Abstract

A fiber-optic collimator that emits a Gaussian beam with its beam waist at a certain distance after the exit face of the lens is labeled a self-imaging collimator. For such a collimator, the waist of the emitted Gaussian beam and its location are partly dependent on the properties of the gradient-index (GRIN) lens. Parameters for the self-imaging collimator are formulated in terms of the parameters of a GRIN lens (e.g., pitch, core refractive index, gradient index, length) and the optical wavelength. Next, by use of the Gaussian beam approximation, a general expression for the coupling power loss between two self-imaging-type single-mode fiber (SMF) collimators is, for the first time to our knowledge, derived as a function of three types of misalignment, namely, separation, lateral offset, and angular tilt misalignment. A coupling experiment between two self-imaging collimators with changing separation distance is successfully performed and matches the proposed self-imaging mechanism coupling loss theory. In addition, using a prism, lateral offset, as well as angular tilt, misalignments are experimentally simulated for a two self-imaging collimator coupling condition by a single collimator reflective test geometry. Experimental results agree well with the proposed loss formulas for self-imaging GRIN lenses. Hence, for the first time to our knowledge, the mathematical foundations are laid for employing self-imaging-type fiber collimators in SMF-based free-space systems allowing optimal design for ultra-low-loss coupling.

© 2003 Optical Society of America

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References

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  1. R. März, Integrated Optics: Design and Modeling (Artech House, Norwood, Mass., 1995).
  2. M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
    [CrossRef]
  3. W. J. Tomlinson, “Application of GRIN-rod lenses in optical fiber communication systems,” Appl. Opt. 19, 1127–1138 (1980).
    [CrossRef]
  4. N. A. Riza, S. Yuan, “Low optical interchannel crosstalk, fast switching time, polarization independent 2 × 2 fiber optic switch using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342 (1998).
    [CrossRef]
  5. N. Madamopoulus, N. A. Riza, “Directly modulated semiconductor-laser fed photonic delay line with ferroelectric liquid crystals,” Appl. Opt. 37, 1407–1416 (1998).
    [CrossRef]
  6. S. Yuan, N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt. 38, 3214–3222 (1999).
    [CrossRef]
  7. S. Yuan, N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses: errata,” Appl. Opt. 38, 6292 (1999).
    [CrossRef]
  8. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997).
  9. N. A. Riza, N. Madamopoulos, “High signal-to-noise ratio birefringence-compensated optical delay line based on a noise-reduction scheme,” Opt. Lett. 20, 2351–2353 (1995).
    [CrossRef] [PubMed]
  10. N. Madamopoulos, N. A. Riza, “Reversible fiber-optic switched delay module using GRIN lens fiber-optic collimators and ferroelectric liquid crystals,” in Proceedings of the IEEE-LEOS Annual Meeting (Institute of Electrical and Electronic Engineers, New York, 1998), pp. 275–276.
  11. N. Madamopoulos, N. A. Riza, “All-fiber connectorized compact fiber optic delay-line modulus using three-dimensional polarization optics,” Opt. Eng. 39, 2338–2344 (2000).
    [CrossRef]
  12. W. T. Silfvast, Laser Fundamentals (Cambridge U. Press, Cambridge, UK, 1996).
  13. J. C. Palais, “Fiber coupling using graded-index rod lenses,” Appl. Opt. 19, 2011–2018 (1980).
    [CrossRef] [PubMed]
  14. F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, New York, 1996), p. 70.
  15. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 140.
  16. A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).
  17. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [CrossRef]
  18. H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, J. Fox, ed., Vol. 14 of Polytechnic Institute Microwave Research Institute Symposia Series (Polytechnic Brooklyn, Brooklyn, N.Y., 1964), pp. 335–347.
  19. S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
    [CrossRef]
  20. R. W. Gilsdorf, J. C. Palais, “Single-mode fiber coupling efficiency with graded-index rod lenses,” Appl. Opt. 33, 3440–3445 (1994).
    [CrossRef] [PubMed]
  21. W. J. Smith, Modern Optical Engineering: The Design of Optical Systems (McGraw-Hill, New York, 1990).
  22. LightPath Technologies, Product Information CD, Collimators Catalogue, 2603 Challenger Tech Ct., Orlando, Fla. 32826, 2001.
  23. M. J. Mughal, N. A. Riza, “Compact acousto-optic high speed variable attenuator for high power applications,” IEEE Photon. Technol. Lett. 14, 510–512 (2002).
    [CrossRef]
  24. N. A. Riza, R. Akbar, S. Sumriddetchkajorn, F. Perez, M. J. Mughal, “47 dB dynamic range sub-microsecond switching speed variable fiber-optic attenuator for fast transient fiber-optics,” in Photonics in Switching, Vol. 59 of 2001 OSA Trends in Optics and Photonics Series Postconference Digest (Optical Society of America, Washington, D.C., 2001), paper PDP2.
  25. M. J. Mughal, N. A. Riza, “65 dB dynamic range 2.8 microseconds switching speed variable fiber-optic attenuator,” in the Twenty-Seventh European Conference on Optical Communication ECOC’01 (Institute of Electrical and Electronic Engineers, New York, 2001), Vol. 6, pp. 56, 57.

