1. The Metal–Dielectric Single-Cavity Filter

The simplest narrowband thin-film filter is based on the Fabry–Perot interferometer. Traditionally, the Fabry–Perot interferometer uses two identical parallel reflecting surfaces spaced apart by a distance \(d\), resulting in low transmission for most wavelengths except for very narrow transmission bands spaced at constant intervals in terms of wave number.

This concept can be translated into a thin-film assembly consisting of a dielectric layer bounded by two metallic reflecting layers, as shown in Figure 8.3. The dielectric layer acts as the spacer layer, now often referred to as the cavity layer.

Compared to the conventional etalon:

• The substrate’s surface requires a high polish but does not need the extreme tolerances of etalon plates.
• Vapor deposition ensures uniform thin-film layers, eliminating significant thickness variations.
• The thin-film Fabry–Perot filter operates in lower orders than the conventional etalon, as increased thickness introduces roughness, broadening the pass band and reducing peak transmittance.

This type of filter is referred to as a metal–dielectric Fabry–Perot filter or a single-cavity filter, distinct from its all-dielectric counterpart.

Analytical Performance of the Single-Cavity Filter

The performance of the single-cavity filter can be analyzed, incorporating phase shift effects at the reflectors. Using Equation 3.20, the transmittance is given by:

\[
T = \frac{T_a T_b}{[1 + F \sin^2(\delta/2)]},
\]
where:
\[
F = \frac{4R_a R_b}{(1 – R_a)(1 – R_b)},
\]
\[
\delta = \frac{2\pi nd \cos \theta}{\lambda},
\]
and \(R_a\), \(R_b\) are the reflectances, \(T_a\), \(T_b\) the transmittances, and \(\phi_a\), \(\phi_b\) the phase shifts at the reflectors.

The transmittance maxima occur at:

\[
2\pi nd \cos \theta – (\phi_a + \phi_b) = m\lambda, \quad m = 0, \pm 1, \pm 2, \ldots
\]

Switching to wave number (\(\nu = 1/\lambda\)) simplifies the expression:

\[
\frac{1}{\lambda} = \nu = \frac{m}{2nd \cos \theta} + \frac{\phi_a + \phi_b}{4\pi nd \cos \theta}.
\]

The order number (\(m\)) determines the optical thickness required for specific wavelengths.

Resolving Power and Bandwidth

The resolving power of the filter is defined as:

\[
\text{Resolving Power} = \frac{\text{Peak Wavelength}}{\text{Half-width of Pass Band}}.
\]

For small deviations near the transmission peak, the finesse (\(F\)) is:

\[
F = \frac{\pi}{\Delta \delta_h},
\]

where \(\Delta \delta_h\) is the half-peak bandwidth. The resolving power can also be expressed as:

\[
\frac{\nu_0}{\Delta \nu_h} = \frac{\lambda_0}{\Delta \lambda_h},
\]

where \(\nu_0\) and \(\lambda_0\) represent the peak wave number and wavelength, respectively.

Manufacturing Considerations

Key manufacturing considerations include:

1. Metallic Layers: Must be deposited quickly onto cold substrates to ensure optimal performance.

2. Material Selection:
– Visible/Infrared: Silver and cryolite.
– Ultraviolet: Aluminum and magnesium fluoride.

3. Protection: Layers should be covered with a protective slip as soon as possible to prevent degradation, especially in the ultraviolet region.

Figure 8.4 shows performance curves for first-order metal–dielectric single-cavity filters.

Peak Transmittance with Absorption

The peak transmittance in the presence of absorption (\(A\)) is given by:

\[
T_{\text{peak}} = \frac{(T_a T_b)}{1 + (A/T_a)},
\]

where \(A\) represents the absorptance.

If reflectances \(R_a\) and \(R_b\) are mismatched by a small amount \(\Delta_a\), the peak transmittance becomes:

\[
T_{\text{peak}} = \frac{T_a}{(1 – R_a)^2} \left[1 – \frac{\Delta_a^2}{(1 – R_a)^2}\right].
\]

This shows that the single-cavity filter is relatively tolerant to small mismatches in reflectance.

Practical Applications and Limitations

1. High-Order Limitations:
– Orders beyond the third are rarely used due to scattering and diminishing performance.
– Sidebands due to higher-order peaks can be suppressed by absorption glass filters.

2. Ultraviolet Filters:
– Aluminum is preferred over silver.
– Protective dielectric layers slow down oxidation but cannot completely stabilize performance.

Figure 8.5 demonstrates the insensitivity of peak transmission to small reflector mismatches. The single-cavity metal–dielectric filter remains an essential tool for narrowband filtering applications, balancing precision and practical manufacturability.

