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Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 31, Iss. 3 — Mar. 1, 2014
  • pp: 505–509
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Spectral response of fiber-coupled Fabry–Perot etalons

Pavel Ionov  »View Author Affiliations


JOSA A, Vol. 31, Issue 3, pp. 505-509 (2014)
http://dx.doi.org/10.1364/JOSAA.31.000505


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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

1. INTRODUCTION

The use of multimode fibers for delivering light from telescopes to detectors is a frequent practice in remote sensing applications [1

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.

5

5. H. Xia, D. Sun, Y. Yang, F. Shen, J. Dong, and T. Kobayashi, “Fabry-Perot interferometer based Mie Doppler lidar for low tropospheric wind observation,” Appl. Opt. 46, 7120–7131 (2007). [CrossRef]

]. This configuration offers several practical advantages including the mechanical decoupling and remote placement of the telescope optics and detector apparatus, as well as the reduction of the instrument field of view (FOV), which may help lower the background noise in certain lidar applications. Another important benefit is the expected desensitization of the instrument with respect to transmitter-to-receiver alignment errors within the FOV [4

4. D. Hua, M. Uchida, and T. Kobayashi, “Ultraviolet Rayleigh-Mie lidar with Mie-scattering correction by Fabry-Perot etalons for temperature profiling of the troposphere,” Appl. Opt. 44, 1305–1314 (2005). [CrossRef]

,5

5. H. Xia, D. Sun, Y. Yang, F. Shen, J. Dong, and T. Kobayashi, “Fabry-Perot interferometer based Mie Doppler lidar for low tropospheric wind observation,” Appl. Opt. 46, 7120–7131 (2007). [CrossRef]

]. This benefit stems from the fact that the light entering the multimode fiber excites a large number of guided modes, which is expected to produce a nearly flat-top irradiance profile at the fiber output independent of the input irradiance profile. However, experiments conducted in our laboratory and evidence from relevant work on fiber-based beam delivery [6

6. D. Su, A. A. P. Boechat, and J. D. C. Jones, “Beam delivery by large-core fibers: effect of launching conditions on near-field output profile,” Appl. Opt. 31, 5816–5821 (1992). [CrossRef]

] suggest that, in practice, the excitation and mixing of fiber modes is hardly ever complete. Thus, some memory of the initial spatial characteristics of collected light is typically retained. The residual alignment dependence of lidar data caused by such incomplete in-fiber mode mixing is especially important in detection apparatuses that include etalons (e.g., for spectral filtering or referencing), as these optical devices usually exhibit high sensitivity to the light incidence angle. In general, we can assume that the fiber-delivered beam is expressed as a linear combination of guided modes, the specifics of which depend to some extent on alignment conditions. It is thus instructive to study how the spectral response of a fiber-coupled etalon depends on the individual transverse modes that are excited in the fiber.

The configuration of interest in this article is schematically illustrated in Fig. 1. We assume that the beam delivery to the etalon is carried out through a standard, weakly guiding multimode fiber. In such fiber, the core-guided eigenmodes are commonly approximated by the linearly polarized (LP) set [7

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

]. We also assume that the beam exiting the fiber is collimated and then transmitted through a flat-mirror Fabry–Perot etalon. To determine the etalon response, the diffraction of LP modes is calculated by using the binomial approximation to the Rayleigh–Sommerfeld (RS) integral [8

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

,9

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

].

First, a closed-form expression for diffracted LPnm modes is derived in terms of the Hankel transform. Then this representation is used to demonstrate that the spectral response of the system in Fig. 1 can be expressed as a convolution of the etalon and fiber/collimator contributions. In this view, the action of the fiber plus the collimator constitutes a fiber-mode-dependent broadening of the instrument spectral response, which is independent of the etalon characteristics. This conceptual separation provides a valuable aid in the design of this type of instrument.

Fig. 1. Optical configuration under study: multimode fiber, collimator, and Fabry–Perot etalon.

A general numerical treatment of this problem is, in principle, available through the use of a recently published Hankel transform algorithm [10

10. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58 (2004). [CrossRef]

]. However, the weak guidance (low NA) of practical fibers is advantageously leveraged here to express the fiber-mode-dependent part of the spectral response analytically in terms of the radial component of LPnm modes at the output end of the fiber.

2. DIFFRACTION OF LPnm MODE

In this section we show that a binomial approximation to the RS integral for the LPnm fiber mode can be reduced to a closed-form expression [see Eq. (9), below], via the nth-order Hankel transform.

