Spectrometers working on the same extensive continuum source: an estimation of the relative performances of eight instruments

John Meaburn

doi:10.1364/AO.14.002521

Applied Optics
Vol. 14,
Issue 10,
pp. 2521-2526
(1975)
•https://doi.org/10.1364/AO.14.002521

Spectrometers working on the same extensive continuum source: an estimation of the relative performances of eight instruments

John Meaburn

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John Meaburn, "Spectrometers working on the same extensive continuum source: an estimation of the relative performances of eight instruments," Appl. Opt. 14, 2521-2526 (1975)

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Abstract

The SNR's predicted for three types of Hadamard transform spectrometer, a Michelson Fourier spectrometer, SISAM, PEPSIOS, and grating monochromators and a grating spectrograph all observing an extensive continuum source are estimated and compared in both photon and receiver noise limited conditions. This complements a previous paper [ J. Meaburn, Appl. Opt. 12, 279 ( 1973)] dealing with spectrometers for extended line emission sources in a similar way.

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Equations (110)

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			B_c = 0		B_c ≥ B₀
Spectrometer	l_max	R_max	Photon noise ltd.	Rec. noise ltd.	Photon noise ltd.
Grating monochromator (GM)	10⁵	5 × 10⁴	${(K Ω_{GM} T / l)}^{½}$ or for p > 1	$(\frac{K Ω_{GM}}{CONST}) {(\frac{T}{l})}^{½}$	${[K B_{0} Ω_{GM} T / l (B_{0} + B_{c})]}^{½}$
			${(K Ω_{GM} T / l p)}^{½}$	$(\frac{K Ω_{GM}}{CONST}) {(\frac{T}{l p})}^{½}$	${[K B_{0} Ω_{GM} T / l p (B_{0} + B_{c})]}^{½}$
Michelson (M)	10⁶	2 × 10⁶	${(K Ω_{M} T / 2 l)}^{½}$	$(\frac{K Ω_{M}}{2 CONST}) T^{½}$	${[K B_{0} Ω_{M} T / 2 l (B_{0} + B_{c})]}^{½}$
Hadamard transform (HTS₁)	2 × 10³	5 × 10⁴	${(K Ω_{{HTS}_{1}} T / l x_{l})}^{½}$	$(\frac{K Ω_{{HTS}_{1}}}{CONST x_{l}}) T^{½}$	${[K B_{0} Ω_{{HTS}_{1}} T / l x_{l} (B_{0} + B_{c})]}^{½}$
Hadamard transform (HTS₂)	2 × 10³	5 × 10⁴	${(K Ω_{{HTS}_{1}} T / l p x_{l} x_{p})}^{½}$	$(\frac{K Ω_{{HTS}_{1}}}{CONST x_{l} x_{p}}) T^{½}$	${[K B_{0} Ω_{{HTS}_{1}} T / l p x_{l} x_{p} (B_{0} + B_{c})]}^{½}$
Hadamard transform (HTS₃)	2 × 10³	5 × 10⁴	${(K Ω_{{HTS}_{1}} T / l x_{l} x_{p})}^{½}$	$(\frac{K p Ω_{{HTS}_{1}}}{CONST x_{l} x_{p}}) T^{½}$	${[K B_{0} Ω_{{HTS}_{1}} T / l x_{l} x_{p} (B_{0} + B_{c})]}^{½}$
PEPSIOS monochromator (PM)	10²	2 × 10⁶	${(K Ω_{PM} T / l)}^{½}$	$(\frac{K Ω_{PM}}{CONST}) {(\frac{T}{l})}^{½}$	${[K B_{0} Ω_{PM} T / l (B_{o} + B_{c})]}^{½}$
SISAM monochromator (SM)	10³	5 × 10⁴	$\frac{1}{l} {(K Ω_{SM} T / 2)}^{½}$	$(\frac{K Ω_{SM}}{2 CONST}) {(\frac{T}{l})}^{½}$	$\frac{1}{l} {[K B_{0} Ω_{SM} T / 2 (B_{0} + B_{c})]}^{½}$
Grating spectrograph (GS)	10⁴	5 × 10⁴	${(K Ω_{GM} T)}^{½}$	$(\frac{K Ω_{GM}}{CONST}) T^{½}$	${[K B_{0} Ω_{GM} T / (B_{o} + B_{c})]}^{½}$