2002 (1)

M. J. Mughal, N. A. Riza, “Compact acousto-optic high speed variable attenuator for high power applications,” IEEE Photon. Technol. Lett. 14, 510–512 (2002).
[CrossRef]

2000 (1)

N. Madamopoulos, N. A. Riza, “All-fiber connectorized compact fiber optic delay-line modulus using three-dimensional polarization optics,” Opt. Eng. 39, 2338–2344 (2000).
[CrossRef]

1999 (2)

1998 (2)

N. Madamopoulus, N. A. Riza, “Directly modulated semiconductor-laser fed photonic delay line with ferroelectric liquid crystals,” Appl. Opt. 37, 1407–1416 (1998).
[CrossRef]

N. A. Riza, S. Yuan, “Low optical interchannel crosstalk, fast switching time, polarization independent 2 × 2 fiber optic switch using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342 (1998).
[CrossRef]

1997 (1)

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

1995 (1)

1994 (1)

1980 (2)

1979 (1)

S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

1977 (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Akbar, R.

N. A. Riza, R. Akbar, S. Sumriddetchkajorn, F. Perez, M. J. Mughal, “47 dB dynamic range sub-microsecond switching speed variable fiber-optic attenuator for fast transient fiber-optics,” in Photonics in Switching, Vol. 59 of 2001 OSA Trends in Optics and Photonics Series Postconference Digest (Optical Society of America, Washington, D.C., 2001), paper PDP2.

Borella, M. S.

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Gilsdorf, R. W.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 140.

Jue, J. P.

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, J. Fox, ed., Vol. 14 of Polytechnic Institute Microwave Research Institute Symposia Series (Polytechnic Brooklyn, Brooklyn, N.Y., 1964), pp. 335–347.

Madamopoulos, N.

N. Madamopoulos, N. A. Riza, “All-fiber connectorized compact fiber optic delay-line modulus using three-dimensional polarization optics,” Opt. Eng. 39, 2338–2344 (2000).
[CrossRef]

N. A. Riza, N. Madamopoulos, “High signal-to-noise ratio birefringence-compensated optical delay line based on a noise-reduction scheme,” Opt. Lett. 20, 2351–2353 (1995).
[CrossRef] [PubMed]

N. Madamopoulos, N. A. Riza, “Reversible fiber-optic switched delay module using GRIN lens fiber-optic collimators and ferroelectric liquid crystals,” in Proceedings of the IEEE-LEOS Annual Meeting (Institute of Electrical and Electronic Engineers, New York, 1998), pp. 275–276.

Madamopoulus, N.

Makimoto, T.

S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

März, R.

R. März, Integrated Optics: Design and Modeling (Artech House, Norwood, Mass., 1995).

Mughal, M. J.

M. J. Mughal, N. A. Riza, “Compact acousto-optic high speed variable attenuator for high power applications,” IEEE Photon. Technol. Lett. 14, 510–512 (2002).
[CrossRef]

N. A. Riza, R. Akbar, S. Sumriddetchkajorn, F. Perez, M. J. Mughal, “47 dB dynamic range sub-microsecond switching speed variable fiber-optic attenuator for fast transient fiber-optics,” in Photonics in Switching, Vol. 59 of 2001 OSA Trends in Optics and Photonics Series Postconference Digest (Optical Society of America, Washington, D.C., 2001), paper PDP2.