The binomial approximation to the RS integral has been used for other problems pertaining to fiber mode diffraction in free space [11

11. H. Kihm and S.-W. Kim, “Nonparaxial free-space diffraction from oblique end faces of single-mode optical fibers,” Opt. Lett. 29, 2366–2368 (2004). [CrossRef]

,12

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]

]. Young and Wittmann [13

13. M. Young and R. C. Wittmann, “Vector theory of diffraction by a single-mode fiber: application to mode-field diameter measurements,” Opt. Lett. 18, 1715–1717 (1993). [CrossRef]

] derived a version of Eq. (9) applicable to the specific case of D symmetry (a circular aperture with a normally incident plane wave), which results in a zero-order Hankel transform. However, a detailed derivation of the general form of Eq. (9) has not been published to date, to our knowledge, and appears especially useful given the Hankel transform algorithm described in Ref. [10

10. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58 (2004). [CrossRef]

].

With the notation of Fig. 2, the RS integral can be written as [8

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

,9

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

]
U(P1)=1jλΣ0U(P0)cos(n⃗,r⃗01)exp(jkr01)r01ds.
(1)
Here, the initial condition is given by U(P0), which denotes the scalar field at the point P0 located on the exit surface of the fiber, Σ0; P1 is an arbitrary point where the propagating field is calculated; λ is the wavelength; k is the wavenumber (=2π/λ); r⃗01 is the vector connecting points P0 and P1 [see Fig. 2(b)], with r01=|r⃗01|; and n⃗ is normal to the surface Σ0. The argument of the cosine term, cos(n⃗,r⃗01), is the angle between the r⃗01 and n⃗ vectors. In our adopted system of coordinates [Fig. 2(b)], the point P0 is identified by the cylindrical coordinates (r0, φ0, 0), whereas P1 can be denoted by both cylindrical (r1, φ1, z) and spherical (ρ, φ1, θ) coordinates.

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).

Based on the geometrical construct illustrated in Fig. 2(c), we have
cos(n⃗,r⃗01)=zr01=ρcos(θ)r01.
(2)
Moreover, given the functional form of the azimuth-angle dependence of LPnm modes [7

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

],
U(P0)=exp(jnφ0)u0(r0),
(3)
where n=0,1, is the mode order, we can rewrite Eq. (1) as
U(P1)=ρexp(jnφ1)cos(θ)jλ0u0(r0)r0dr002πexp(jnΔφ+jkr01)r012dΔφ.
(4)
Here, the abbreviation Δφ=φ0φ1 is introduced. Further, r01 is explicitly calculated by applying the cosine formula to the OP0P1 triangle and spherical cosine formulas at the origin O [see Fig. 2(c)], which yields
r01=(ρ2+r022ρr0sin(θ)cos(Δφ))1/2.
(5)
The square root term in Eq. (5) can be expanded into the Taylor series with respect to the argument r0/ρ and retaining only the first-order term in phase and zero-order term in amplitude, in a fashion similar to the Fresnel approximation:
r01ρr0sin(θ)cos(Δφ),in the phase
(6a)
r01ρ,in the amplitude term.
(6b)
Substituting Eq. (6) into Eq. (4) yields
U(P1)=exp(jnφ1+jkρ)cos(θ)jλρ0u0(r0)r0dr002πexp(jnΔφ+jkr0sin(θ)cos(Δφ))dΔφ.
(7)
Here, the integral over Δφ is a Bessel function to within a constant phase factor, such that Eq. (7) can be rewritten as
U(P1)=exp(jnφ1jnπ/2+jkρ)cos(θ)jλρ2π0u0(r0)Jn(kr0sin(θ))r0dr0.
(8)
The radial integral can then be expressed in terms of a Hankel transform:
U(P1)=exp(jnφ1jnπ/2+jkρ)cos(θ)jλρHn{u0}(sin(θ)λ),
(9)
where forward and reverse Hankel transforms are defined as
Hn{f}(ν)=2π0f(r)Jn(2πνr)rdr,
(10a)
Hn1{f}(r)=2π0f(ν)Jn(2πνr)νdν.
(10b)
The approximation in Eq. (6) is of the order d/2z, where d is the fiber core diameter. As such, this approximation improves at longer propagation distances, z, and can thus be viewed as a far-field approximation in contrast to the Fresnel approximation in its common form [8

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

], which is of order r/2z (corresponding to the numerical aperture of the fiber) in the case of Fig. 1. To elucidate this aspect, a Gaussian beam of width w0=1mm and λ=1.06μm is propagated according to Eq. (9) for distances of 3 and 10 times the Rayleigh range, zR=πw02/λ. As shown in Fig. 3, excellent agreement with the analytical formula is observed at 10zR.

