Effective Refractive Indices of Metal–Dielectric Interference Filters

D. J. Hemingway; P. H. Lissberger

doi:10.1364/AO.6.000471

Applied Optics
Vol. 6,
Issue 3,
pp. 471-476
(1967)
•https://doi.org/10.1364/AO.6.000471

Effective Refractive Indices of Metal–Dielectric Interference Filters

D. J. Hemingway and P. H. Lissberger

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D. J. Hemingway and P. H. Lissberger, "Effective Refractive Indices of Metal–Dielectric Interference Filters," Appl. Opt. 6, 471-476 (1967)

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Abstract

It is shown that the performance of metal–dielectric interference filters of the Fabry–Perot and Spacerless Induced Transmission types in parallel radiation incident obliquely can be described in terms of that of a single equivalent dielectric layer of refractive index μ_E. Expressions for μ_E are deduced in terms of the optical constants of the layer materials, and their validity is established by the satisfactory agreement that is shown to exist between experimental and calculated values of μ_E.

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	Silver thickness (Å)	Optical thickness of matching layer M quarterwaves
	Silver thickness (Å)	ZnS (μ_H)	MgF₂ (μ_L)
Calculated from general expressions	300	0.762	1.859
	600	0.649	1.782
	900	0.626	1.765
Approximation Eqs. (22) and (23)		0.622	1.762
Approximation Eqs. (22) and (23)		(K = 0)	(K = 1)
Assumed refractive indices	0.075 + 3.41 i	2.30	1.34

Filter type	V	VP	VQ
F.P. (spacer index μ₀)	1	$\frac{1}{{μ_{0}}^{2}}$	$(1 + \frac{μ_{0}}{S a_{0}}) \frac{λ_{0}}{μ_{0}} \frac{\partial μ_{0}}{\partial λ} - \frac{λ_{0}}{S b_{0}} \frac{\partial k}{\partial λ}$
I.T. (μ_M = μ_H)	$M + \frac{μ_{L}}{μ_{H} - μ_{L}}$	$\frac{1}{{μ_{H}}^{2}} [M + \frac{{μ_{H}}^{2} + {μ_{L}}^{2} - μ_{H} μ_{L}}{μ_{L} (μ_{H} - μ_{L})}]$	$(M + \frac{{μ_{L}}^{2}}{{μ_{H}}^{2} - {μ_{L}}^{2}} + \frac{μ_{H}}{a_{H}}) \frac{λ_{0}}{μ_{H}} \frac{\partial μ_{H}}{\partial λ} + (\frac{μ_{L} μ_{H}}{{μ_{H}}^{2} - {μ_{L}}^{2}}) \frac{λ_{0}}{μ_{L}} \frac{\partial μ_{L}}{\partial λ} - \frac{λ_{0}}{b_{H}} \frac{\partial k}{\partial λ}$
I.T. (μ_M = μ_L)	$M + \frac{μ_{L}}{μ_{H} - μ_{L}}$	$\frac{1}{{μ_{L}}^{2}} [M + \frac{{μ_{L}}^{2}}{μ_{H} (μ_{H} - μ_{L})}]$	$\frac{μ_{L} μ_{H}}{(μ^{2} H - {μ_{L}}^{2})} \frac{λ_{0}}{μ_{H}} \frac{\partial μ_{H}}{\partial λ} + (M + \frac{{μ_{L}}^{2}}{{μ_{H}}^{2} - {μ_{L}}^{2}} + \frac{μ_{L}}{a_{L}}) \frac{λ_{0}}{μ_{L}} \frac{\partial μ_{L}}{\partial λ} - \frac{λ_{0}}{b_{L}} \frac{\partial k}{\partial λ}$