M. J. Mughal, N. A. Riza, “65 dB dynamic range 2.8 microseconds switching speed variable fiber-optic attenuator,” in the Twenty-Seventh European Conference on Optical Communication ECOC’01 (Institute of Electrical and Electronic Engineers, New York, 2001), Vol. 6, pp. 56, 57.

Mukherjee, B.

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Nemoto, S.

S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Palais, J. C.

Pedrotti, F. L.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, New York, 1996), p. 70.

Pedrotti, L. S.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, New York, 1996), p. 70.

Perez, F.

N. A. Riza, R. Akbar, S. Sumriddetchkajorn, F. Perez, M. J. Mughal, “47 dB dynamic range sub-microsecond switching speed variable fiber-optic attenuator for fast transient fiber-optics,” in Photonics in Switching, Vol. 59 of 2001 OSA Trends in Optics and Photonics Series Postconference Digest (Optical Society of America, Washington, D.C., 2001), paper PDP2.

Ramamurthy, B.

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Riza, N. A.

M. J. Mughal, N. A. Riza, “Compact acousto-optic high speed variable attenuator for high power applications,” IEEE Photon. Technol. Lett. 14, 510–512 (2002).
[CrossRef]

N. Madamopoulos, N. A. Riza, “All-fiber connectorized compact fiber optic delay-line modulus using three-dimensional polarization optics,” Opt. Eng. 39, 2338–2344 (2000).
[CrossRef]

S. Yuan, N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses: errata,” Appl. Opt. 38, 6292 (1999).
[CrossRef]

S. Yuan, N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt. 38, 3214–3222 (1999).
[CrossRef]

N. Madamopoulus, N. A. Riza, “Directly modulated semiconductor-laser fed photonic delay line with ferroelectric liquid crystals,” Appl. Opt. 37, 1407–1416 (1998).
[CrossRef]

N. A. Riza, S. Yuan, “Low optical interchannel crosstalk, fast switching time, polarization independent 2 × 2 fiber optic switch using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342 (1998).
[CrossRef]

N. A. Riza, N. Madamopoulos, “High signal-to-noise ratio birefringence-compensated optical delay line based on a noise-reduction scheme,” Opt. Lett. 20, 2351–2353 (1995).
[CrossRef] [PubMed]

N. A. Riza, R. Akbar, S. Sumriddetchkajorn, F. Perez, M. J. Mughal, “47 dB dynamic range sub-microsecond switching speed variable fiber-optic attenuator for fast transient fiber-optics,” in Photonics in Switching, Vol. 59 of 2001 OSA Trends in Optics and Photonics Series Postconference Digest (Optical Society of America, Washington, D.C., 2001), paper PDP2.

M. J. Mughal, N. A. Riza, “65 dB dynamic range 2.8 microseconds switching speed variable fiber-optic attenuator,” in the Twenty-Seventh European Conference on Optical Communication ECOC’01 (Institute of Electrical and Electronic Engineers, New York, 2001), Vol. 6, pp. 56, 57.

N. Madamopoulos, N. A. Riza, “Reversible fiber-optic switched delay module using GRIN lens fiber-optic collimators and ferroelectric liquid crystals,” in Proceedings of the IEEE-LEOS Annual Meeting (Institute of Electrical and Electronic Engineers, New York, 1998), pp. 275–276.

Silfvast, W. T.

W. T. Silfvast, Laser Fundamentals (Cambridge U. Press, Cambridge, UK, 1996).

Smith, W. J.

W. J. Smith, Modern Optical Engineering: The Design of Optical Systems (McGraw-Hill, New York, 1990).

Sumriddetchkajorn, S.

N. A. Riza, R. Akbar, S. Sumriddetchkajorn, F. Perez, M. J. Mughal, “47 dB dynamic range sub-microsecond switching speed variable fiber-optic attenuator for fast transient fiber-optics,” in Photonics in Switching, Vol. 59 of 2001 OSA Trends in Optics and Photonics Series Postconference Digest (Optical Society of America, Washington, D.C., 2001), paper PDP2.

Tomlinson, W. J.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

Yuan, S.

Appl. Opt. (6)

Bell Syst. Tech. J. (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Electron. Lett. (1)

N. A. Riza, S. Yuan, “Low optical interchannel crosstalk, fast switching time, polarization independent 2 × 2 fiber optic switch using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. J. Mughal, N. A. Riza, “Compact acousto-optic high speed variable attenuator for high power applications,” IEEE Photon. Technol. Lett. 14, 510–512 (2002).
[CrossRef]

Opt. Eng. (1)

N. Madamopoulos, N. A. Riza, “All-fiber connectorized compact fiber optic delay-line modulus using three-dimensional polarization optics,” Opt. Eng. 39, 2338–2344 (2000).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Proc. IEEE (1)

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Other (12)

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997).

