ABCD transfer matrix model of Gaussian beam propagation in Fabry-Perot etalons

David Martin-Sanchez; Jing Li; Dylan M. Marques; Dylan M. Marques; Edward Z. Zhang; Edward Z. Zhang; Peter R. T. Munro; Paul C. Beard; Paul C. Beard; James A. Guggenheim; James A. Guggenheim; James A. Guggenheim

doi:10.1364/OE.477563

Optics Express
Vol. 30,
Issue 26,
pp. 46404-46417
(2022)
•https://doi.org/10.1364/OE.477563

ABCD transfer matrix model of Gaussian beam propagation in Fabry-Perot etalons

David Martin-Sanchez, Jing Li, Dylan M. Marques, Edward Z. Zhang, Peter R. T. Munro, Paul C. Beard, and James A. Guggenheim

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David Martin-Sanchez, Jing Li, Dylan M. Marques, Edward Z. Zhang, Peter R. T. Munro, Paul C. Beard, and James A. Guggenheim, "ABCD transfer matrix model of Gaussian beam propagation in Fabry-Perot etalons," Opt. Express 30, 46404-46417 (2022)

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Abstract

A numerical model of Gaussian beam propagation in planar Fabry-Perot (FP) etalons is presented. The model is based on the ABCD transfer matrix method. This method is easy to use and interpret, and readily connects models of lenses, mirrors, fibres and other optics to aid simulating complex multi-component etalon systems. To validate the etalon model, its predictions were verified using a previously validated model based on Fourier optics. To demonstrate its utility, three different etalon systems were simulated. The results suggest the model is valid and versatile and could aid in designing and understanding a range of systems containing planar FP etalons. The method could be extended to model higher order beams, other FP type devices such as plano-concave resonators, and more complex etalon systems such as those involving tilted components.

Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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Data availability

The code is open-source and freely available in Ref. [41].

41. D. Martin-Sanchez, D. M. Marques, E. Z. Zhang, P. R. T. Munro, P. C. Beard, and J. A. Guggenheim, “ABCD model for FP, etalons script,”GitHub (2022). https://github.com/marsandav/ABCDmodel

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Fig. 1. Multiple-beam interference in a Fabry-Perot etalon. Note, normally-incident beams are assumed in this work; the diagonal rays are for illustrative purpose only.

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Fig. 2. Total reflected field, |U_out|, calculated using the ABCD model (blue line) as compared to the AAF (red dashed line) for an FP etalon (h = 102 µm, n = 1.444, R₁ = R₂ = 97%) illuminated at normal incidence by a Gaussian beam with a spot size of (a) 2ω₀ = 50 µm (top row) and (b) 2ω₀ = 250 µm, using 200 radial spatial sampling points, indicating the number of partial beams calculated before convergence. Simulations performed at the resonance wavelength.

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Fig. 3. ITFs of an FP etalon (h = 102 µm, n = 1.444, R₁ = R₂ = 97%) calculated using the Angular Airy Function model (dashed lines) and the ABCD model (solid lines) illuminated at normal incidence by four Gaussian beams of different focal spot size (2ω₀) as per the figure subtitles. The ITFs were simulated in reflection mode (top row) and transmission mode (bottom row). An infinitely large detector was assumed, and the infinite extent of the field was approximated by 10ω₀. The spectral sampling interval rate was 1 pm and the radial profile was sampled using 200 radial spatial sampling points.

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Fig. 4. Comparison of fringe width (left), visibility (centre) and finesse (right) simulated with the Angular Airy Function (dots) and the ABCD model (lines) using several spot sizes considering an infinite aperture detector. FP etalon made of fused silica (n = 1.444 at λ=1550 nm), h = 102 µm, and variable mirror reflectivities, R₁ = R₂ between 95.8% and 98%. Visibility is defined as V=|I_max-I_min|/(I_max + I_min) and finesse as the free spectral range divided by the FWHM.

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Fig. 5. a) Schematic diagram of the modelled optical system consisting of an optical fibre (MFD = 10.4 µm) illuminating an FP etalon (h = 102 µm, n = 1.444, R₁ = R₂ = 97%) using a collimator and an objective in a 4f configuration where the reflected beam is directed on to a fibre-coupled detector; and b) Comparison of ITFs in reflection mode for fibre based detection and free space detection (blue and red, respectively), for different spot sizes.

