Abstract

A model of the reflected fringe system for an ideal plane-parallel, low-finesse Fabry–Perot (FP) cavity illuminated by a multimode optical fiber has been developed and experimentally validated. This showed that the phase dispersion within the cavity arising from the divergent nature of the incident illumination significantly degrades the visibility of the reflected fringes. Departures from the ideal FP cavity are also examined. The effect on fringe visibility when the plane of the FP cavity is tilted with respect to the fiber axis and when the cavity surfaces are no longer perfectly parallel to each other has been explored. The analysis described is relevant to the design and the optimization of multimode optical-fiber sensors that use FP sensing cavities.

© 1999 Optical Society of America

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References

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  1. T. Yoshino, K. Kurosawa, K. Itoh, T. Ose, “Fiber-optic Fabry–Perot interferometer and its sensor applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1612–1621 (1982).
    [CrossRef]
  2. R. A. Wolthius, G. L. Mitchell, E. Saaski, J. C. Hartl, M. A. Afromowitz, “Development of medical pressure and temperature sensors employing optical spectrum modulation,” IEEE Trans. Biomed. Eng. 38, 974–981 (1991).
    [CrossRef]
  3. A. K. Murphy, M. F. Gunter, A. M. Vengsarker, R. O. Claus, “Quadrature phase-shifted, extrinsic Fabry–Perot optical fiber sensors,” Opt. Lett. 16, 273–275 (1991).
    [CrossRef] [PubMed]
  4. V. Arya, M. J. de Vries, K. A. Murphy, A. Wang, R. O. Claus, “Exact analysis of the EFPI optical fiber sensor using Kirchhoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
    [CrossRef]
  5. E. R. Cox, B. E. Jones, “Fiber optic colour sensors based on Fabry–Perot interferometry,” in Proceedings of the First International Conference on Optical Fiber Sensors (Optical Society of America, Washington, D.C., 1983), pp. 122–126.
  6. P. C. Beard, T. N. Mills, “A miniature optical fibre ultrasonic hydrophone using a Fabry–Perot polymer film interferometer,” Electron. Lett. 33, 801–803 (1997).
    [CrossRef]
  7. A. J. Coleman, E. Draguioti, R. Tiptaf, N. Shotri, J. E. Saunders, “Acoustic performance and clinical use of a fibreoptic hydrophone,” Ultrasound Med. Biol. 24, 143–151 (1998).
    [CrossRef] [PubMed]
  8. P. C. Beard, T. N. Mills, “Extrinsic optical fiber ultrasound sensor with a thin polymer film as a low-finesse Fabry–Perot interferometer,” Appl. Opt. 35, 663–675 (1996).
    [CrossRef] [PubMed]
  9. P. C. Beard, F. Pérennès, E. Draguioti, T. N. Mills, “Optical fiber photoacoustic–photothermal probe,” Opt. Lett. 23, 1235–1237 (1998).
    [CrossRef]
  10. T. Liu, D. Brooks, A. Martin, R. A. Badcock, B. Ralph, G. F. Fernando, “A multimode fibre based extrinsic Fabry–Perot interferometric strain sensor,” J. Smart Mater. Structures 6, 464–469 (1997).
    [CrossRef]
  11. V. Arya, M. J. de Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the performance of imperfect fiber endfaces on the performance of extrinsic Fabry–Perot interferometric optical fiber sensors,” Opt. Eng. 35, 2262–2265 (1996).
    [CrossRef]
  12. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), pp. 364–366.
  13. J. L. Santos, A. P. Leite, D. A. Jackson, “Optical fiber sensing with a low-finesse Fabry–Perot cavity,” Appl. Opt. 31, 7361–7366 (1992).
    [CrossRef] [PubMed]

1998

A. J. Coleman, E. Draguioti, R. Tiptaf, N. Shotri, J. E. Saunders, “Acoustic performance and clinical use of a fibreoptic hydrophone,” Ultrasound Med. Biol. 24, 143–151 (1998).
[CrossRef] [PubMed]

P. C. Beard, F. Pérennès, E. Draguioti, T. N. Mills, “Optical fiber photoacoustic–photothermal probe,” Opt. Lett. 23, 1235–1237 (1998).
[CrossRef]

1997

T. Liu, D. Brooks, A. Martin, R. A. Badcock, B. Ralph, G. F. Fernando, “A multimode fibre based extrinsic Fabry–Perot interferometric strain sensor,” J. Smart Mater. Structures 6, 464–469 (1997).
[CrossRef]

P. C. Beard, T. N. Mills, “A miniature optical fibre ultrasonic hydrophone using a Fabry–Perot polymer film interferometer,” Electron. Lett. 33, 801–803 (1997).
[CrossRef]

