Spectral response of fiber-coupled Fabry&#x2013;Perot etalons

Pavel Ionov

doi:10.1364/JOSAA.31.000505

1. INTRODUCTION

The use of multimode fibers for delivering light from telescopes to detectors is a frequent practice in remote sensing applications [1–5]. 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 [4,5]. 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] 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]. 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 [8,9].

First, a closed-form expression for diffracted ${LP}_{nm}$ 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.

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A general numerical treatment of this problem is, in principle, available through the use of a recently published Hankel transform algorithm [10]. 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 ${LP}_{nm}$ modes at the output end of the fiber.

2. DIFFRACTION OF ${LP}_{nm}$ MODE

In this section we show that a binomial approximation to the RS integral for the ${LP}_{nm}$ fiber mode can be reduced to a closed-form expression [see Eq. (9), below], via the $n$ th-order Hankel transform.

The binomial approximation to the RS integral has been used for other problems pertaining to fiber mode diffraction in free space [11,12]. Young and Wittmann [13] derived a version of Eq. (9) applicable to the specific case of $D_{\infty}$ 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].

With the notation of Fig. 2, the RS integral can be written as [8,9]

U (P_{1}) = \frac{1}{j λ} \iint_{Σ_{0}} U (P_{0}) \cos (n⃗, {r⃗}_{01}) \frac{\exp (j k r_{01})}{r_{01}} d s .

Here, the initial condition is given by

U (P_{0})

, which denotes the scalar field at the point

P_{0}

located on the exit surface of the fiber,

Σ_{0}

;

P_{1}

is an arbitrary point where the propagating field is calculated;

λ

is the wavelength;

k

is the wavenumber (

= 2 π / λ

);

{r⃗}_{01}

is the vector connecting points

P_{0}

and

P_{1}

[see Fig. 2(b)], with

r_{01} = | {r⃗}_{01} |

; and

n⃗

is normal to the surface

Σ_{0}

. The argument of the cosine term,

\cos (n⃗, {r⃗}_{01})

, is the angle between the

{r⃗}_{01}

and

n⃗

vectors. In our adopted system of coordinates [Fig. 2(b)], the point

P_{0}

is identified by the cylindrical coordinates (

r_{0}

,

φ_{0}

, 0), whereas

P_{1}

can be denoted by both cylindrical (

r_{1}

,

φ_{1}

,

z

) and spherical (

ρ

,

φ_{1}

,

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 $Σ_{0}$ by $U (P_{0})$ , and the field $U (P_{1})$ is calculated at a point $P_{1}$ . (b) Coordinate systems adopted to derive Eq. (9), (c) geometrical construct used to obtain Eq. (2).

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Based on the geometrical construct illustrated in Fig. 2(c), we have

\cos (n⃗, {r⃗}_{01}) = \frac{z}{r_{01}} = \frac{ρ \cos (θ)}{r_{01}} .

Moreover, given the functional form of the azimuth-angle dependence of

{LP}_{nm}

modes [7],

U (P_{0}) = \exp (j n φ_{0}) u_{0} (r_{0}),

where

n = 0, 1, \dots

is the mode order, we can rewrite Eq. (1) as

U (P_{1}) = \frac{ρ \exp (j n φ_{1}) \cos (θ)}{j λ} \int_{0}^{\infty} u_{0} (r_{0}) r_{0} d r_{0} \int_{0}^{2 π} \frac{\exp (j n Δ φ + j k r_{01})}{r_{01}^{2}} d Δ φ .

Here, the abbreviation

Δ φ = φ_{0} - φ_{1}

is introduced. Further,

r_{01}

is explicitly calculated by applying the cosine formula to the

{OP}_{0} P_{1}

triangle and spherical cosine formulas at the origin O [see Fig. 2(c)], which yields

r_{01} = {(ρ^{2} + r_{0}^{2} - 2 ρ r_{0} \sin (θ) \cos (Δ φ))}^{1 / 2} .

