]

Hayes, D. J.

W. R. Cox, C. Guan, and D. J. Hayes, “Microjet Printing of Micro-optical Interconnects and Sensors,” SPIE Phot. West Proc., V.3852, 2000, 400.
[Crossref]

Herzig, H.P.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

Ph. Nussbaum, R. Völkel, H.P. Herzig, M. Eisner, and S. Haselback, “Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystem,” Pure Appl. Opt. 6, 1997, 617–636.
[Crossref]

Jack Curtis, A

L Emkey William and A Jack Curtis, “Multimode fiber-lens optical coupler” 20.10.1987.

Krackhardt, U.

J. Bähr, U. Krackhardt, and K.-H. Brenner, “Fabrication and Testing of planar Micro Lens Arrays by Ion Exchange Technique in Glass,” SPIE-Int. Soc. Opt. Eng. Proceedings of Spie 4455, 2001, 281–92.

Manzardo, O.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

Marxer, C.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

International Patent Application: W. Noell, M. Zickar, C. Marxer, and N. de Rooij, “Optical Coupling Element and Method of Manufacturing the Optical Coupling Element,” PCT/IB2004/001611.

M. Zickar, W. Noell, A. Oudalov, C. Marxer, and N. de Rooij, “4×4 & 8×8 Optical Matrix Switch with 250 µm Mirror Pitch,” OMEMS ‘04, Takamatsu, Japan, 162–163.

Marxer et, C.

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

Noell, W.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

M. Zickar, W. Noell, A. Oudalov, C. Marxer, and N. de Rooij, “4×4 & 8×8 Optical Matrix Switch with 250 µm Mirror Pitch,” OMEMS ‘04, Takamatsu, Japan, 162–163.

International Patent Application: W. Noell, M. Zickar, C. Marxer, and N. de Rooij, “Optical Coupling Element and Method of Manufacturing the Optical Coupling Element,” PCT/IB2004/001611.

Nussbaum, Ph.

Ph. Nussbaum, R. Völkel, H.P. Herzig, M. Eisner, and S. Haselback, “Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystem,” Pure Appl. Opt. 6, 1997, 617–636.
[Crossref]

Oudalov, A.

M. Zickar, W. Noell, A. Oudalov, C. Marxer, and N. de Rooij, “4×4 & 8×8 Optical Matrix Switch with 250 µm Mirror Pitch,” OMEMS ‘04, Takamatsu, Japan, 162–163.

Riza, A.

S. Yuan and A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt.38, No. 15, 1999.

Saleh, B

B Saleh and M Teich, Foundametals of photonics, “New York, Wiley, 1991”.

Sasaki, Y.

F. Zhu, Y. Sato, Y. Sasaki, and K. Hamanaka, “Fiber Collimator Arrays for Optical Devices/Systems,” Proc. SPIE 4564, 2001, 175–182.
[Crossref]

Sato, Y.

F. Zhu, Y. Sato, Y. Sasaki, and K. Hamanaka, “Fiber Collimator Arrays for Optical Devices/Systems,” Proc. SPIE 4564, 2001, 175–182.
[Crossref]

Teich, M

B Saleh and M Teich, Foundametals of photonics, “New York, Wiley, 1991”.

Thio, C.

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

Toshiyoshi, H.

H. Toshiyoshi and H. Fujita, “Electrostatic Micro Torsion Mirrors for an Optical Switch Matrix,” IEEE Journal of Microelectromechanical Systems 5, 1996, 231–7.
[Crossref]

Valk, B.

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

Vogel, P.

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

Völkel, R.

Ph. Nussbaum, R. Völkel, H.P. Herzig, M. Eisner, and S. Haselback, “Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystem,” Pure Appl. Opt. 6, 1997, 617–636.
[Crossref]

Weible, K.

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

Yuan, S.

S. Yuan and A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt.38, No. 15, 1999.

Zhu, F.

F. Zhu, Y. Sato, Y. Sasaki, and K. Hamanaka, “Fiber Collimator Arrays for Optical Devices/Systems,” Proc. SPIE 4564, 2001, 175–182.
[Crossref]

Zickar, M.

International Patent Application: W. Noell, M. Zickar, C. Marxer, and N. de Rooij, “Optical Coupling Element and Method of Manufacturing the Optical Coupling Element,” PCT/IB2004/001611.

M. Zickar, W. Noell, A. Oudalov, C. Marxer, and N. de Rooij, “4×4 & 8×8 Optical Matrix Switch with 250 µm Mirror Pitch,” OMEMS ‘04, Takamatsu, Japan, 162–163.

IEEE J. Sel. Top. Quantum Electron. (1)

W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, and N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148–154.
[Crossref]

IEEE Journal of Microelectromechanical Systems (1)

H. Toshiyoshi and H. Fujita, “Electrostatic Micro Torsion Mirrors for an Optical Switch Matrix,” IEEE Journal of Microelectromechanical Systems 5, 1996, 231–7.
[Crossref]

JMEMS (1)

C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, and P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277–285.