	Photon noise (B_c = 0) limited	Receiver noise limited
$\frac{{(SNR)}_{M}}{{(SNR)}_{{HTS}_{1}}}$	${(\frac{Ω_{M} x_{l}}{2 Ω_{{HTS}_{1}}})}^{½} \sim 10$	$(\frac{Ω_{M}}{2 Ω_{{HTS}_{1}}} x_{l}) \sim 100$
$\frac{{(SNR)}_{M}}{{(SNR)}_{{HTS}_{3}}}$	${(\frac{Ω_{M} x_{l} x_{p}}{2 Ω_{{HTS}_{1}}})}^{½} \sim 14$	$(\frac{Ω_{M} x_{l} x_{p}}{2 Ω_{{HTS}_{1}} p}) \sim 2$
$\frac{{(SNR)}_{M}}{{(SNR)}_{GM}}$	${(\frac{Ω_{M}}{2 Ω_{GM}})}^{½} \sim 7$	$(\frac{Ω_{M}}{2 Ω_{GM}}) l^{½} \sim 50 l^{½}$
$\frac{{(SNR)}_{{HTS}_{1}}}{{(SNR)}_{GM}}$	${(\frac{Ω_{{HTS}_{1}}}{Ω_{GM} x_{l}})}^{½} \sim 0.7$	$(\frac{Ω_{{HTS}_{1}}}{x_{l} Ω_{GM}}) l^{½} \sim 0.5 l^{½}$
$\frac{{(SNR)}_{{HTS}_{2}}}{{(SNR)}_{GM}}$	${(\frac{Ω_{{HTS}_{1}}}{Ω_{GM} x_{l} x_{p}})}^{½} \sim 0.5$	$(\frac{Ω_{{HTS}_{1}}}{Ω_{GM} x_{l} x_{p}}) l^{½} \sim 0.25 l^{½}$
$\frac{{(SNR)}_{PM}}{{(SNR)}_{M}}$	${(\frac{2 Ω_{PM}}{Ω_{M}})}^{½} \sim 1.4$	$(\frac{2 Ω_{PM}}{Ω_{M}}) {(\frac{1}{l})}^{½} \sim {(\frac{4}{l})}^{½}$
$\frac{{(SNR)}_{{HTS}_{1}}}{{(SNR)}_{GS}}$	${(\frac{Ω_{{HTS}_{1}}}{Ω_{GM} l x_{l}})}^{½} \sim 0.7 {(\frac{1}{l})}^{½}$	$\frac{Ω_{{HTS}_{1}}}{Ω_{GM} x_{l}} \sim 0.5$
$\frac{{(SNR)}_{SM}}{{(SNR)}_{M}}$	${(\frac{Ω_{SM}}{Ω_{M} l})}^{½} \sim {(\frac{1}{l})}^{½}$	$\frac{Ω_{SM}}{Ω_{M}} {(\frac{1}{l})}^{½} \sim {(\frac{1}{l})}^{½}$
$\frac{{(SNR)}_{M}}{{(SNR)}_{GS}}$	${(\frac{Ω_{M}}{2 Ω_{GM}} \frac{1}{l})}^{½} \sim 7 {(\frac{1}{l})}^{½}$	$(\frac{Ω_{M}}{2 Ω_{GM}}) \sim 50$

			B_c = 0		B_c ≥ B₀
Spectrometer	l_max	R_max	Photon noise ltd.	Rec. noise ltd.	Photon noise ltd.
Grating monochromator (GM)	10⁵	5 × 10⁴	${(K Ω_{GM} T / l)}^{½}$ or for p > 1	$(\frac{K Ω_{GM}}{CONST}) {(\frac{T}{l})}^{½}$	${[K B_{0} Ω_{GM} T / l (B_{0} + B_{c})]}^{½}$
			${(K Ω_{GM} T / l p)}^{½}$	$(\frac{K Ω_{GM}}{CONST}) {(\frac{T}{l p})}^{½}$	${[K B_{0} Ω_{GM} T / l p (B_{0} + B_{c})]}^{½}$
Michelson (M)	10⁶	2 × 10⁶	${(K Ω_{M} T / 2 l)}^{½}$	$(\frac{K Ω_{M}}{2 CONST}) T^{½}$	${[K B_{0} Ω_{M} T / 2 l (B_{0} + B_{c})]}^{½}$
Hadamard transform (HTS₁)	2 × 10³	5 × 10⁴	${(K Ω_{{HTS}_{1}} T / l x_{l})}^{½}$	$(\frac{K Ω_{{HTS}_{1}}}{CONST x_{l}}) T^{½}$	${[K B_{0} Ω_{{HTS}_{1}} T / l x_{l} (B_{0} + B_{c})]}^{½}$
Hadamard transform (HTS₂)	2 × 10³	5 × 10⁴	${(K Ω_{{HTS}_{1}} T / l p x_{l} x_{p})}^{½}$	$(\frac{K Ω_{{HTS}_{1}}}{CONST x_{l} x_{p}}) T^{½}$	${[K B_{0} Ω_{{HTS}_{1}} T / l p x_{l} x_{p} (B_{0} + B_{c})]}^{½}$
Hadamard transform (HTS₃)	2 × 10³	5 × 10⁴	${(K Ω_{{HTS}_{1}} T / l x_{l} x_{p})}^{½}$	$(\frac{K p Ω_{{HTS}_{1}}}{CONST x_{l} x_{p}}) T^{½}$	${[K B_{0} Ω_{{HTS}_{1}} T / l x_{l} x_{p} (B_{0} + B_{c})]}^{½}$
PEPSIOS monochromator (PM)	10²	2 × 10⁶	${(K Ω_{PM} T / l)}^{½}$	$(\frac{K Ω_{PM}}{CONST}) {(\frac{T}{l})}^{½}$	${[K B_{0} Ω_{PM} T / l (B_{o} + B_{c})]}^{½}$
SISAM monochromator (SM)	10³	5 × 10⁴	$\frac{1}{l} {(K Ω_{SM} T / 2)}^{½}$	$(\frac{K Ω_{SM}}{2 CONST}) {(\frac{T}{l})}^{½}$	$\frac{1}{l} {[K B_{0} Ω_{SM} T / 2 (B_{0} + B_{c})]}^{½}$
Grating spectrograph (GS)	10⁴	5 × 10⁴	${(K Ω_{GM} T)}^{½}$	$(\frac{K Ω_{GM}}{CONST}) T^{½}$	${[K B_{0} Ω_{GM} T / (B_{o} + B_{c})]}^{½}$