1996

V. Arya, M. J. de Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the performance of imperfect fiber endfaces on the performance of extrinsic Fabry–Perot interferometric optical fiber sensors,” Opt. Eng. 35, 2262–2265 (1996).
[CrossRef]

P. C. Beard, T. N. Mills, “Extrinsic optical fiber ultrasound sensor with a thin polymer film as a low-finesse Fabry–Perot interferometer,” Appl. Opt. 35, 663–675 (1996).
[CrossRef] [PubMed]

1995

V. Arya, M. J. de Vries, K. A. Murphy, A. Wang, R. O. Claus, “Exact analysis of the EFPI optical fiber sensor using Kirchhoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

1992

1991

A. K. Murphy, M. F. Gunter, A. M. Vengsarker, R. O. Claus, “Quadrature phase-shifted, extrinsic Fabry–Perot optical fiber sensors,” Opt. Lett. 16, 273–275 (1991).
[CrossRef] [PubMed]

R. A. Wolthius, G. L. Mitchell, E. Saaski, J. C. Hartl, M. A. Afromowitz, “Development of medical pressure and temperature sensors employing optical spectrum modulation,” IEEE Trans. Biomed. Eng. 38, 974–981 (1991).
[CrossRef]

1982

T. Yoshino, K. Kurosawa, K. Itoh, T. Ose, “Fiber-optic Fabry–Perot interferometer and its sensor applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1612–1621 (1982).
[CrossRef]

Afromowitz, M. A.

R. A. Wolthius, G. L. Mitchell, E. Saaski, J. C. Hartl, M. A. Afromowitz, “Development of medical pressure and temperature sensors employing optical spectrum modulation,” IEEE Trans. Biomed. Eng. 38, 974–981 (1991).
[CrossRef]

Arya, V.

V. Arya, M. J. de Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the performance of imperfect fiber endfaces on the performance of extrinsic Fabry–Perot interferometric optical fiber sensors,” Opt. Eng. 35, 2262–2265 (1996).
[CrossRef]

V. Arya, M. J. de Vries, K. A. Murphy, A. Wang, R. O. Claus, “Exact analysis of the EFPI optical fiber sensor using Kirchhoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Athreya, M.

V. Arya, M. J. de Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the performance of imperfect fiber endfaces on the performance of extrinsic Fabry–Perot interferometric optical fiber sensors,” Opt. Eng. 35, 2262–2265 (1996).
[CrossRef]

Badcock, R. A.

T. Liu, D. Brooks, A. Martin, R. A. Badcock, B. Ralph, G. F. Fernando, “A multimode fibre based extrinsic Fabry–Perot interferometric strain sensor,” J. Smart Mater. Structures 6, 464–469 (1997).
[CrossRef]

Beard, P. C.

Brooks, D.

T. Liu, D. Brooks, A. Martin, R. A. Badcock, B. Ralph, G. F. Fernando, “A multimode fibre based extrinsic Fabry–Perot interferometric strain sensor,” J. Smart Mater. Structures 6, 464–469 (1997).
[CrossRef]

Claus, R. O.

V. Arya, M. J. de Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the performance of imperfect fiber endfaces on the performance of extrinsic Fabry–Perot interferometric optical fiber sensors,” Opt. Eng. 35, 2262–2265 (1996).
[CrossRef]

V. Arya, M. J. de Vries, K. A. Murphy, A. Wang, R. O. Claus, “Exact analysis of the EFPI optical fiber sensor using Kirchhoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

A. K. Murphy, M. F. Gunter, A. M. Vengsarker, R. O. Claus, “Quadrature phase-shifted, extrinsic Fabry–Perot optical fiber sensors,” Opt. Lett. 16, 273–275 (1991).
[CrossRef] [PubMed]

Coleman, A. J.

A. J. Coleman, E. Draguioti, R. Tiptaf, N. Shotri, J. E. Saunders, “Acoustic performance and clinical use of a fibreoptic hydrophone,” Ultrasound Med. Biol. 24, 143–151 (1998).
[CrossRef] [PubMed]

Cox, E. R.

E. R. Cox, B. E. Jones, “Fiber optic colour sensors based on Fabry–Perot interferometry,” in Proceedings of the First International Conference on Optical Fiber Sensors (Optical Society of America, Washington, D.C., 1983), pp. 122–126.

de Vries, M. J.

V. Arya, M. J. de Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the performance of imperfect fiber endfaces on the performance of extrinsic Fabry–Perot interferometric optical fiber sensors,” Opt. Eng. 35, 2262–2265 (1996).
[CrossRef]

V. Arya, M. J. de Vries, K. A. Murphy, A. Wang, R. O. Claus, “Exact analysis of the EFPI optical fiber sensor using Kirchhoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Draguioti, E.