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

3. COLLIMATION OF LPnm MODE

In order to explicitly write the field after an ideal collimator in Fig. 1, it is convenient to rewrite Eq. (9) as a product of three terms. Also, since the constant phase term in Eq. (9) is not important in many applications, we can drop it with little loss of generality. Thus, Eq. (9) is rewritten as
U(P1)=exp(jnφ1)exp(jkρ)λρu1(θ).
(11)
Here, the angular term dependent on φ1 is identical to that of the LPnm mode, confirming that the approximation does not break the Dn symmetry of the system.

The second term, exp(jkρ)/λρ, is a spherical wave. Finally, u1(θ) is an amplitude term expressing the radial field dependence at the collimator plane Σ1 and is explicitly given by
u1(θ)=cos(θ)Hn{u0}(sin(θ)λ).
(12)
The term u1(θ) takes on real values for real u0(r0), as is the case of LP modes. Based on Eq. (11), only the second (spherical) phase term exp(jkρ) can be compensated with a regular lens system, because both share D axial symmetry. Conversely, the first phase term, exp(jnφ1), exhibits Dn symmetry and cannot be corrected with regular lenses. Thus, since u1(θ) is real for LP modes, we can conclude that the ideal collimator would have a phase of exp(jkρ), which corresponds to an ideal lens with a focal length F=z. After such a collimator the field becomes
U(Σ2)=exp(jnφ2)u2(r2),
(13)
where
u2(r2)=Fλ(F2+r22)Hn{u0}(r2λ(F2+r22)0.5).
(14)

Before considering how such a collimated field diffracts, it is instructive to examine the case of a small NA1, or θ1. In this limit, the collimated field [Eqs. (13) and (14)] can simply be written as
U(Σ2)exp(jnφ2)Hn{u0}(r2/λF)/λF.
(15)
In this expression, the radial component of the collimated beam simply becomes a Hankel transform of the radial component of the LPnm mode.

The far-field expression for the collimated beam can be found by recognizing that the geometry is identical to that of Fig. 2(b), such that we can immediately use Eq. (11) or (15). Since θ1 for a collimated beam, Eq. (15) is a good approximation, and the far-field radial component of the collimated field can be written as
u3(θ)=Hn{u2}(θλ).
(16)
If the fiber NA1, successive application of Eqs. (15) and (16) produces u0(θF) for u3(θ): in other words, an ideal collimator forms a perfect image of the LP mode at infinity for low-NA fiber. However, for large-NA fibers, the cosine term in Eq. (12) cannot be ignored, and consequently, Eq. (14) must be calculated numerically to yield accurate results.

4. FIBER-COUPLED FABRY–PEROT ETALON

A common approach to the analysis of a Fabry–Perot etalon response is to decompose the field into plane waves [14

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

]. However, in the cylindrically symmetric geometry of Fig. 1, Bessel waves are more naturally applied. Such waves are also exact solutions of the wave equation (see, for instance, Ref. [15

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

]):
gn(kr)=exp(jωt+jnφ+jkzz)Jn(krr),where
(17a)
kz2+kr2=k2=(2πλ)2.
(17b)

As in the case of plane waves, Bessel waves can be viewed as propagating at an angle to the optical axis of the etalon, such that the wavenumber k can be decomposed into the components parallel and perpendicular to the optical axis (kz and kr, respectively). By introducing amplitude transmission (t1, t2) and reflectivity (r1, r2) for the etalon mirrors (denoted by the indices “1” and “2”), we can express the overall etalon transmission [Te(kz)] as a geometrical series [14

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

], which, for a Bessel wave with a given kz, reads as
Te(kz)=|t1t21r1r2exp(2jkzL)|2.
(18)
By expressing the field u2 entering the etalon in terms of Bessel waves with a Hankel transform and taking advantage of Parseval’s theorem, we can express the optical intensity after the etalon as
It(Lnm)=0|H{u2}(ν)|2Te(2πλ2ν2)νdν.
(19)

Since transverse spatial frequency ν1/λ in practical cases of collimated etalon input, the square root in Eq. (19) can be expanded into a Taylor series, keeping only the first term dependent on ν:
It(Lnm)=0|H{u2}(ν)|2Te(2π(λ10.5λν2))νdν.
(20)