N. Madamopoulos, N. A. Riza, “Reversible fiber-optic switched delay module using GRIN lens fiber-optic collimators and ferroelectric liquid crystals,” in Proceedings of the IEEE-LEOS Annual Meeting (Institute of Electrical and Electronic Engineers, New York, 1998), pp. 275–276.

R. März, Integrated Optics: Design and Modeling (Artech House, Norwood, Mass., 1995).

W. T. Silfvast, Laser Fundamentals (Cambridge U. Press, Cambridge, UK, 1996).

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, New York, 1996), p. 70.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 140.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

W. J. Smith, Modern Optical Engineering: The Design of Optical Systems (McGraw-Hill, New York, 1990).

LightPath Technologies, Product Information CD, Collimators Catalogue, 2603 Challenger Tech Ct., Orlando, Fla. 32826, 2001.

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, J. Fox, ed., Vol. 14 of Polytechnic Institute Microwave Research Institute Symposia Series (Polytechnic Brooklyn, Brooklyn, N.Y., 1964), pp. 335–347.

N. A. Riza, R. Akbar, S. Sumriddetchkajorn, F. Perez, M. J. Mughal, “47 dB dynamic range sub-microsecond switching speed variable fiber-optic attenuator for fast transient fiber-optics,” in Photonics in Switching, Vol. 59 of 2001 OSA Trends in Optics and Photonics Series Postconference Digest (Optical Society of America, Washington, D.C., 2001), paper PDP2.

M. J. Mughal, N. A. Riza, “65 dB dynamic range 2.8 microseconds switching speed variable fiber-optic attenuator,” in the Twenty-Seventh European Conference on Optical Communication ECOC’01 (Institute of Electrical and Electronic Engineers, New York, 2001), Vol. 6, pp. 56, 57.

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Figures (20)

Fig. 1
Fig. 1

Propagation of the Gaussian beam through a GRIN lens. The solid curves represent the beam width of the Gaussian beam. The complex radii of curvature at the end of the SMF, at the left- and right-hand side of the GRIN lens, and at the location of the beam waist are denoted as q 0, q 1, q 2, and q 3, respectively.

Fig. 2
Fig. 2

Distance d between the edge of the GRIN lens and the beam waist as a function of the pitch of the GRIN lens according to Eq. (9). For the gap L between the SMF and the GRIN lens a value of 1 μm was taken. For a pitch of approximately 0.26, the beam-waist distance has its maximum value of 40 mm.

Fig. 3
Fig. 3

SMF directly coupled to a quarter-pitch lens and the beam width of the Gaussian beam. The beam waist is located at the edge (called the exit face) of the GRIN lens.

Fig. 4
Fig. 4

General case in which the beam waist does not coincide with the edge of the GRIN lens. The beam waist is located at z = d 1.

Fig. 5
Fig. 5

GRIN lens 2 in coordinate frame 2. The location of the beam waist is z′ = d 1.

Fig. 6
Fig. 6

Three types of misalignment: (a) separation misalignment (Z 0), (b) lateral offset misalignment (X 0), (c) angular tilt misalignment (θ). d 1 = d 2 = 100 mm, w T = w R = 0.5 mm, λ = 1550 nm.

Fig. 7
Fig. 7

Relationship between the coupling loss and the separation misalignment, where there exists a mirror symmetry in which Z 0 = 2d 1. For a separation distance of Z 0 = 2d 1, there is theoretically no loss. Lateral offset and angular tilt misalignments are assumed to be nonexistent.

Fig. 8
Fig. 8

GRIN-GRIN coupling case in which we combined the three types of misalignment, i.e., separation misalignment (Z 0), lateral offset misalignment (X 0), and angular tilt misalignment (θ).

Fig. 9
Fig. 9

Coupling loss as a function of the lateral offset misalignment for different values of the separation misalignment with no angular tilt misalignment. The plots belonging to Z 0 = d 1 and Z 0 = 3d 1 coincide. Also the plots belonging to Z 0 = 0 and Z 0 = 3d 1 are exactly the same. The loss equals zero when Z 0 = 2d 1 and X 0 = 0.