The square root term in Eq. (5) can be expanded into the Taylor series with respect to the argument

r_{0} / ρ

and retaining only the first-order term in phase and zero-order term in amplitude, in a fashion similar to the Fresnel approximation:

r_{01} \approx ρ - r_{0} \sin (θ) \cos (Δ φ), in the phase

Substituting Eq. (6) into Eq. (4) yields

U (P_{1}) = \frac{\exp (j n φ_{1} + j k ρ) \cos (θ)}{j λ ρ} \int_{0}^{\infty} u_{0} (r_{0}) r_{0} d r_{0} \int_{0}^{2 π} \exp (j n Δ φ + j k r_{0} \sin (θ) \cos (Δ φ)) d Δ φ .

Here, the integral over

Δ φ

is a Bessel function to within a constant phase factor, such that Eq. (7) can be rewritten as

U (P_{1}) = \frac{\exp (j n φ_{1} - j n π / 2 + j k ρ) \cos (θ)}{j λ ρ} 2 π \int_{0}^{\infty} u_{0} (r_{0}) J_{n} (k r_{0} \sin (θ)) r_{0} d r_{0} .

The radial integral can then be expressed in terms of a Hankel transform:

U (P_{1}) = \frac{\exp (j n φ_{1} - j n π / 2 + j k ρ) \cos (θ)}{j λ ρ} H_{n} {u_{0}} (\frac{\sin (θ)}{λ}),

where forward and reverse Hankel transforms are defined as

H_{n} {f} (ν) = 2 π \int_{0}^{\infty} f (r) J_{n} (2 π ν r) r d r,

The approximation in Eq. (6) is of the order

d / 2 z

, 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], which is of order

r / 2 z

(corresponding to the numerical aperture of the fiber) in the case of Fig. 1. To elucidate this aspect, a Gaussian beam of width

w_{0} = 1 mm

and

λ = 1.06 μm

is propagated according to Eq. (9) for distances of 3 and 10 times the Rayleigh range,

z_{R} = π w_{0}^{2} / λ

. As shown in Fig. 3, excellent agreement with the analytical formula is observed at

10 z_{R}

.

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

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3. COLLIMATION OF ${LP}_{nm}$ 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 (P_{1}) = \exp (j n φ_{1}) \frac{\exp (j k ρ)}{λ ρ} u_{1} (θ) .

Here, the angular term dependent on

φ_{1}

is identical to that of the

{LP}_{nm}

mode, confirming that the approximation does not break the

D_{n}

symmetry of the system.

The second term, $\exp (j k ρ) / λ ρ$ , is a spherical wave. Finally, $u_{1} (θ)$ is an amplitude term expressing the radial field dependence at the collimator plane $Σ_{1}$ and is explicitly given by

u_{1} (θ) = \cos (θ) H_{n} {u_{0}} (\frac{\sin (θ)}{λ}) .

The term

u_{1} (θ)

takes on real values for real

u_{0} (r_{0})

, as is the case of LP modes. Based on Eq. (11), only the second (spherical) phase term

\exp (- j k ρ)

can be compensated with a regular lens system, because both share

D_{\infty}

axial symmetry. Conversely, the first phase term,

\exp (j n φ_{1})

, exhibits

D_{n}

symmetry and cannot be corrected with regular lenses. Thus, since

u_{1} (θ)

is real for LP modes, we can conclude that the ideal collimator would have a phase of

\exp (- j k ρ)

, which corresponds to an ideal lens with a focal length

F = z

. After such a collimator the field becomes

U (Σ_{2}) = \exp (j n φ_{2}) u_{2} (r_{2}),

where

u_{2} (r_{2}) = \frac{F}{λ (F^{2} + r_{2}^{2})} H_{n} {u_{0}} (\frac{r_{2}}{λ {(F^{2} + r_{2}^{2})}^{0.5}}) .

Before considering how such a collimated field diffracts, it is instructive to examine the case of a small $NA ≪ 1$ , or $θ ≪ 1$ . In this limit, the collimated field [Eqs. (13) and (14)] can simply be written as

U (Σ_{2}) \approx \exp (j n φ_{2}) H_{n} {u_{0}} (r_{2} / λ F) / λ F .

In this expression, the radial component of the collimated beam simply becomes a Hankel transform of the radial component of the

{LP}_{nm}

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 $θ ≪ 1$ for a collimated beam, Eq. (15) is a good approximation, and the far-field radial component of the collimated field can be written as

u_{3} (θ) = H_{n} {u_{2}} (\frac{θ}{λ}) .