	Photon noise (B_c = 0) limited	Receiver noise limited
$\frac{{(SNR)}_{M}}{{(SNR)}_{{HTS}_{1}}}$	${(\frac{Ω_{M} x_{l}}{2 Ω_{{HTS}_{1}}})}^{½} \sim 10$	$(\frac{Ω_{M}}{2 Ω_{{HTS}_{1}}} x_{l}) \sim 100$
$\frac{{(SNR)}_{M}}{{(SNR)}_{{HTS}_{3}}}$	${(\frac{Ω_{M} x_{l} x_{p}}{2 Ω_{{HTS}_{1}}})}^{½} \sim 14$	$(\frac{Ω_{M} x_{l} x_{p}}{2 Ω_{{HTS}_{1}} p}) \sim 2$
$\frac{{(SNR)}_{M}}{{(SNR)}_{GM}}$	${(\frac{Ω_{M}}{2 Ω_{GM}})}^{½} \sim 7$	$(\frac{Ω_{M}}{2 Ω_{GM}}) l^{½} \sim 50 l^{½}$
$\frac{{(SNR)}_{{HTS}_{1}}}{{(SNR)}_{GM}}$	${(\frac{Ω_{{HTS}_{1}}}{Ω_{GM} x_{l}})}^{½} \sim 0.7$	$(\frac{Ω_{{HTS}_{1}}}{x_{l} Ω_{GM}}) l^{½} \sim 0.5 l^{½}$
$\frac{{(SNR)}_{{HTS}_{2}}}{{(SNR)}_{GM}}$	${(\frac{Ω_{{HTS}_{1}}}{Ω_{GM} x_{l} x_{p}})}^{½} \sim 0.5$	$(\frac{Ω_{{HTS}_{1}}}{Ω_{GM} x_{l} x_{p}}) l^{½} \sim 0.25 l^{½}$
$\frac{{(SNR)}_{PM}}{{(SNR)}_{M}}$	${(\frac{2 Ω_{PM}}{Ω_{M}})}^{½} \sim 1.4$	$(\frac{2 Ω_{PM}}{Ω_{M}}) {(\frac{1}{l})}^{½} \sim {(\frac{4}{l})}^{½}$
$\frac{{(SNR)}_{{HTS}_{1}}}{{(SNR)}_{GS}}$	${(\frac{Ω_{{HTS}_{1}}}{Ω_{GM} l x_{l}})}^{½} \sim 0.7 {(\frac{1}{l})}^{½}$	$\frac{Ω_{{HTS}_{1}}}{Ω_{GM} x_{l}} \sim 0.5$
$\frac{{(SNR)}_{SM}}{{(SNR)}_{M}}$	${(\frac{Ω_{SM}}{Ω_{M} l})}^{½} \sim {(\frac{1}{l})}^{½}$	$\frac{Ω_{SM}}{Ω_{M}} {(\frac{1}{l})}^{½} \sim {(\frac{1}{l})}^{½}$
$\frac{{(SNR)}_{M}}{{(SNR)}_{GS}}$	${(\frac{Ω_{M}}{2 Ω_{GM}} \frac{1}{l})}^{½} \sim 7 {(\frac{1}{l})}^{½}$	$(\frac{Ω_{M}}{2 Ω_{GM}}) \sim 50$

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Applied Optics