Cross-correlation discrimination for optical cavity modes

C. C. Aleksoff; B. D. Guenther

doi:10.1364/AO.15.000206

Applied Optics
Vol. 15,
Issue 1,
pp. 206-217
(1976)
•https://doi.org/10.1364/AO.15.000206

Cross-correlation discrimination for optical cavity modes

C. C. Aleksoff and B. D. Guenther

Not Accessible

Your library or personal account may give you access

Get PDF
Email
Share
Get Citation
Copy Citation Text
C. C. Aleksoff and B. D. Guenther, "Cross-correlation discrimination for optical cavity modes," Appl. Opt. 15, 206-217 (1976)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Abstract

Matched spatial filters were constructed for various modes of a spherical Fabry-Perot interferometer (FPSI). The cross correlation between many of these modes was obtained experimentally as well as theoretically. The orthogonality between various modes was demonstrated and used in order to discriminate against all orthogonal modes. A discrimination funtion that predicts the performance of the matched-filtering process is defined and analytically evaluated for rectangular and cylindrical Gaussian modes. It is shown that arbitrary discrimination strength can be obtained at the sacrifice of output signal strength. It was experimentally demonstrated that a scanning FPSI used in conjunction with only one zero-order-mode filter could be used over nearly its entire domain of stable cavity spacings and still maintain a full (c/2L) free spectral range.

Full Article | PDF Article

More Like This

Quality control of small mechanical pieces using optical correlation techniques

G. Indebetouw, T. Tschudi, and G. Herziger
Appl. Opt. 15(2) 516-522 (1976)

Annular converging wave cavity

R. A. Chodzko, S. B. Mason, and E. F. Cross
Appl. Opt. 15(9) 2137-2144 (1976)

Lensless matched filter: operating principle, sensitivity to spectrum shift, and third-order holographic aberrations

Marc J. Bage and Michael P. Beddoes
Appl. Opt. 15(11) 2830-2839 (1976)

Previous Article Next Article

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Figures (14)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Tables (3)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Equations (66)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

$Ψ_{m n} (x, y) = Ψ_{m} (x) Ψ_{n} (y)$		(A)
$Ψ_{m} (x) = {(2^{m} m! π^{1 / 2})}^{- 1 / 2} H_{m} (x) exp (- x^{2} / 2)$		(B)
$H_{m} (x) = {(- 1)}^{m} exp (x^{2}) \frac{d^{m} exp (- x^{2})}{d x^{m}}$	[7–193(7)]	(C)
	[16–785 (22.11.7)]
$Ψ_{m} (x) = {(\frac{2}{m})}^{1 / 2} x Ψ_{m - 1} (x) - {(\frac{m + 1}{m})}^{1 / 2} Ψ_{m - 2} (x) m ⩾ 2$		(D)
	[7–193(10)]
	[16–782(22.7.13)]
$\begin{array}{l} \begin{array}{l} Ψ_{0} (x) & = π^{- 1 / 4} exp (- x^{2} / 2) \\ Ψ_{1} (x) & = \sqrt 2 x Ψ_{0} (x) \end{array} \\ F [Ψ_{m n} (x, y)] = 2 π {(- i)}^{m + n} Ψ_{m n} (2 π f_{x}, 2 π f_{y}) \end{array}$	[7–193(8)]	(E)
	[17–39(7), 94(8)]
	[18–838(7.376.1)]
$\int \int_{- \infty}^{\infty} Ψ_{m n} (x, y) Ψ_{P q} (x, y) dxdy = δ_{m P} δ_{n q}$	[16–775(22.2.14)]	(F)