Fig. 10
Fig. 10

Loss as a function of the angular tilt misalignment for different values of the separation misalignment. The lateral offset misalignment is fixed at zero. When there is no angular tilt misalignment, the loss for both the pairs Z 0 = d 1 and Z 0 = 3d 1 as well Z 0 = 0 and Z 0 = 4d 1 is the same. There is no loss when Z 0 = 2d 1, and the angular tilt misalignment is equal to zero.

Fig. 11
Fig. 11

Overview of the mirror experiment setup. A circulator is used to send the light from the source toward the GRIN lens and the reflected light toward the detector. This detector is connected to a powermeter that measures the reflected power. i/o, in/out.

Fig. 12
Fig. 12

Coupling loss result of the mirror experiment when we adjust the mirror for GRIN lens 1. From these results it follows that the beam-waist distance is equal to 4 cm.

Fig. 13
Fig. 13

Coupling loss result of the mirror experiment with no adjustment to the mirror. According to this experiment, GRIN lens 1 has a beam-waist distance of 4 cm.

Fig. 14
Fig. 14

(a) For a large separation distance a certain misalignment θ of the mirror leads to a small overlap area or a large loss. (b) For a smaller separation distance the same misalignment θ results in a larger overlap, thus a smaller loss.

Fig. 15
Fig. 15

Results for the razor blade experiment for GRIN lens 1. For GRIN lens 1 the beam waist is 0.479 mm.

Fig. 16
Fig. 16

Comparison of the experimental data with our theoretical result for separation coupling loss between two GRIN lenses. The theoretical curve was shifted 0.35 dB upward to make a better comparison. The experiment shows good agreement with theory.

Fig. 17
Fig. 17

Prism is reflecting a light beam (a) without offset and (b) with offset. When there is no lateral offset, the reflected beam exactly coincides with the incoming beam. The offset of the reflected beam is twice as large as the offset of the prism relative to the incoming beam.

Fig. 18
Fig. 18

Comparison of the experimental data with the theoretical results for the normalized coupling loss that is due to lateral offset for a separation distance of (a) 8 cm, (b) 110 cm, and (c) 150 cm. The experimental data match the theory.

Fig. 19
Fig. 19

Geometry of the angular tilt setup. By placing a convex lens exactly at its focal distance f after the GRIN lens, we can focus the reflected Gaussian beam on the exit face of the GRIN lens under a certain angle θ. The distance L between the convex lens and the prism can be adjusted to vary the imaginary separation distance Z 0.

Fig. 20
Fig. 20

Comparison of the experimental data with the theoretical results for the normalized coupling loss that is due to angular tilt for (a) L = 70 cm and (b) L = 15 cm.

Equations (87)