P. C. Beard, F. Pérennès, E. Draguioti, T. N. Mills, “Optical fiber photoacoustic–photothermal probe,” Opt. Lett. 23, 1235–1237 (1998).
[CrossRef]

A. J. Coleman, E. Draguioti, R. Tiptaf, N. Shotri, J. E. Saunders, “Acoustic performance and clinical use of a fibreoptic hydrophone,” Ultrasound Med. Biol. 24, 143–151 (1998).
[CrossRef] [PubMed]

Fernando, G. F.

T. Liu, D. Brooks, A. Martin, R. A. Badcock, B. Ralph, G. F. Fernando, “A multimode fibre based extrinsic Fabry–Perot interferometric strain sensor,” J. Smart Mater. Structures 6, 464–469 (1997).
[CrossRef]

Gunter, M. F.

Hartl, J. C.

R. A. Wolthius, G. L. Mitchell, E. Saaski, J. C. Hartl, M. A. Afromowitz, “Development of medical pressure and temperature sensors employing optical spectrum modulation,” IEEE Trans. Biomed. Eng. 38, 974–981 (1991).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), pp. 364–366.

Itoh, K.

T. Yoshino, K. Kurosawa, K. Itoh, T. Ose, “Fiber-optic Fabry–Perot interferometer and its sensor applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1612–1621 (1982).
[CrossRef]

Jackson, D. A.

Jones, B. E.

E. R. Cox, B. E. Jones, “Fiber optic colour sensors based on Fabry–Perot interferometry,” in Proceedings of the First International Conference on Optical Fiber Sensors (Optical Society of America, Washington, D.C., 1983), pp. 122–126.

Kurosawa, K.

T. Yoshino, K. Kurosawa, K. Itoh, T. Ose, “Fiber-optic Fabry–Perot interferometer and its sensor applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1612–1621 (1982).
[CrossRef]

Leite, A. P.

Liu, T.

T. Liu, D. Brooks, A. Martin, R. A. Badcock, B. Ralph, G. F. Fernando, “A multimode fibre based extrinsic Fabry–Perot interferometric strain sensor,” J. Smart Mater. Structures 6, 464–469 (1997).
[CrossRef]

Martin, A.

T. Liu, D. Brooks, A. Martin, R. A. Badcock, B. Ralph, G. F. Fernando, “A multimode fibre based extrinsic Fabry–Perot interferometric strain sensor,” J. Smart Mater. Structures 6, 464–469 (1997).
[CrossRef]

Mills, T. N.

Mitchell, G. L.

R. A. Wolthius, G. L. Mitchell, E. Saaski, J. C. Hartl, M. A. Afromowitz, “Development of medical pressure and temperature sensors employing optical spectrum modulation,” IEEE Trans. Biomed. Eng. 38, 974–981 (1991).
[CrossRef]

Murphy, A. K.

Murphy, K. A.

V. Arya, M. J. de Vries, K. A. Murphy, A. Wang, R. O. Claus, “Exact analysis of the EFPI optical fiber sensor using Kirchhoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Ose, T.

T. Yoshino, K. Kurosawa, K. Itoh, T. Ose, “Fiber-optic Fabry–Perot interferometer and its sensor applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1612–1621 (1982).
[CrossRef]

Pérennès, F.

Ralph, B.

T. Liu, D. Brooks, A. Martin, R. A. Badcock, B. Ralph, G. F. Fernando, “A multimode fibre based extrinsic Fabry–Perot interferometric strain sensor,” J. Smart Mater. Structures 6, 464–469 (1997).
[CrossRef]

Saaski, E.

R. A. Wolthius, G. L. Mitchell, E. Saaski, J. C. Hartl, M. A. Afromowitz, “Development of medical pressure and temperature sensors employing optical spectrum modulation,” IEEE Trans. Biomed. Eng. 38, 974–981 (1991).
[CrossRef]

Santos, J. L.

Saunders, J. E.

A. J. Coleman, E. Draguioti, R. Tiptaf, N. Shotri, J. E. Saunders, “Acoustic performance and clinical use of a fibreoptic hydrophone,” Ultrasound Med. Biol. 24, 143–151 (1998).
[CrossRef] [PubMed]

Shotri, N.

A. J. Coleman, E. Draguioti, R. Tiptaf, N. Shotri, J. E. Saunders, “Acoustic performance and clinical use of a fibreoptic hydrophone,” Ultrasound Med. Biol. 24, 143–151 (1998).
[CrossRef] [PubMed]

Tiptaf, R.