Filter		Spacer or matching layer^a			λ₀ (Å)	μ_E

No.	Type	Refractive index	S	M		Experimental	rms error ±	Theoretical
1	F.P.	L	3.79	—	5935	1.51	0.01	1.39
1	”	”	4.72	—	4760	1.508	0.008	1.39
2	F.P.	H	3.62	—	5465	2.79	0.02	2.65
2	”	”	4.52	—	4538	3.01	0.02	2.79
3	F.P.	H	3.62	—	5542	2.81	0.02	2.64
3	”	”	4.53	—	4590	2.99	0.02	2.77
4	F.P.	H	3.63	—	5580	2.71	0.02	2.64
4	”	”	4.53	—	4615	2.76	0.03	2.76
5	F.P.	H	0.62	—	5570	3.4	0.1	3.20
6	I.T.	L	—	1.79	5440	1.61	0.02	1.58
7	I.T.	L	—	1.79	5490	1.587	0.003	1.58
8^a	I.T.	L	—	1.78	5500	1.587	0.007	1.58
9^a	I.T.	L	—	1.78	5519	1.71	0.01	1.81
10	I.T.	L	—	1.78	5340	1.79	0.03	1.81
11	I.T.	H	—	0.66	5472	2.131	0.005	1.96
12	I.T.	H	—	0.66	5542	2.05	0.02	1.96
13	I.T.	H	—	0.66	5550	2.02	0.03	1.95

	Silver thickness (Å)	Optical thickness of matching layer M quarterwaves
	Silver thickness (Å)	ZnS (μ_H)	MgF₂ (μ_L)
Calculated from general expressions	300	0.762	1.859
	600	0.649	1.782
	900	0.626	1.765
Approximation Eqs. (22) and (23)		0.622	1.762
Approximation Eqs. (22) and (23)		(K = 0)	(K = 1)
Assumed refractive indices	0.075 + 3.41 i	2.30	1.34

Filter type	V	VP	VQ
F.P. (spacer index μ₀)	1	$\frac{1}{{μ_{0}}^{2}}$	$(1 + \frac{μ_{0}}{S a_{0}}) \frac{λ_{0}}{μ_{0}} \frac{\partial μ_{0}}{\partial λ} - \frac{λ_{0}}{S b_{0}} \frac{\partial k}{\partial λ}$
I.T. (μ_M = μ_H)	$M + \frac{μ_{L}}{μ_{H} - μ_{L}}$	$\frac{1}{{μ_{H}}^{2}} [M + \frac{{μ_{H}}^{2} + {μ_{L}}^{2} - μ_{H} μ_{L}}{μ_{L} (μ_{H} - μ_{L})}]$	$(M + \frac{{μ_{L}}^{2}}{{μ_{H}}^{2} - {μ_{L}}^{2}} + \frac{μ_{H}}{a_{H}}) \frac{λ_{0}}{μ_{H}} \frac{\partial μ_{H}}{\partial λ} + (\frac{μ_{L} μ_{H}}{{μ_{H}}^{2} - {μ_{L}}^{2}}) \frac{λ_{0}}{μ_{L}} \frac{\partial μ_{L}}{\partial λ} - \frac{λ_{0}}{b_{H}} \frac{\partial k}{\partial λ}$
I.T. (μ_M = μ_L)	$M + \frac{μ_{L}}{μ_{H} - μ_{L}}$	$\frac{1}{{μ_{L}}^{2}} [M + \frac{{μ_{L}}^{2}}{μ_{H} (μ_{H} - μ_{L})}]$	$\frac{μ_{L} μ_{H}}{(μ^{2} H - {μ_{L}}^{2})} \frac{λ_{0}}{μ_{H}} \frac{\partial μ_{H}}{\partial λ} + (M + \frac{{μ_{L}}^{2}}{{μ_{H}}^{2} - {μ_{L}}^{2}} + \frac{μ_{L}}{a_{L}}) \frac{λ_{0}}{μ_{L}} \frac{\partial μ_{L}}{\partial λ} - \frac{λ_{0}}{b_{L}} \frac{\partial k}{\partial λ}$

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Applied Optics