It is convenient to rewrite Eq. (20) by defining a frequency shift Δf=0.5λν2c and a function Tf(Δf):
It/I=0Tf(Δf)Te(2π(fΔf)/c)dΔf,where
(21a)
Tf(Δf)=|H{u2}(2Δf/λc)|2/0|H{u2}(2Δf/λc)|2dΔf.
(21b)
In this notation, frequency-dependent transmission of the fiber-coupled etalon is a convolution of the response of the etalon itself and the function Tf(Δf) [Eq. (21a)]. Thus, the function Tf(Δf) behaves as a spectral broadening, which is independent of the etalon parameters. An important consequence of this unexpected result is that the mode-dependent instrumental function Tf of the fiber and collimator can be studied independently of the etalon. In applications, such as Doppler lidar [2

2. B. M. Gentry, H. Chen, and S. X. Li, “Wind measurements with 355  nm molecular Doppler lidar,” Opt. Lett. 25, 1231–1233 (2000). [CrossRef]

,5

5. H. Xia, D. Sun, Y. Yang, F. Shen, J. Dong, and T. Kobayashi, “Fabry-Perot interferometer based Mie Doppler lidar for low tropospheric wind observation,” Appl. Opt. 46, 7120–7131 (2007). [CrossRef]

], where high spectral accuracy is required, this mode dependence of the instrumental function presents a challenge. On the other hand, independence of Tf of the parameters of the plane-parallel Fabry–Perot etalon also means that an experimentalist has no ability to influence the spectral effect of the mode by clever selection of the parameters.

In the case of a low-NA fiber (NA1), the expression for Tf [Eq. (21b)] can be simplified. Substituting Eq. (15) into Eq. (21b), and remembering that the forward and inverse Hankel transforms have identical form, we get
Tf(Δf)|u0(F2Δf/f)|2/0|u0(F2Δf/f)|2dΔf,forNA1.
(22)
Thus, in the case of a low-NA fiber, Tf, is expressed only through the radial component of the LPnm mode, u0.

In a general case, the low-NA approximation of Eq. (22) is unjustified and Eqs. (14) and (21b) need to be evaluated numerically. While numerical implementation of the Hankel transform is not as well established as the Fourier transform, a recently published algorithm [10

10. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58 (2004). [CrossRef]

] can be easily implemented in NumPy, a numerical Python library [16], which is the approach used to obtain the numerical results presented here.

Two representative numerical calculations of Tf are given in Fig. 4 and Fig. 5 for fibers with numerical apertures of 0.22 and 0.12, respectively. One practically important property of broadening function Tf is immediately seen in Figs. 4 and 5. The broadening is one-sided. An important consequence of this is that the combined transmission spectrum exhibits a mode-dependent shift of approximately half of its width. This is generally an undesirable outcome. For instance, in a double-edge measurement [2

2. B. M. Gentry, H. Chen, and S. X. Li, “Wind measurements with 355  nm molecular Doppler lidar,” Opt. Lett. 25, 1231–1233 (2000). [CrossRef]

], the shift is a more significant error source than the broadening, as the symmetric line width variations are canceled out by the double-edge technique. Also, the shape of the broadening function Tf in Figs. 4 and 5 is not symmetric around its center frequency either. Since the double-edge technique relies on the assumption of the symmetry of the interferometer’s spectral response, both types of asymmetry lead to systematic measurement errors that depend on fiber mode launching conditions and, thus, on lidar alignment.

Fig. 4. Fiber–collimator instrumental function Tf, for high-order mode in NA=0.22 fiber.
Fig. 5. Fiber–collimator instrumental function Tf, for high-order mode in NA=0.12 fiber.

Not surprisingly, comparison of Fig. 4 and Fig. 5 reveals that the approximate formula in Eq. (22) clearly performs better in the case of a lower-NA fiber. However, even for an NA of 0.22 the approximate result still captures the qualitative behavior of the broadening function Tf versus frequency. Thus, frequency shift and broadening are of the order of 0.5f(d/F)2 and can be visualized with the help of the result in Eq. (22) for the purpose of gaining qualitative understanding.

5. CONCLUSION

It was demonstrated that a combined effect of a multimode fiber plus a collimator in front of a Fabry–Perot etalon leads to a mode-dependent asymmetric spectral broadening and shift. In the limit of the low-NA fiber, the effect can be understood in terms of the radial component of LPnm mode [Eq. (22)]. In a more general case the diffraction problem has to be treated numerically, but explicitly accounting for the system symmetry reduces the problem to a single dimension [Eq. (14)]. However, even for an NA of 0.22 the approximate formula captures the qualitative behavior quite well. Probably the most important and surprising conclusion is that the broadening caused by the multimode fiber is not dependent on the parameters of the plane-parallel Fabry–Perot etalon that follows it. This has important implications in the design of high-spectral-resolution remote sensing instruments, such as double-edge Doppler lidar.