If the fiber

NA ≪ 1

, successive application of Eqs. (15) and (16) produces

u_{0} (θ F)

for

u_{3} (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]. 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]):

g_{n} (k_{r}) = \exp (j ω t + j n φ + j k_{z} z) J_{n} (k_{r} r), where

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 ( $k_{z}$ and $k_{r}$ , respectively). By introducing amplitude transmission ( $t_{1}$ , $t_{2}$ ) and reflectivity ( $r_{1}$ , $r_{2}$ ) for the etalon mirrors (denoted by the indices “1” and “2”), we can express the overall etalon transmission [ $T_{e} (k_{z})$ ] as a geometrical series [14], which, for a Bessel wave with a given $k_{z}$ , reads as

T_{e} (k_{z}) = {| \frac{t_{1} t_{2}}{1 - r_{1} r_{2} \exp (2 j k_{z} L)} |}^{2} .

By expressing the field

u_{2}

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

I_{t} (L_{nm}) = \int_{0}^{\infty} {| H {u_{2}} (ν) |}^{2} T_{e} (2 π \sqrt{λ^{- 2} - ν^{2}}) ν d ν .

Since transverse spatial frequency $ν ≪ 1 / λ$ 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 $ν$ :

I_{t} (L_{nm}) = \int_{0}^{\infty} {| H {u_{2}} (ν) |}^{2} T_{e} (2 π (λ^{- 1} - 0.5 λ ν^{2})) ν d ν .

It is convenient to rewrite Eq. (20) by defining a frequency shift $Δ f = 0.5 λ ν^{2} c$ and a function $T_{f} (Δ f)$ :

I_{t} / I = \int_{0}^{\infty} T_{f} (Δ f) T_{e} (2 π (f - Δ f) / c) d Δ f, where

In this notation, frequency-dependent transmission of the fiber-coupled etalon is a convolution of the response of the etalon itself and the function

T_{f} (Δ f)

[Eq. (21a)]. Thus, the function

T_{f} (Δ f)

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

T_{f}

of the fiber and collimator can be studied independently of the etalon. In applications, such as Doppler lidar [2,5], where high spectral accuracy is required, this mode dependence of the instrumental function presents a challenge. On the other hand, independence of

T_{f}

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 ( $NA ≪ 1$ ), the expression for $T_{f}$ [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

T_{f} (Δ f) \approx {| u_{0} (F \sqrt{2 Δ f / f}) |}^{2} / \int_{0}^{\infty} {| u_{0} (F \sqrt{2 Δ f / f}) |}^{2} d Δ f, for NA ≪ 1 .

Thus, in the case of a low-NA fiber,

T_{f}

, is expressed only through the radial component of the

{LP}_{nm}

mode,

u_{0}

.

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] 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 $T_{f}$ 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 $T_{f}$ 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], 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 $T_{f}$ 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 $T_{f}$ , for high-order mode in $NA = 0.22$ fiber.

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Fig. 5. Fiber–collimator instrumental function $T_{f}$ , for high-order mode in $NA = 0.12$ fiber.

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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 $T_{f}$ versus frequency. Thus, frequency shift and broadening are of the order of $0.5 f {(d / F)}^{2}$ 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 ${LP}_{nm}$ 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.

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16. http://numpy.scipy.org/.

Spectral response of fiber-coupled Fabry–Perot etalons

Abstract

1. INTRODUCTION

2. DIFFRACTION OF ${LP}_{nm}$ MODE

3. COLLIMATION OF ${LP}_{nm}$ MODE

4. FIBER-COUPLED FABRY–PEROT ETALON

5. CONCLUSION

ACKNOWLEDGMENTS

REFERENCES

Cited By

Figures (5)

Equations (26)

Journal of the Optical Society of America A

Abstract

1. INTRODUCTION

2. DIFFRACTION OF LPnm MODE

3. COLLIMATION OF LPnm MODE

4. FIBER-COUPLED FABRY–PEROT ETALON

5. CONCLUSION

ACKNOWLEDGMENTS

REFERENCES

Cited By

Figures (5)

Equations (26)

Journal of the Optical Society of America A

2. DIFFRACTION OF ${LP}_{nm}$ MODE

3. COLLIMATION OF ${LP}_{nm}$ MODE