$Φ_{P q} (r, θ) = R_{P q} (r) Θ_{q} (θ)$		(A)
$Θ_{q} (θ) = {[π (1 + δ_{0 q})]}^{- 1 / 2} cos (q θ)$		(B)
$where δ_{0 q} = {\begin{array}{l} 1 if q = 0 \\ 0 if otherwise \end{array}$
$R_{P q} (r) = {[2 P! / (P + q)!]}^{1 / 2} r q {L_{P}}^{(q)} (r^{2}) exp (- r^{2} / 2)$		(C)
${L_{P}}^{(q)} (t) = \frac{e^{t}}{P! t^{P}} \frac{d^{P}}{d t^{P}} [exp (- t) t^{P + q}]$	[7–188(5)]	(D)
	[16–785(22.11.6)]
$\begin{array}{l} R_{P q} (r) = \frac{2 P + q - 1 - r^{2}}{{[P (P + q)]}^{1 / 2}} R_{P - 1, q} (r) - {[\frac{(P - 1) (P + q - 1)}{P (P + q)}]}^{1 / 2} R_{P - 2, q} (r) \end{array}$	$P ⩾ 2$ [7–188(8)] [16–782(22.7.12)]	(E)
$\begin{matrix} R_{0 q} (r) = {(\frac{2}{q!})}^{1 / 2} r^{q} exp (- r^{2} / 2) \end{matrix}$
$\begin{matrix} R_{1 q} (r) = {[\frac{2}{(q + 1)!}]}^{1 / 2} r^{q} (q + 1 - r^{2}) exp (- r^{2} / 2) \end{matrix}$	[7–188(6)]
$H [R_{P q} (r); ρ] = {(- 1)}^{P} R_{P q} (ρ)$	[15–42(3)]	(F)
$F [Φ_{P q} (r, θ)] = 2 π {(- i)}^{2 P + q} Φ_{P q} (2 π f_{r}, ϕ)$	[15–42(3)]	(G)
$\int_{0}^{2 π} \int_{0}^{\infty} Φ_{P q} (r, θ) Φ_{m n} (r, θ) rdrd θ = δ_{P m} δ_{q n}$	[16–775(22.2.12)]	(H)
	[16–78(4.3.140)]

$\begin{array}{l} \begin{matrix} Ψ_{m} {(x)}_{x}^{} Ψ_{n} (x) = 2^{- 1 / 2} R_{n, m - n} (2^{- 1 / 2} x) n ⩽ m & [15 - 292 (30)] \end{matrix} \\ Ψ_{0} {(x)}_{x}^{} Ψ_{m} (x) = {(m! 2^{m})}^{- 1 / 2} x^{m} exp (- x^{2} / 4) \end{array}$	(A)
$Ψ_{0} {(A x)}_{x}^{*} Ψ_{m} (B x) = {(\frac{2 π}{A^{2} + B^{2}})}^{1 / 2} {(\frac{A^{2} - B^{2}}{A^{2} + B^{2}})}^{m / 2} Ψ_{0} [\frac{A B^{2} x}{{(B^{4} - A^{4})}^{1 / 2}}] Ψ_{m} [\frac{A^{2} B x}{{(A^{4} - B^{4})}^{1 / 2}}] [15 - 290 (17)]$	(B)
$Φ_{00} (A r, θ) * Φ_{P q} (B r, θ) = (\frac{2 π}{A^{2} + B^{2}}) {(\frac{A^{2} - B^{2}}{A^{2} + B^{2}})}^{\frac{2 P + q}{2}} Φ_{00} (\frac{A B^{2} r}{{(B^{4} - A^{4})}^{1 / 2}}, θ) Φ_{P q} [\frac{A^{2} B r}{{(A^{4} - B^{4})}^{1 / 2}}, θ] [15 - 43 (5)]$	(C)
$Φ_{00} (r, θ) * Φ_{P q} (r, θ) = \sqrt 2 {[(P + q)! P! (1 + δ_{0 q})]}^{- 1 / 2} cos (q θ) {(\begin{matrix} r \\ - \\ 2 \end{matrix})}^{2 P + q} exp (- r^{2} / 4) [15 - 42 (3)]$
$Φ_{0 q} (r, θ) * Φ_{0 n} (r, θ) = \frac{cos [(q + n) θ] R_{0 q} (r / 2) R_{0 n} (r / 2) + {(- 1)}^{n} cos [(q - n) θ] R_{n, q - n} (r / 2) R_{00} (r / 2)}{2 {[(1 + δ_{0 q}) (1 + δ_{0 n})]}^{1 / 2}} q ⩾ n [15 - 29 (10), 30 (13)]$	(D)