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Fig. 6. a) Schematic diagram of an FP etalon integrated in an optical fibre illuminated by a diverging beam focussed using a GRIN lens; b) reflection-mode ITFs obtained using the ABCD model for different lengths of GRIN lens (h_g = 0 µm, h_g = 50 µm and h_g = 100 µm); and c) ITFs obtained for different lengths of NCF (z_off = 10, 30 and 60 µm). For the simulations, the etalon parameters were n₀ = 1.444, h_g = 102 µm, z_off = 10 µm, and MFD = 10.4 µm

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Fig. 7. a) Schematic diagram of the modelled optical system consisting of two cascaded FP etalons and a pair of thin lenses in 4f configuration illuminated by a single mode fibre; and b) Simulated ITF in reflection mode of the system, when h₁ = 100 µm, h₂ = 400 µm, R₁ = R₂ = R₃ = 20%, n = 1.444, and 2ω0 = 30 µm.

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Tables (1)

Table 1. ABCD matrices of some optical elements [18].

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Equations (18)

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q (z_{1}) \to [\begin{array}{cc} A & B \\ C & D \end{array}] \to q (z) = \frac{A q (z_{1}) + B}{C q (z_{1}) + D}

\frac{1}{q (z)} = \frac{1}{R (z)} - i \frac{2}{k ω^{2} (z)},

U_{o u t} (r, z; M) = \frac{1}{A + B / q (z_{1})} \exp (- i k \frac{r^{2}}{2 q (z)}),

U_{o u t} (r, z) = \frac{ω_{0}}{ω (z)} \exp (- \frac{r^{2}}{ω^{2} (z)}) \exp (- i k z - \frac{i k r^{2}}{2 R (z)} - i ζ (z)),

M = M_{N} \dots M_{2} M_{1}

M_{i n} = M_{r e f r} .

M_{c a v} = M_{r e f l}^{\leftarrow} M_{p r o p}^{\leftarrow} M_{r e f l} M_{p r o p} .

M_{o u t} = {\begin{array}{cc} M_{r e f r} M_{p r o p}, & t r a n s m i s s i o n \\ M_{r e f r}^{\leftarrow} M_{p r o p}^{\leftarrow} M_{r e f l} M_{p r o p}, & r e f l e c t i o n \end{array}

M_{F P, m} = M_{o u t} (M_{c a v})^{m} M_{i n} .

M_{s y s t, m} = M_{detection} M_{F P, m} M_{illumination},

U_{o u t} (r, z) = \sum_{m = 0}^{\infty} A_{m} U_{m} (r, z; M_{s y s t, m}), A_{m} = {\begin{array}{cc} t_{1} t_{2} {(r_{1} r_{2})}^{m} & m \geq 0, t r a n s m i s s i o n \\ t_{1} r_{2} t_{1} {(r_{1} r_{2})}^{m - 1} & m > 0, r e f l e c t i o n \\ r_{1} & m = 0, r e f l e c t i o n \end{array}

I T F (λ) = \frac{I_{o u t} (λ)}{I_{i n} (λ)},

I_{o u t} = \int_{0}^{\infty} {| U_{o u t} (r, z) |}^{2} 2 π r d r,

I_{o u t} = {| \int_{0}^{\infty} U_{o u t} (r, z) \exp (- \frac{4 r^{2}}{M F D^{2}}) 2 π r d r |}^{2},

M_{s y s t, m} = M_{4 f}^{\leftarrow} M_{F P, m} M_{4 f},

M_{4 f} = M_{p r o p}^{f_{o b j}} M_{l e n s}^{f_{o b j}} M_{p r o p}^{f_{o b j} + f_{c o l}} M_{l e n s}^{f_{c o l}} M_{p r o p}^{f_{c o l}} = [\begin{array}{cc} - f_{o b j} / f_{c o l} & 0 \\ 0 & - f_{c o l} / f_{o b j} \end{array}] .

M_{s y s t, m} = M_{p r o p}^{z_{o f f} \leftarrow} M_{G R I N}^{\leftarrow} M_{F P, m} M_{G R I N} M_{p r o p}^{z_{o f f}} .

M_{s y s t, m, n} = M_{4 f}^{\leftarrow} M_{F P 2, n} M_{F P 1, m} M_{4 f},

Dylan M. Marques	https://orcid.org/0000-0003-2666-5900
Peter R. T. Munro	https://orcid.org/0000-0001-9306-3673
James A. Guggenheim	https://orcid.org/0000-0003-4295-0525

Abstract

Data availability

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Tables (1)

Equations (18)

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