Proc. SPIE (1)

F. Zhu, Y. Sato, Y. Sasaki, and K. Hamanaka, “Fiber Collimator Arrays for Optical Devices/Systems,” Proc. SPIE 4564, 2001, 175–182.
[Crossref]

Pure Appl. Opt. (1)

Ph. Nussbaum, R. Völkel, H.P. Herzig, M. Eisner, and S. Haselback, “Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystem,” Pure Appl. Opt. 6, 1997, 617–636.
[Crossref]

SPIE-Int. Soc. Opt. Eng. Proceedings of Spie (1)

J. Bähr, U. Krackhardt, and K.-H. Brenner, “Fabrication and Testing of planar Micro Lens Arrays by Ion Exchange Technique in Glass,” SPIE-Int. Soc. Opt. Eng. Proceedings of Spie 4455, 2001, 281–92.

Other (9)

W. R. Cox, C. Guan, and D. J. Hayes, “Microjet Printing of Micro-optical Interconnects and Sensors,” SPIE Phot. West Proc., V.3852, 2000, 400.
[Crossref]

Corning SMF-28 product information.

Corning Product Information: http://www.corning.com

L Emkey William and A Jack Curtis, “Multimode fiber-lens optical coupler” 20.10.1987.

Webpage: http://www.oki.com/en/press/2002/z02062e.html

S. Yuan and A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt.38, No. 15, 1999.

B Saleh and M Teich, Foundametals of photonics, “New York, Wiley, 1991”.

International Patent Application: W. Noell, M. Zickar, C. Marxer, and N. de Rooij, “Optical Coupling Element and Method of Manufacturing the Optical Coupling Element,” PCT/IB2004/001611.

M. Zickar, W. Noell, A. Oudalov, C. Marxer, and N. de Rooij, “4×4 & 8×8 Optical Matrix Switch with 250 µm Mirror Pitch,” OMEMS ‘04, Takamatsu, Japan, 162–163.

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Figures (14)

Fig. 1.
Fig. 1.

Trajectory of a ray and refractive index distribution of a step-index fiber (a) and a GRIN fiber with parabolic index distribution (b).

Fig. 2.
Fig. 2.

Schematic of the collimating setup. A light beam emerges from the SMF and is collimated by GRIN lens 1. GRIN lens 2 refocuses the beam into the other SMF.

Fig. 3.
Fig. 3.

GRIN lenses with a diameter of 125 µm passively aligned to an optical fiber ribbon and a MEMS mirror matrix: The microfabricated U-grooves and holding springs assure a perfect passive alignment between the micro-optical parts.

Fig. 4.
Fig. 4.

Measurement of the refractive index distribution of the selected MM fiber.

Fig. 5.
Fig. 5.

Calculation of the beam waist in a collimator setup. The light beam emerging from the SMF1 diverges and is collimated by GRIN lens 1.

Fig. 6.
Fig. 6.

Coupling loss as function of GRIN lens length and coupling distance: a) 200 µm, b) 250 µm, c) 300 µm, d) 350 µm. A larger gap between SM fiber and GRIN lens increases the coupling length and shorter GRIN lenses need to be taken.

Fig. 7.
Fig. 7.

Simulated effect of the gap width between SMF and GRIN lenses on the coupling loss. A large gap leads to a smaller overall loss which is due to the overall broadening of the beam waist. However, a small gap can result in a smaller loss for a fixed, short coupling length.

Fig. 8.
Fig. 8.

Coupling loss as function of GRIN lens length and coupling distance for GRIN fiber with core diameter of 110 µm: (a) NA=0.2, (b) NA=0.25, (c) NA=0.3 µm, (d) NA=0.3 with corrected gap SMF-GRIN. Coupling length of 8 mm with losses <3 dB are achievable. A smaller NA yields a larger length tolerance which facilitates the lens fabrication.

Fig. 9.
Fig. 9.

Fabrication sequence for GRIN lens rods: (a) dicing of the V-grooves, (b) gluing of the GRIN fiber pieces, (c) dicing of the GRIN lenses.

Fig. 10.
Fig. 10.

Surface roughness of the GRIN lens after dicing. The roughness was measured to be about 1 µm RMS.

Fig. 11.
Fig. 11.

(a) AFM image of a diced GRIN lens after dicing. (b) GRIN lens after etching for 60 minutes in BHF. The crater-like surface is probably caused by micro-cracks in the glass after dicing.

Fig. 12.
Fig. 12.

GRIN lens before (left) and after (right) mechanical polishing.

Fig. 13.
Fig. 13.

Comparison between simulations and measurements of the GRIN lenses in index matching oil: (a) Simulation of the losses with a gap size of 250 µm, (b) measured losses of the collimation setup with gap size of 250 µm.

Fig. 14.
Fig. 14.

Comparison of the losses for two different GRIN lens and gap lengths. Even though the beam diameter exceeds the GRIN lens core the setup yields low losses over a large coupling length.

Tables (1)

Tables Icon

Table 1. Summarized parameters of the selected MM fiber.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

n ( r ) = n 0 2 ( 1 α 2 r 2 )
f 1 n 0 α · sin ( α d ) .
( x out u out ) = ( A B C D ) ( x in u in ) .
( A B C D ) = ( 1 d 0 1 )
( A B C D ) = ( 1 0 0 n 1 n 2 )
( A B C D ) = ( cos ( α Z ) 1 α sin ( α Z ) α sin ( α Z ) cos ( α Z ) )
α NA n 0 r core
q j + 1 = A q j + B C q j + D
η = E i ( x , y ) E f * ( x , y ) dx dy 2 E i ( x , y ) E i * ( x , y ) dx dy E f ( x , y ) E f * ( x , y ) dx dy

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