$Ψ_{m n} (x, y) = Ψ_{m} (x) Ψ_{n} (y)$		(A)
$Ψ_{m} (x) = {(2^{m} m! π^{1 / 2})}^{- 1 / 2} H_{m} (x) exp (- x^{2} / 2)$		(B)
$H_{m} (x) = {(- 1)}^{m} exp (x^{2}) \frac{d^{m} exp (- x^{2})}{d x^{m}}$	[7–193(7)]	(C)
	[16–785 (22.11.7)]
$Ψ_{m} (x) = {(\frac{2}{m})}^{1 / 2} x Ψ_{m - 1} (x) - {(\frac{m + 1}{m})}^{1 / 2} Ψ_{m - 2} (x) m ⩾ 2$		(D)
	[7–193(10)]
	[16–782(22.7.13)]
$\begin{array}{l} \begin{array}{l} Ψ_{0} (x) & = π^{- 1 / 4} exp (- x^{2} / 2) \\ Ψ_{1} (x) & = \sqrt 2 x Ψ_{0} (x) \end{array} \\ F [Ψ_{m n} (x, y)] = 2 π {(- i)}^{m + n} Ψ_{m n} (2 π f_{x}, 2 π f_{y}) \end{array}$	[7–193(8)]	(E)
	[17–39(7), 94(8)]
	[18–838(7.376.1)]
$\int \int_{- \infty}^{\infty} Ψ_{m n} (x, y) Ψ_{P q} (x, y) dxdy = δ_{m P} δ_{n q}$	[16–775(22.2.14)]	(F)

$Φ_{P q} (r, θ) = R_{P q} (r) Θ_{q} (θ)$		(A)
$Θ_{q} (θ) = {[π (1 + δ_{0 q})]}^{- 1 / 2} cos (q θ)$		(B)
$where δ_{0 q} = {\begin{array}{l} 1 if q = 0 \\ 0 if otherwise \end{array}$
$R_{P q} (r) = {[2 P! / (P + q)!]}^{1 / 2} r q {L_{P}}^{(q)} (r^{2}) exp (- r^{2} / 2)$		(C)
${L_{P}}^{(q)} (t) = \frac{e^{t}}{P! t^{P}} \frac{d^{P}}{d t^{P}} [exp (- t) t^{P + q}]$	[7–188(5)]	(D)
	[16–785(22.11.6)]
$\begin{array}{l} R_{P q} (r) = \frac{2 P + q - 1 - r^{2}}{{[P (P + q)]}^{1 / 2}} R_{P - 1, q} (r) - {[\frac{(P - 1) (P + q - 1)}{P (P + q)}]}^{1 / 2} R_{P - 2, q} (r) \end{array}$	$P ⩾ 2$ [7–188(8)] [16–782(22.7.12)]	(E)
$\begin{matrix} R_{0 q} (r) = {(\frac{2}{q!})}^{1 / 2} r^{q} exp (- r^{2} / 2) \end{matrix}$
$\begin{matrix} R_{1 q} (r) = {[\frac{2}{(q + 1)!}]}^{1 / 2} r^{q} (q + 1 - r^{2}) exp (- r^{2} / 2) \end{matrix}$	[7–188(6)]
$H [R_{P q} (r); ρ] = {(- 1)}^{P} R_{P q} (ρ)$	[15–42(3)]	(F)
$F [Φ_{P q} (r, θ)] = 2 π {(- i)}^{2 P + q} Φ_{P q} (2 π f_{r}, ϕ)$	[15–42(3)]	(G)
$\int_{0}^{2 π} \int_{0}^{\infty} Φ_{P q} (r, θ) Φ_{m n} (r, θ) rdrd θ = δ_{P m} δ_{q n}$	[16–775(22.2.12)]	(H)
	[16–78(4.3.140)]

Abstract

Cited By

Figures (14)

Tables (3)

Equations (66)

Applied Optics