2. The All-Dielectric Single-Cavity Filter

To enhance the performance of a Fabry–Perot etalon, metallic reflecting layers are replaced with all-dielectric multilayers. The result is the all-dielectric single-cavity filter, illustrated schematically in Figure 8.6. This filter resembles a conventional etalon with dielectric coatings and a solid thin-film cavity layer, or spacer.

The characteristics of the all-dielectric filter largely align with those of the metal–dielectric filter, but with notable differences:

• The substrate polish must be of high quality but does not require the extreme flatness necessary for etalon plates.
• Coating machines ensure uniform film thickness over reasonable substrate contours.

Bandwidth Calculation

For sufficiently high reflectance in the multilayers, the finesse (\(F\)) is given by:

\[
F = \frac{4R}{(1 – R)^2},
\]

and the fractional half-width, \(\Delta \lambda_h / \lambda_0\), becomes:

\[
\frac{\Delta \lambda_h}{\lambda_0} = \frac{1}{\pi m F},
\]

where \(m\) is the order number.

Transmittance Cases

There are two primary cases to consider:
1. High-Index Cavity Layer:
\[
T = \frac{4n_L^2}{n_s n_H^{2x}},
\]
where \(x\) is the number of high-index layers in each stack.

The fractional half-width is:
\[
\frac{\Delta \lambda_h}{\lambda_0} = \frac{n_L^2}{n_s n_H^{2x} \pi m}.
\]

2. Low-Index Cavity Layer:
\[
T = \frac{4n_H^2}{n_s n_L^{2x}},
\]

The fractional half-width is:
\[
\frac{\Delta \lambda_h}{\lambda_0} = \frac{n_H^2}{n_s n_L^{2x} \pi m}.
\]

These equations ignore phase change dispersion at reflection. The phase variation reduces the bandwidth and increases the resolving power of the filter.

Phase Dispersion and Bandwidth Refinement

Following Seeley’s analysis, the first-order fractional half-width expressions from Equations 8.15 and 8.16 can be refined by introducing a factor \((n_H – n_L)/n_H\), which accounts for the reduced bandwidth due to phase change dispersion.

The half-peak points are given by:

\[
F \sin^2 \left( \frac{2 \pi D}{\lambda} – \phi \right) = 1,
\]

where \(g = \lambda_0 / \lambda = \nu / \nu_0\) is used to simplify the analysis. At small changes in \(g\):

\[
\frac{2 \pi D}{\lambda} = m \pi (1 + \Delta g), \quad \phi = \phi_0 + \frac{d\phi}{dg} \Delta g.
\]

The fractional half-width is:

\[
\Delta g = \frac{1}{F \pi m} \left( 1 – \frac{d\phi}{dg} \right)^{-1}.
\]

Thus, the half-width in terms of wavelength is:

\[
\frac{\Delta \lambda_h}{\lambda_0} = \frac{1}{F \pi m} \left( 1 – \frac{d\phi}{dg} \right)^{-1}.
\]

Matrix Representation of Quarter-Wave Layers

The matrix for a dielectric quarter-wave layer is:

\[
\begin{bmatrix}
\cos \delta & i \sin \delta / \eta \\
i \eta \sin \delta & \cos \delta
\end{bmatrix},
\]

where \(\eta\) is the optical admittance, and for near quarter-wave layers:

\[
\delta = \frac{\pi}{2} + \epsilon, \quad \cos \delta \approx -\epsilon, \quad \sin \delta \approx 1.
\]

The matrix becomes:

\[
\begin{bmatrix}
-\epsilon & i / \eta \\
i \eta & -\epsilon
\end{bmatrix}.
\]

This analysis applies to quarter-wave multilayer stacks, assuming a high-index layer is adjacent to the substrate. Separate cases for even and odd numbers of layers need to be analyzed further for specific designs.

The all-dielectric single-cavity filter offers enhanced performance compared to metal–dielectric filters, with precise control over bandwidth and resolving power. The calculated half-width depends on reflectance finesse, cavity order, and dispersion effects, providing design flexibility for high-performance narrowband filtering.

Case 1: Even Number (2x) of Layers

The resultant multilayer matrix for an even number of layers is derived as follows. The structure is represented by the product of matrices for individual layers:

\[
\begin{bmatrix}
B & C \\
L & H
\end{bmatrix}
=
[\text{Layer}_1][\text{Layer}_2]\ldots[\text{Layer}_{2x}] \begin{bmatrix}
1 \\
\eta_{\text{sub}}
\end{bmatrix}
\]

Layer Matrices

For the low-index (\(L\)) and high-index (\(H\)) layers, the matrices are defined as:

\[
[L] =
\begin{bmatrix}
-\epsilon & i / \eta_L \\
i \eta_L & -\epsilon
\end{bmatrix},
\quad
[H] =
\begin{bmatrix}
-\epsilon & i / \eta_H \\
i \eta_H & -\epsilon
\end{bmatrix},
\]

where \(\epsilon\) represents the deviation from the ideal quarter-wave condition.

