Abstract

In many remote sensing applications one or multiple Fabry–Perot etalons are used as high-spectral-resolution filter elements. These etalons are often coupled to a receiving telescope with a multimode fiber, leading to subtle effects of the fiber mode order on the overall spectral response of the system. A theoretical model is developed to treat the spectral response of the combined system: fiber, collimator, and etalon. The method is based on a closed-form expression of the diffracted mode in terms of a Hankel transform. In this representation, it is shown how the spectral effect of the fiber and collimator can be separated from the details of the etalon and can be viewed as a mode-dependent spectral broadening and shift.

© 2014 Optical Society of America

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References

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  1. J. A. R. Rall, J. Campbell, J. B. Abshire, and J. D. Spinhirne, “Automatic weather station (AWS) lidar,” in IEEE 2001 International Geoscience and Remote Sensing Symposium (IEEE, 2001), Vol. 7, pp. 3065–3067.
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    [CrossRef]
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    [CrossRef]
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  12. L. Li and F. Guo, “An improved definition of divergence half-angle for the far-field of fiber,” Optik 123, 283–285 (2012).
    [CrossRef]
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    [CrossRef]
  14. O. Svelto, Principles of Lasers (Plenum, 1998), Chaps. 4.5 and 4.7.
  15. G. Arfken, Mathematical Methods for Physicists (Academic, 1985), Chaps. 2 and 11.
  16. http://numpy.scipy.org/ .

2012 (1)

L. Li and F. Guo, “An improved definition of divergence half-angle for the far-field of fiber,” Optik 123, 283–285 (2012).
[CrossRef]

2007 (1)

2005 (1)

2004 (2)

2000 (1)

1993 (1)

1992 (1)

1991 (1)

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Abshire, J. B.

J. A. R. Rall, J. Campbell, J. B. Abshire, and J. D. Spinhirne, “Automatic weather station (AWS) lidar,” in IEEE 2001 International Geoscience and Remote Sensing Symposium (IEEE, 2001), Vol. 7, pp. 3065–3067.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, 1985), Chaps. 2 and 11.

Boechat, A. A. P.

Campbell, J.

J. A. R. Rall, J. Campbell, J. B. Abshire, and J. D. Spinhirne, “Automatic weather station (AWS) lidar,” in IEEE 2001 International Geoscience and Remote Sensing Symposium (IEEE, 2001), Vol. 7, pp. 3065–3067.

Chen, H.

Dong, J.

Eloranta, E. W.

Gentry, B. M.

Ghatak, A.

A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics (Cambridge University, 1998).

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts, 2005), Chap. 3.

Grund, C. J.

Guizar-Sicairos, M.

Guo, F.

L. Li and F. Guo, “An improved definition of divergence half-angle for the far-field of fiber,” Optik 123, 283–285 (2012).
[CrossRef]

Gutiérrez-Vega, J. C.

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Hua, D.

Jones, J. D. C.

Kihm, H.

Kim, S.-W.

Kobayashi, T.

Li, L.

L. Li and F. Guo, “An improved definition of divergence half-angle for the far-field of fiber,” Optik 123, 283–285 (2012).
[CrossRef]

Li, S. X.

Rall, J. A. R.

J. A. R. Rall, J. Campbell, J. B. Abshire, and J. D. Spinhirne, “Automatic weather station (AWS) lidar,” in IEEE 2001 International Geoscience and Remote Sensing Symposium (IEEE, 2001), Vol. 7, pp. 3065–3067.

Shen, F.

Spinhirne, J. D.

J. A. R. Rall, J. Campbell, J. B. Abshire, and J. D. Spinhirne, “Automatic weather station (AWS) lidar,” in IEEE 2001 International Geoscience and Remote Sensing Symposium (IEEE, 2001), Vol. 7, pp. 3065–3067.

Su, D.

Sun, D.

Svelto, O.

O. Svelto, Principles of Lasers (Plenum, 1998), Chaps. 4.5 and 4.7.

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics (Cambridge University, 1998).

Uchida, M.

Wittmann, R. C.

Xia, H.

Yang, Y.

Young, M.

