Theory of alignment monitoring by diffraction from superimposed dual gratings

Yasuhiro Torii; Yoshihiko Mizushima

doi:10.1364/JOSA.68.001716

Journal of the Optical Society of America
Vol. 68,
Issue 12,
pp. 1716-1731
(1978)
•https://doi.org/10.1364/JOSA.68.001716

Theory of alignment monitoring by diffraction from superimposed dual gratings

Yasuhiro Torii and Yoshihiko Mizushima

Not Accessible

Your library or personal account may give you access

Get PDF
Email
Share
Get Citation
Copy Citation Text
Yasuhiro Torii and Yoshihiko Mizushima, "Theory of alignment monitoring by diffraction from superimposed dual gratings," J. Opt. Soc. Am. 68, 1716-1731 (1978)

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

Abstract

Theoretical considerations are presented on characteristics of the optical-alignment technique using dual gratings. The general formulation of the diffraction pattern is derived for superimposition of dual amplitude–phase gratings, as functions of aperture ratios for gratings and separation gaps between dual gratings. Numerical analyses reveal characteristics for precise positioning control or alignment technique, which utilize the detection of the minimum value of the first-order diffraction beam or the detection of the zero cross point of the difference in intensities between +1- and −1-order diffraction light for dual gratings. Alignment errors due to variations in an incident light angle, displacement in detector position, and fabrication errors for the gratings are also discussed.

Full Article | PDF Article

More Like This

Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves

K. Knop
J. Opt. Soc. Am. 68(9) 1206-1210 (1978)

Diffraction regimes of transmission gratings

R. Magnusson and T. K. Gaylord
J. Opt. Soc. Am. 68(6) 809-814 (1978)

On an asymptotic theory of diffraction gratings used in the scalar domain

Erwin G. Loewen, Michel Nevière, and Daniel Maystre
J. Opt. Soc. Am. 68(4) 496-502 (1978)

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 (12)

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 (67)

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

G₁		G₂		nth-order diffraction light intensity: $I_{o} (\frac{n}{p} - \frac{s i n θ}{λ})$
a₁	b₁	a₂	b₂
a₁	b₁	a₂	b₂	\|[b₂(a₁ − b₁)e⁻ⁱ²^πnY + b₁(a₂ − b₂) Ĉ_n + 2(a₁ − b₁)(a₂ − b₂)e⁻^iπn⁽^Y⁻^Nm^/4)S(Y; n)\|²
1	γ	1	γ	4(1 − γ)²\|γĈ_n cos(πnY) + (1 − γ)e^iπMn²/4S(Y;n)\|²
1	0	1	0	4 \|S(Y;n)\|²
e^iδ^₁	1	e^iδ^₂	1	\|[(e^iδ^₁ − 1)e⁻ⁱ²^πnY + (e^iδ^₂− 1)] Ĉ_n + 2(e^iδ^₁ − 1) (e^iδ^₂ − 1) (e⁻^iπn⁽^Y⁻^nM^/4)S(Y;n)\|²
e^iδ	1	e^iδ	1	$16 {sin}^{2} (\frac{δ}{2}) ∣ Ĉ_{n} cos (π n Y) + (e^{i δ} - 1) e^{i π M n^{2} / 4} S (Y; n) ∣^{2}$
1	γ	e^iδ	1	\|[(1 − γ)e⁻ⁱ²^πnY + γ(e^iδ − 1)] Ĉ_n + 2(1 − γ)(e^iδ − 1)e⁻^iπn⁽^Y⁻^Mn^/4)S(Y;n)\|²
1	0	e^iδ	1	\|Ĉ_n + 2(e^iδ − 1) e^iπn⁽^Y⁺^nM^/4)S(Y;n)\|²

(a)

G₁		G₂		± 1-order diffraction light intensity: $I_{o} (\pm \frac{1}{p} - \frac{sin θ}{λ})$
a₁	b₁	a₂	b₂
a₁	b₁	a₂	b₂	\|[b₂(a₁ − b₁)e^∓i2^πnY^_±1 + b₁(a₂ − b₂)]\| C_±1 + 2(a₁ − b₁)(a₂ − b₂)e^∓^iπY^_±1S′(Y_±1)\|²
1	γ	1	γ	4(1 − γ)²\|γĈ_±1 cos(πY_±1) + (1 − γ)S′(Y_±1)\|²
1	0	1	0	4 \|S′(Y_±1)\|²
e^iδ^₁	1	e^iδ^₂	1	\|[(e^iδ^₁ − 1)e^∓2^iπnY^_±1 + (eⁱ^Δ^b^₂− 1)] Ĉ_±1 + 2(e^iδ^₁ − 1) (e^iδ^₂ − 1)e^∓^iπnY^_±1S′(Y_±1)\|²
e^iδ	1	e^iδ	1	16 sin²(δ/2) \|Ĉ_±1 cos (πY_±1) + (e^iδ − 1)S′(Y_±1)\|²
1	γ	e^iδ	1	\|[(1 − γ)e^∓ⁱ²^πnY^±₁ + γ(e^iδ − 1)] Ĉ_±1 + 2(1 − γ)(e^iδ − 1)e^∓^πY^_±1S′(y_±1)\|²
1	0	e^iδ	1	\|Ĉ_±1 + 2(e^iδ − 1) e^±^iπY^_±1S′(Y_±1)\|²