Resultant Matrix

The total matrix after \(2x\) layers is:

\[
\begin{bmatrix}
B & C \\
L & H
\end{bmatrix}
=
\begin{bmatrix}
M_{11} & M_{12} \\
M_{21} & M_{22}
\end{bmatrix}
\begin{bmatrix}
1 \\
\eta_{\text{sub}}
\end{bmatrix}.
\]

Derivation of Matrix Components

We calculate the components \(M_{11}, M_{12}, M_{21}, M_{22}\) while neglecting all terms of second and higher orders in \(\epsilon\).

1. For \(M_{11}\):
\[
M_{11} = (-1)^x \left( \frac{\eta_H}{\eta_L} \right)^x
\]

2. For \(M_{22}\):
\[
M_{22} = (-1)^x \left( \frac{\eta_L}{\eta_H} \right)^x
\]

3. For \(M_{12}\):
The first-order terms in \(\epsilon\) dominate:
\[
M_{12} = (-1)^x \left( \frac{\eta_L}{\eta_H} \right)^{x} \left[ -\epsilon \left( \frac{\eta_H}{\eta_L} \right) + \frac{\epsilon}{\eta_L} + \ldots \right].
\]

4. For \(M_{21}\):
A similar expression to \(M_{12}\) emerges:
\[
M_{21} = (-1)^x \left( \frac{\eta_H}{\eta_L} \right)^x \left[ -\epsilon \left( \frac{\eta_L}{\eta_H} \right) + \frac{\epsilon}{\eta_H} + \ldots \right].
\]

Simplifications for Large \(x\) and Small \(\eta_L / \eta_H\)

When \(x\) is large and \((\eta_L / \eta_H)^x\) is negligible compared to 1, the terms simplify:

\[
M_{12} \approx (-1)^x \left( \frac{\eta_L}{\eta_H} \right)^x \frac{\epsilon}{\eta_L},
\quad
M_{21} \approx (-1)^x \left( \frac{\eta_H}{\eta_L} \right)^x \frac{\epsilon}{\eta_H}.
\]

These approximations provide practical expressions for the multilayer system’s matrix elements under the stated conditions.

Case II: Odd Number (2x + 1) of Layers

For an odd number of layers (\(2x + 1\)), the resultant matrix is derived as follows:

Matrix Representation

The multilayer matrix for the system can be expressed as:
\[
\begin{bmatrix}
B & C \\
H & L
\end{bmatrix}
=
[L][H][L][H]\ldots[L][H][H]_{\text{sub}}
\begin{bmatrix}
1 \\
\eta_{\text{sub}}
\end{bmatrix}
\]
or equivalently:
\[
\begin{bmatrix}
B & C \\
H & L
\end{bmatrix}
=
\begin{bmatrix}
N_{11} & N_{12} \\
N_{21} & N_{22}
\end{bmatrix}
\begin{bmatrix}
1 \\
\eta_{\text{sub}}
\end{bmatrix},
\]
where \(\begin{bmatrix} N_{11} & N_{12} \\ N_{21} & N_{22} \end{bmatrix}\) represents the resultant matrix for the odd number of layers.

Matrix Derivation

For \(2x + 1\) layers, the system is essentially the matrix for \(2x\) layers (as derived in Case I) multiplied by the additional high-index (\(H\)) layer. The additional layer introduces a term represented as:

\[
\begin{bmatrix}
-\epsilon & i / \eta_H \\
i \eta_H & -\epsilon
\end{bmatrix}.
\]

The resulting matrix for \(2x + 1\) layers is then:

\[
\begin{bmatrix}
N_{11} & N_{12} \\
N_{21} & N_{22}
\end{bmatrix}
=
\begin{bmatrix}
M_{11} & M_{12} \\
M_{21} & M_{22}
\end{bmatrix}
\begin{bmatrix}
-\epsilon & i / \eta_H \\
i \eta_H & -\epsilon
\end{bmatrix}.
\]

Matrix Components

The components of the resultant matrix are calculated as follows:

1. For \(N_{11}\):
\[
N_{11} = M_{11}(-\epsilon) + M_{12}(i \eta_H),
\]
simplifying to:
\[
N_{11} = (-1)^x \left(\frac{\eta_H}{\eta_L}\right)^x (-\epsilon) – (-1)^x \left(\frac{\eta_L}{\eta_H}\right)^x \epsilon.
\]