ACKNOWLEDGMENTS

The Author thanks Fabio Di Teodoro for many fruitful discussions and review of the early draft. This work was supported by The Aerospace Corporation’s Sustained Experimentation and Research for Program Applications and Independent Research and Development Programs.

REFERENCES

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.

2.

B. M. Gentry, H. Chen, and S. X. Li, “Wind measurements with 355  nm molecular Doppler lidar,” Opt. Lett. 25, 1231–1233 (2000). [CrossRef]

3.

C. J. Grund and E. W. Eloranta, “Fiber-optic scrambler reduces the bandpass range dependence of Fabry-Perot etalons used for spectral analysis of lidar backscatter,” Appl. Opt. 30, 2668–2670 (1991). [CrossRef]

4.

D. Hua, M. Uchida, and T. Kobayashi, “Ultraviolet Rayleigh-Mie lidar with Mie-scattering correction by Fabry-Perot etalons for temperature profiling of the troposphere,” Appl. Opt. 44, 1305–1314 (2005). [CrossRef]

5.

H. Xia, D. Sun, Y. Yang, F. Shen, J. Dong, and T. Kobayashi, “Fabry-Perot interferometer based Mie Doppler lidar for low tropospheric wind observation,” Appl. Opt. 46, 7120–7131 (2007). [CrossRef]

6.

D. Su, A. A. P. Boechat, and J. D. C. Jones, “Beam delivery by large-core fibers: effect of launching conditions on near-field output profile,” Appl. Opt. 31, 5816–5821 (1992). [CrossRef]

7.

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

8.

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

9.

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

10.

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58 (2004). [CrossRef]

11.

H. Kihm and S.-W. Kim, “Nonparaxial free-space diffraction from oblique end faces of single-mode optical fibers,” Opt. Lett. 29, 2366–2368 (2004). [CrossRef]

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]

13.

M. Young and R. C. Wittmann, “Vector theory of diffraction by a single-mode fiber: application to mode-field diameter measurements,” Opt. Lett. 18, 1715–1717 (1993). [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/.

OCIS Codes
(120.2230) Instrumentation, measurement, and metrology : Fabry-Perot
(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments
(260.1960) Physical optics : Diffraction theory
(230.2285) Optical devices : Fiber devices and optical amplifiers
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: November 25, 2013
Manuscript Accepted: December 16, 2013
Published: February 13, 2014

Citation
Pavel Ionov, "Spectral response of fiber-coupled Fabry–Perot etalons," J. Opt. Soc. Am. A 31, 505-509 (2014)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-3-505


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References

  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.
  2. B. M. Gentry, H. Chen, and S. X. Li, “Wind measurements with 355  nm molecular Doppler lidar,” Opt. Lett. 25, 1231–1233 (2000). [CrossRef]
  3. C. J. Grund and E. W. Eloranta, “Fiber-optic scrambler reduces the bandpass range dependence of Fabry-Perot etalons used for spectral analysis of lidar backscatter,” Appl. Opt. 30, 2668–2670 (1991). [CrossRef]
  4. D. Hua, M. Uchida, and T. Kobayashi, “Ultraviolet Rayleigh-Mie lidar with Mie-scattering correction by Fabry-Perot etalons for temperature profiling of the troposphere,” Appl. Opt. 44, 1305–1314 (2005). [CrossRef]
  5. H. Xia, D. Sun, Y. Yang, F. Shen, J. Dong, and T. Kobayashi, “Fabry-Perot interferometer based Mie Doppler lidar for low tropospheric wind observation,” Appl. Opt. 46, 7120–7131 (2007). [CrossRef]
  6. D. Su, A. A. P. Boechat, and J. D. C. Jones, “Beam delivery by large-core fibers: effect of launching conditions on near-field output profile,” Appl. Opt. 31, 5816–5821 (1992). [CrossRef]
  7. A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics (Cambridge University, 1998).
  8. J. Goodman, Introduction to Fourier Optics (Roberts, 2005), Chap. 3.
  9. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979). [CrossRef]
  10. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58 (2004). [CrossRef]
  11. H. Kihm and S.-W. Kim, “Nonparaxial free-space diffraction from oblique end faces of single-mode optical fibers,” Opt. Lett. 29, 2366–2368 (2004). [CrossRef]
  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]
  13. M. Young and R. C. Wittmann, “Vector theory of diffraction by a single-mode fiber: application to mode-field diameter measurements,” Opt. Lett. 18, 1715–1717 (1993). [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/ .

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