(b)

G₁		G₂		± 1-order diffraction light intensity for r = 0.5: $I_{o} (\pm \frac{1}{p} - \frac{sin θ}{λ})$
a₁	b₁	a₂	b₂
a₁	b₁	a₂	b₂	${(\frac{1}{2 π})}^{2} ∣ (a_{1} - b_{1}) (a_{2} + b_{2}) e^{\mp i 2 π Y_{\pm 1}} + (a_{1} + b_{1}) (a_{2} - b_{2}) ∣^{2}$
1	γ	1	γ	${(\frac{1}{π})}^{2} {(1 - γ^{2})}^{2} {cos}^{2} (π Y_{\pm 1})$
1	0	1	0	${(\frac{1}{π})}^{2} {cos}^{2} (π Y_{\pm 1})$
e^iδ^₁	1	e^iδ^₂	1	${(\frac{2}{π})}^{2} {{sin}^{2} (\frac{δ_{1} - δ_{2}}{2}) + sin (δ_{1}) sin (δ_{2}) {cos}^{2} (π Y_{\pm 1})}$
e^iδ	1	e^iδ	1	${(\frac{2}{π})}^{2} {sin}^{2} (δ) {cos}^{2} (π Y_{\pm 1})$
1	γ	e^iδ	1	${(\frac{1}{2 π})}^{2} ∣ (1 - γ) (1 + e^{i δ}) e^{\mp i 2 π Y_{t 1}} + (e^{i δ} - 1) (1 + γ) ∣^{2}$
1	0	e^iδ	1	${(\frac{1}{π})}^{2} {1 \mp sin (δ) sin (2 π Y_{\pm 1})}$

G₁		G₂		nth-order diffraction light intensity: $I_{o} (\frac{n}{p} - \frac{sin θ}{λ})$
a₁	b₁	a₂	b₂	p(1 − r) < d < rp
a₁	b₁	a₂	b₂	${sin}^{2} (π n r) ∣ a_{2} (a_{1} - b_{1}) exp (- i 2 π n \frac{d}{p}) + a_{1} (a_{2} - b_{2}) ∣^{2}$
1	γ	1	γ	$4 {(1 - γ)}^{2} {sin}^{2} (π n r) {cos}^{2} (π n \frac{d}{p})$
1	0	1	0	$4 {sin}^{2} (π n r) {cos}^{2} (π n \frac{d}{p})$
e^iδ^₁	1	e^iδ^₂	1	4 sin²(πnr) {sin²(δ₁/2) + sin²(δ₂/2) + 2 sin(δ₁/2) sin(δ₂/2) cos(2πnd/p + δ₁/2 − δ₂/2)}
e^iδ	1	e^iδ	1	16 sin²(πnr) sin²(δ/2) cos²(πnd/p)
1	γ	e^iδ	1	sin²(πnr) {(1 − γ)² + 4 sin²(δ/2) − 4(1 − γ) sin(δ/2) sin(2πnd − δ/2)}
1	0	e^iδ	1	sin²(πnr) {1 + 4 sin²(δ/2) − 4 sin(δ/2) sin(2πnd − δ/2)}

G₁		G₂		nth-order diffraction light intensity: $I_{o} (\frac{n}{p} - \frac{s i n θ}{λ})$
a₁	b₁	a₂	b₂
a₁	b₁	a₂	b₂	\|[b₂(a₁ − b₁)e⁻ⁱ²^πnY + b₁(a₂ − b₂) Ĉ_n + 2(a₁ − b₁)(a₂ − b₂)e⁻^iπn⁽^Y⁻^Nm^/4)S(Y; n)\|²
1	γ	1	γ	4(1 − γ)²\|γĈ_n cos(πnY) + (1 − γ)e^iπMn²/4S(Y;n)\|²
1	0	1	0	4 \|S(Y;n)\|²
e^iδ^₁	1	e^iδ^₂	1	\|[(e^iδ^₁ − 1)e⁻ⁱ²^πnY + (e^iδ^₂− 1)] Ĉ_n + 2(e^iδ^₁ − 1) (e^iδ^₂ − 1) (e⁻^iπn⁽^Y⁻^nM^/4)S(Y;n)\|²
e^iδ	1	e^iδ	1	$16 {sin}^{2} (\frac{δ}{2}) ∣ Ĉ_{n} cos (π n Y) + (e^{i δ} - 1) e^{i π M n^{2} / 4} S (Y; n) ∣^{2}$
1	γ	e^iδ	1	\|[(1 − γ)e⁻ⁱ²^πnY + γ(e^iδ − 1)] Ĉ_n + 2(1 − γ)(e^iδ − 1)e⁻^iπn⁽^Y⁻^Mn^/4)S(Y;n)\|²
1	0	e^iδ	1	\|Ĉ_n + 2(e^iδ − 1) e^iπn⁽^Y⁺^nM^/4)S(Y;n)\|²

Abstract

Cited By

Figures (12)

Tables (3)

Equations (67)

Journal of the Optical Society of America