A. J. Coleman, E. Draguioti, R. Tiptaf, N. Shotri, J. E. Saunders, “Acoustic performance and clinical use of a fibreoptic hydrophone,” Ultrasound Med. Biol. 24, 143–151 (1998).
[CrossRef] [PubMed]

Vengsarker, A. M.

Wang, A.

V. Arya, M. J. de Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the performance of imperfect fiber endfaces on the performance of extrinsic Fabry–Perot interferometric optical fiber sensors,” Opt. Eng. 35, 2262–2265 (1996).
[CrossRef]

V. Arya, M. J. de Vries, K. A. Murphy, A. Wang, R. O. Claus, “Exact analysis of the EFPI optical fiber sensor using Kirchhoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Wolthius, R. A.

R. A. Wolthius, G. L. Mitchell, E. Saaski, J. C. Hartl, M. A. Afromowitz, “Development of medical pressure and temperature sensors employing optical spectrum modulation,” IEEE Trans. Biomed. Eng. 38, 974–981 (1991).
[CrossRef]

Yoshino, T.

T. Yoshino, K. Kurosawa, K. Itoh, T. Ose, “Fiber-optic Fabry–Perot interferometer and its sensor applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1612–1621 (1982).
[CrossRef]

Appl. Opt.

Electron. Lett.

P. C. Beard, T. N. Mills, “A miniature optical fibre ultrasonic hydrophone using a Fabry–Perot polymer film interferometer,” Electron. Lett. 33, 801–803 (1997).
[CrossRef]

IEEE Trans. Biomed. Eng.

R. A. Wolthius, G. L. Mitchell, E. Saaski, J. C. Hartl, M. A. Afromowitz, “Development of medical pressure and temperature sensors employing optical spectrum modulation,” IEEE Trans. Biomed. Eng. 38, 974–981 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

T. Yoshino, K. Kurosawa, K. Itoh, T. Ose, “Fiber-optic Fabry–Perot interferometer and its sensor applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1612–1621 (1982).
[CrossRef]

J. Smart Mater. Structures

T. Liu, D. Brooks, A. Martin, R. A. Badcock, B. Ralph, G. F. Fernando, “A multimode fibre based extrinsic Fabry–Perot interferometric strain sensor,” J. Smart Mater. Structures 6, 464–469 (1997).
[CrossRef]

Opt. Eng.

V. Arya, M. J. de Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the performance of imperfect fiber endfaces on the performance of extrinsic Fabry–Perot interferometric optical fiber sensors,” Opt. Eng. 35, 2262–2265 (1996).
[CrossRef]

Opt. Fiber Technol.

V. Arya, M. J. de Vries, K. A. Murphy, A. Wang, R. O. Claus, “Exact analysis of the EFPI optical fiber sensor using Kirchhoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Opt. Lett.

Ultrasound Med. Biol.

A. J. Coleman, E. Draguioti, R. Tiptaf, N. Shotri, J. E. Saunders, “Acoustic performance and clinical use of a fibreoptic hydrophone,” Ultrasound Med. Biol. 24, 143–151 (1998).
[CrossRef] [PubMed]

Other

E. R. Cox, B. E. Jones, “Fiber optic colour sensors based on Fabry–Perot interferometry,” in Proceedings of the First International Conference on Optical Fiber Sensors (Optical Society of America, Washington, D.C., 1983), pp. 122–126.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), pp. 364–366.

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Figures (11)

Fig. 1
Fig. 1

Schematic of a plane-parallel low-finesse FPI illuminated by a multimode optical fiber. θ d1 and θ d are the angles of the most diverging rays in the media between the fiber and the FPI and in the FP cavity, respectively. θ1 and θ are the angles of a particular ray in the media between the fiber and the FPI and in the FP cavity, respectively.

Fig. 2
Fig. 2

Visibility versus maximum internal beam divergence θ d for different values of the cavity thickness l, z 0 = 0, n = 1.64, n 1 = n f = 1.47, n 2 = 1.33, and λ = 850 nm.

Fig. 3
Fig. 3

Experimental setup for the validation of FP transfer function and visibility, R 1 = R 2 = 0.04, θ d = 3.7°, z 0 = 22 cm, step-index fiber with 380-µm core diameter and NA = 0.12, n = n 1 = n 2 = 1.

Fig. 4
Fig. 4

Transfer function of FP obtained by wavelength tuning.

Fig. 5
Fig. 5

Comparison between theory and experiment of visibility versus phase dispersion.