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. A (1)

Opt. Lett. (3)

Optik (1)

L. Li and F. Guo, “An improved definition of divergence half-angle for the far-field of fiber,” Optik 123, 283–285 (2012).
[CrossRef]

Other (6)

A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics (Cambridge University, 1998).

J. Goodman, Introduction to Fourier Optics (Roberts, 2005), Chap. 3.

J. A. R. Rall, J. Campbell, J. B. Abshire, and J. D. Spinhirne, “Automatic weather station (AWS) lidar,” in IEEE 2001 International Geoscience and Remote Sensing Symposium (IEEE, 2001), Vol. 7, pp. 3065–3067.

O. Svelto, Principles of Lasers (Plenum, 1998), Chaps. 4.5 and 4.7.

G. Arfken, Mathematical Methods for Physicists (Academic, 1985), Chaps. 2 and 11.

http://numpy.scipy.org/ .

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Figures (5)

Fig. 1.
Fig. 1.

Optical configuration under study: multimode fiber, collimator, and Fabry–Perot etalon.

Fig. 2.
Fig. 2.

(a) Geometry used in the derivation of the Rayleigh–Sommerfeld diffraction integral [see Eq. (1)]. Initial conditions are given on the surface Σ0 by U(P0), and the field U(P1) is calculated at a point P1. (b) Coordinate systems adopted to derive Eq. (9), (c) geometrical construct used to obtain Eq. (2).

Fig. 3.
Fig. 3.

Gaussian beam propagation with Eq. (9) for 3 and 10 Rayleigh ranges.

Fig. 4.
Fig. 4.

Fiber–collimator instrumental function Tf, for high-order mode in NA=0.22 fiber.

Fig. 5.
Fig. 5.

Fiber–collimator instrumental function Tf, for high-order mode in NA=0.12 fiber.

Equations (26)

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U(P1)=1jλΣ0U(P0)cos(n⃗,r⃗01)exp(jkr01)r01ds.
cos(n⃗,r⃗01)=zr01=ρcos(θ)r01.
U(P0)=exp(jnφ0)u0(r0),
U(P1)=ρexp(jnφ1)cos(θ)jλ0u0(r0)r0dr002πexp(jnΔφ+jkr01)r012dΔφ.
r01=(ρ2+r022ρr0sin(θ)cos(Δφ))1/2.
r01ρr0sin(θ)cos(Δφ),in the phase
r01ρ,in the amplitude term.
U(P1)=exp(jnφ1+jkρ)cos(θ)jλρ0u0(r0)r0dr002πexp(jnΔφ+jkr0sin(θ)cos(Δφ))dΔφ.
U(P1)=exp(jnφ1jnπ/2+jkρ)cos(θ)jλρ2π0u0(r0)Jn(kr0sin(θ))r0dr0.
U(P1)=exp(jnφ1jnπ/2+jkρ)cos(θ)jλρHn{u0}(sin(θ)λ),
Hn{f}(ν)=2π0f(r)Jn(2πνr)rdr,
Hn1{f}(r)=2π0f(ν)Jn(2πνr)νdν.
U(P1)=exp(jnφ1)exp(jkρ)λρu1(θ).
u1(θ)=cos(θ)Hn{u0}(sin(θ)λ).
U(Σ2)=exp(jnφ2)u2(r2),
u2(r2)=Fλ(F2+r22)Hn{u0}(r2λ(F2+r22)0.5).
U(Σ2)exp(jnφ2)Hn{u0}(r2/λF)/λF.
u3(θ)=Hn{u2}(θλ).
gn(kr)=exp(jωt+jnφ+jkzz)Jn(krr),where
kz2+kr2=k2=(2πλ)2.
Te(kz)=|t1t21r1r2exp(2jkzL)|2.
It(Lnm)=0|H{u2}(ν)|2Te(2πλ2ν2)νdν.
It(Lnm)=0|H{u2}(ν)|2Te(2π(λ10.5λν2))νdν.
It/I=0Tf(Δf)Te(2π(fΔf)/c)dΔf,where
Tf(Δf)=|H{u2}(2Δf/λc)|2/0|H{u2}(2Δf/λc)|2dΔf.
Tf(Δf)|u0(F2Δf/f)|2/0|u0(F2Δf/f)|2dΔf,forNA1.

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