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

nr=n01-Ar22,
Mgap=1L01.
MGRIN=cosAZ1n0AsinAZ-n0AsinAZcosAZ.
Md=1d01.
q0=i πnw02λ=iz0.
w0a0.65+1.619V-3/2+2.879V-6, 1.2<V<2.405.
q3=αγ+βδγ2+δ2+d+i βγ-αδγ2+δ2,
α=L cosAZ+1n0AsinAZ,
β=z0 cosAZ,
γ=-n0AL sinAZ+cosAZ,
δ=-n0Az0 sinAZ.
Req3=0
d=-αγ+βδγ2+δ2
w=wT=λ Imq3πn1/2=λ βγ-αδγ2+δ2πn1/2.
E˜xx, y, z=E1wTwzexp-ikz-ηz-r21w2z+i k2Rz,
k=2πnλ,
ηz=tan-1λzπnwT2,
w2z=wT21+λzπnwT22,
Rz=z1+πnwT2λz2.
Exx, y, z=E˜xx, y, z-d1=E1wTwz-d1exp-ikz-d1-ηz-d1-r21w2z-d1+i k2Rz-d1.
Exx, y, z=E˜x, y, z+d1=E1wTwz+d1exp-ikz+d1-ηz+d1-r21w2z+d1+i k2Rz+d1,
ηc=2πE12wT2  Exx, y, z|z=0×Ex*x, y, z|z=0dxdy.
x=x, y=y, z=z+Z0, r=r.
Exx, y, z|z=0=E1wTwZ0-d1exp-ikZ0-d1-ηZ0-d1-r21w2Z0-d1+i k2RZ0-d1.
 exp-αr2dxdy=πα,
ηc=2 expik2d1-Z0+ηZ0-d1-ηd1wZ0-d1wd11w2Z0-d1+1w2d1+ik21RZ0-d1-1Rd1.
T=4w2d1w2Z0-d1+2+w2Z0-d1w2d1+w2Z0-d1w2d1k241RZ0-d1-1Rd12.
x=x cos θ-z sin θ+X0,
z=x sin θ+z cos θ+Z0,
y=y,
r2=x2+y2=x cos θ-z sin θ+X02+y2.
w12x sin θ+Z0-d1w12Z0-d1,
R1x sin θ+Z0-d1R1Z0-d1,
η1x sin θ+Z0-d1η1Z0-d1.
Exx, y, z|z=0E1wTw1Z0-d1exp-ikx sin θ+Z0-d1-η1Z0-d1×exp-x+X02×1w12Z0-d1+i k2R1Z0-d1.
Exx, y, z=E1wTw2z+d2exp-ikz+d2-η2z+d2-r21w22z+d2+i k2R2z+d2.
η2z=tan-1λzπnwR2,
w22z=wR21+λzπnwR22,
R2z=z1+πnwR2λz2.
-+exp-αx2+βx+γdx=παexpβ2-4αγ4α,
ηc=2w1Z0-d1w2d2Fexpiψ1expG2-4FH4F.
F=Fr+iFi=1w12Z0-d1+1w22d2+i k21R1Z0-d1-1R2d2,
G=Gr+iGi=2X0w12Z0-d1+ikX0R1Z0-d1+sin θ,
H=Hr+iHi=X02w12Z0-d1+i kX022R1Z0-d1,
ψ1=kd1+d2-Z0+η1Z0-d1-η2d2.
J=ReG2-4FH4F=FrGr2-Gi2-4FrHr+Fi2GrGi-4FiHr4Fr2+Fi2,
ψ2=ImG2-4FH4F=-FiGr2-Gi2+4FiHi+Fr2GrGi-4FrHi4Fr2+Fi2.
ηc=2w1Z0-d1w2d2Fexpiψ1+ψ2expJ.
L=-10 log4 exp2Jw12Z0-d1w22d2|F|2.
Loffset=LZ0, X0, θ=0-LZ0, X0=0, θ=0=AoffsetX02,
Aoffset=-5ln 101|F|2Fr4w14Z0-d1-k2R12Z0-d1-4Frw12Z0-d1+Fi4w12Z0-d1kR1Z0-d1-Fi.
Ltilt=LZ0, X0=0, θ-LZ0, X0=0, θ=0=Atilt sin2 θAtiltθ2,
Atilt=5Frk2ln 10|F|2.
L=-10 log4wRwT+wTwR2.
L=-10 logPmP0,
θ=tan-12X0f
Ybw=-B0D+A0Cz02B1D+A1Cz02,
wR=λπnD2+C2z02A1Ybw+A0D-B1Ybw+B0Cz021/2,
A1=2fLf-1,
A0=1-2Lf,
B1=1-2X+Lf+2LXf2,
B0=X+2L-2LXf,
C=2fLf-1,
D=1-2X+Lf+2LXf2,
z0=πnwT2λ.
Mk=AkBkCkDk
qk+1=Akqk+BkCkqk+Dk.
q0=iz0,
z0=πnwT2λ.
M1=1X01.
M2=10-1f1
M3=12L01
M4=1Y01
M5=M4 · M2 · M3 · M2 · M1=ABCD,
A=2fLf-1Y+1-2Lf,
B=1-2X+Lf+2LXf2Y+X+2L-2LXf,
C=2fLf-1,
D=1-2X+Lf+2LXf2.
q1=BD+ACz02D2+C2z02+i AD-BCz02D2+C2z02.
1q=1R-i λπnw2,
Req1=B1Y+B0D+A1Y+A0Cz02D2+C2z02=0,
A1=2fLf-1,
A0=1-2Lf,
B1=1-2X+Lf+2LXf2,
B0=X+2L-2LXf.
Y=Ybw=-B0D+A0Cz02B1D+A1Cz02.
w=wR=-λπn Im1/q11/2=λπnD2+C2z02A1Ybw+A0D-B1Ybw+B0Cz021/2.

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