Digital synthesis of multistage etalons based upon unequal cavity lengths

Faiza Iftikhar; Usman A. Khan; M. Imran Cheema

doi:10.1364/JOSAB.390617

Journal of the Optical Society of America B
Vol. 37,
Issue 6,
pp. 1630-1636
(2020)
•https://doi.org/10.1364/JOSAB.390617

Digital synthesis of multistage etalons based upon unequal cavity lengths

Faiza Iftikhar, Usman A. Khan, and M. Imran Cheema

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Faiza Iftikhar, Usman A. Khan, and M. Imran Cheema, "Digital synthesis of multistage etalons based upon unequal cavity lengths," J. Opt. Soc. Am. B 37, 1630-1636 (2020)

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Abstract

Fabry–Perot fiber etalons (FPEs) built from three or more reflectors are attractive for a variety of applications, including communication and sensing. In order to realize a desired transmission profile, usually multistage FPEs are designed with equal cavity lengths followed by a determination of the required reflectivities. As seen in previous works, however, fabricated reflectors may slightly differ from the designs, leading to a departure from the desired transmission profile of the FPE. Here, we show a novel digital synthesis of multistage etalons with off-the-shelf reflectors and unequal lengths of involved cavities. We find that, in contrast to equal cavity lengths, unequal lengths provide more poles in the $z$ domain and hence more flexibility to achieve a desired multicavity FPE transmission response. For given reflectivities and by determining correct unequal lengths of a multicavity FPE with our synthesis technique, we demonstrate two design examples involving free spectral range enhancement and passband shaping. This work is generalizable to ring resonators and mirror- and fiber-Bragg-grating-based cavities, enabling the design and optimization of cavity systems for a wide range of applications, including lasers, sensors, and filters.

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Inputs: $T_{est} (z), x_{m_{min}}, x_{m_{max}}, n_{max}, r_{n}$
Initialize: $n = 2$
Output: $n, x_{m}$
1:	function DIGISYNTH( $T_{est} (z), x_{m_{m i n}}, x_{m_{m a x}}, n_{m a x}, r_{n}$ )
2:	while $n < n_{m a x}$ do
3:	get $x_{m}$ corresponding to each $n$
4:	while scanning each $x_{m}$ from $x_{m_{m i n}}$ to $x_{m_{m a x}}$ do
5:	$a, b = c o e f (n, r_{n}, x_{m})$
6:	$t_{n} (z) = z p 2 t f (b, a)$
7:	$M S E = p e m (t_{n} (z), T_{est} (z))$
8:	if $M S E < 10^{- 7}$ then
9:	store $n, x_{m}$
10:	end if
11:	end while
12:	end while
13:	return all $n$ and corresponding $x_{m}$
14:	end function

$x_{1}$	$x_{2}$	MSE	$l_{1}$	$l_{2}$	PR (dB)
90	6	$4.92 \times 10^{- 7}$	62.1	4.1	−8.95
90	42	$5.70 \times 10^{- 7}$	62.1	29	−8.24
90	66	$6.15 \times 10^{- 7}$	62.1	45.5	−7.75
90	78	$8.14 \times 10^{- 7}$	62.1	53.8	−6.9

$x_{1}$	$x_{2}$	$x_{3}$	MSE	$l_{1}$	$l_{2}$	$l_{3}$	PR (dB)
90	6	42	$1.88 \times 10^{- 8}$	62.1	4.1	29	−25.04
90	54	6	$1.03 \times 10^{- 7}$	62.1	37.2	4.1	−15.81
90	66	66	$2.22 \times 10^{- 7}$	62.1	45.5	45.5	−7.57
90	78	66	$9.84 \times 10^{- 7}$	62.1	53.8	45.5	−4.89

$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	MSE	$l_{1}$	$l_{2}$	$l_{3}$	$l_{4}$	PR (dB)
90	18	6	30	$1.48 \times 10^{- 9}$	62.1	12.41	4.1	20.69	−39.96
90	78	6	42	$1.29 \times 10^{- 8}$	62.1	53.79	4.1	28.97	−25.63
90	90	66	66	$5.53 \times 10^{- 8}$	62.1	62.1	45.5	45.5	−18.73
90	54	42	96	$8.05 \times 10^{- 8}$	62.1	37.24	28.97	66.21	−16.99

Inputs: $T_{est} (z), x_{m_{min}}, x_{m_{max}}, n_{max}, r_{n}$
Initialize: $n = 2$
Output: $n, x_{m}$
1:	function DIGISYNTH( $T_{est} (z), x_{m_{m i n}}, x_{m_{m a x}}, n_{m a x}, r_{n}$ )
2:	while $n < n_{m a x}$ do
3:	get $x_{m}$ corresponding to each $n$
4:	while scanning each $x_{m}$ from $x_{m_{m i n}}$ to $x_{m_{m a x}}$ do
5:	$a, b = c o e f (n, r_{n}, x_{m})$
6:	$t_{n} (z) = z p 2 t f (b, a)$
7:	$M S E = p e m (t_{n} (z), T_{est} (z))$
8:	if $M S E < 10^{- 7}$ then
9:	store $n, x_{m}$
10:	end if
11:	end while
12:	end while
13:	return all $n$ and corresponding $x_{m}$
14:	end function

Abstract

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Figures (10)

Tables (4)

Equations (6)

Journal of the Optical Society of America B