Fig. 6
Fig. 6

Geometry describing a tilt between the fiber end face and the FPI. ε t is the tilt angle, z 0 is the distance between the untilted fiber end face and the FPI, and z i is the distance between the virtual origin of the light cone of angle θ d1 and the FPI. The ellipse centered on O′ represents the beam footprint on the front side of the FPI.

Fig. 7
Fig. 7

Visibility versus phase dispersion for different values of the tilt angle ε t to beam divergence θ d1 ratio. R 1 = R 2 = 0.04, θ d1 = θ d = 4°, z 0 = 22 cm, n = n 1 = n 2 = 1.

Fig. 8
Fig. 8

Visibility versus tilt angle for a 50-µm-thick polymer cavity in optical contact with the fiber tip for two different values of the divergence θ d (z 0 = 0, n = 1.64, n 1 = n f = 1.47, n 2 = 1.33, l = 50 µm, and λ = 850 nm).

Fig. 9
Fig. 9

Geometry showing a FP cavity with a wedge between the two surfaces. ε w is the wedge angle, l 0 is the equivalent ideal FP cavity thickness (i.e., with no wedge), r 0 is the radius of the beam circular footprint on the front side of the FPI, and Δl 0 and Δl π are the changes in thickness of the FP cavity for α = 0 and α = π, respectively. The ellipse centered in O′ represents the projection of the beam footprint on the back side of the FPI into the OxOy plane.

Fig. 10
Fig. 10

Visibility versus phase dispersion for different values of the wedge angle ε w . R 1 = R 2 = 0.04, θ d1 = θ d = 4°, z 0 = 22 cm, n = n 1 = n 2 = 1.

Fig. 11
Fig. 11

Visibility versus wedge angle for a 50-µm-thick cavity in optical contact with the fiber tip for two different values of the divergence θ d (z 0 = 0, n = 1.64, n 1 = n f = 1.47, n 2 = 1.33, l = 50 µm, and λ = 850 nm).

Equations (30)

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sin θd1=nn1sin θd.
ϕθ=4πnlλcos θ,
IRiθ=I0ΔϕR1+1-R12R2+2R1R21/2×1-R1cos ϕθ,
Δϕ=ϕmax-ϕmin=4πnlλ1-cos θd=ϕ01-cos θd,
Dϕ=1for ϕ0-Δϕ<ϕ<ϕ0Dϕ=0elsewhere.
IR=Δϕ DϕIRiϕdϕ=R1+1-R12R2+2R1R21/21-R1Δϕϕ0-Δϕϕ0cos ϕdϕI0.
IR=R1+1-R12R2+2R1R21/21-R1×sinΔϕ/2Δϕ/2cosϕ0-Δϕ2I0,
γ=2R1R21/21-R1R1+1-R12R2|sin Δϕ/2|Δϕ/2=γ0|sin Δϕ/2|Δϕ/2,
l=λ22n1+cos θdΔλ,
zi=z0+atan θd1,
x-x02b2+y2c2=1,
x0=zi2tanθd1+εt-tanθd1-εt,
b=zi2tanθd1+εt+tanθd1-εt,
c=zi tan θd1.
rα=x0c2 cos α+c2 cos2 α+b2-x02sin2 α1/2c2 cos2 α+b2 sin2 α.
x0=zi tan εt, b=zi tan θd1, giving rα=zi tan θd1εtθd1cos α+1-εtθd1sin α21/2.
θ1α=arctanrαzi=θd1εtθd1cos α+1-εtθd1sin α2,
θα=arcsinn1nsin θ1α.
Dϕ=1 for ϕ0-Δϕα<ϕ<ϕ0 with Δϕα=ϕ01-cos θαDϕ=0 elsewhere.
IR=1π0πϕ0-Δϕαϕ0 DϕIRiϕdϕdα,
IR=R1+1-R12R2+2R1R21/21-R1×1π0πsinΔϕα/2Δϕα/2cosϕ0-Δϕα2dαI0.
x-x02b2+y2c2=1,
x0=-r0+l0 tan θdtan θd tan εw1-tan2 θd tan2 εw,
b=r0+l0 tan θd11-tan2 θd tan2 εw,
c=r0+l0 tan θd,
r0=zi tan θd1=a+z0 tan θd1.
rα=r0+l0 tan θd×-tan θd tan εw cos α+1-tan θd tan εw cos α21-tan2 θd tan2 εw1/21-tan θd tan εw cos α2.
r=r0+l0 tan θd.
Δlα=r tan εw cos α.
Dϕα=1 for ϕαϕ0; ϕ0-Δϕα with Δϕα=ϕ01-1-Δlαl0cos θdDϕα=0 elsewhere.

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