2. For \(N_{12}\):
\[
N_{12} = M_{11}(i / \eta_H) + M_{12}(-\epsilon),
\]
simplifying to:
\[
N_{12} = (-1)^x \left(\frac{\eta_H}{\eta_L}\right)^x \frac{i}{\eta_H} – (-1)^x \left(\frac{\eta_L}{\eta_H}\right)^x \epsilon.
\]

3. For \(N_{21}\):
\[
N_{21} = M_{21}(-\epsilon) + M_{22}(i \eta_H),
\]
simplifying to:
\[
N_{21} = (-1)^x \left(\frac{\eta_L}{\eta_H}\right)^x (-\epsilon) – (-1)^x \left(\frac{\eta_H}{\eta_L}\right)^x \epsilon.
\]

4. For \(N_{22}\):
\[
N_{22} = M_{21}(i / \eta_H) + M_{22}(-\epsilon),
\]
simplifying to:
\[
N_{22} = (-1)^x \left(\frac{\eta_L}{\eta_H}\right)^x \frac{i}{\eta_H} – (-1)^x \left(\frac{\eta_H}{\eta_L}\right)^x \epsilon.
\]

Simplifications

By neglecting terms of order \((\eta_L / \eta_H)^x\) compared to \((\eta_H / \eta_L)^x\), the simplified components are:

1. \(N_{11}\):
\[
N_{11} \approx (-1)^x \left[ \left(\frac{\eta_H}{\eta_L}\right)^x (-\epsilon) + \text{higher-order terms} \right].
\]

2. \(N_{12}\):
\[
N_{12} \approx (-1)^x \frac{i}{\eta_H} \left(\frac{\eta_H}{\eta_L}\right)^x.
\]

3. \(N_{21}\):
\[
N_{21} \approx (-1)^x (-\epsilon) \left(\frac{\eta_L}{\eta_H}\right)^x.
\]

4. \(N_{22}\):
\[
N_{22} \approx (-1)^x \frac{i}{\eta_H} \left(\frac{\eta_L}{\eta_H}\right)^x.
\]

These results highlight the behavior of the matrix components for an odd number of layers (\(2x + 1\)), considering the dominance of higher-index layers in practical scenarios.

Phase Shift: Case I

We now compute the phase shift on reflection. Let the admittance of the incident medium be denoted as \(\eta_0\). The phase shift is derived as follows:

Reflection Coefficient

The reflection coefficient \(\rho\) is given by:
\[
\rho = \frac{\eta_0 – \eta_{\text{eff}}}{\eta_0 + \eta_{\text{eff}}}
\]
where \(\eta_{\text{eff}}\) is the effective admittance of the multilayer system.

Using the multilayer matrix:
\[
\begin{bmatrix}
B & C \\
H & L
\end{bmatrix}
=
\begin{bmatrix}
M_{11} & M_{12} \\
M_{21} & M_{22}
\end{bmatrix}
\begin{bmatrix}
1 \\
\eta_{\text{sub}}
\end{bmatrix},
\]
we find \(\eta_{\text{eff}}\) as:
\[
\eta_{\text{eff}} = \frac{M_{11} + M_{12}\eta_{\text{sub}}}{M_{21} + M_{22}\eta_{\text{sub}}}.
\]

Reflection Coefficient in Terms of Matrix Elements

Substituting for \(\eta_{\text{eff}}\), the reflection coefficient becomes:
\[
\rho = \frac{\eta_0(M_{21} + M_{22}\eta_{\text{sub}}) – (M_{11} + M_{12}\eta_{\text{sub}})}{\eta_0(M_{21} + M_{22}\eta_{\text{sub}}) + (M_{11} + M_{12}\eta_{\text{sub}})}.
\]

Phase Shift

The phase shift \(\phi\) is defined as:
\[
\tan \phi = \frac{\text{Im}(\rho)}{\text{Re}(\rho)}.
\]
Substituting for \(\rho\), this becomes:
\[
\tan \phi = \frac{2\eta_0(M_{12}M_{21} – M_{11}M_{22})}{\eta_0^2(M_{21}^2 + M_{22}^2) – (M_{11}^2 + M_{12}^2)}.
\]

Using the relations:
– \(M_{11}\), \(M_{22}\) are real.
– \(M_{12}\), \(M_{21}\) are imaginary.

The expression simplifies to:
\[
\tan \phi = \frac{-2\eta_0M_{12}M_{21}}{\eta_0^2(M_{21}^2 + M_{22}^2) – (M_{11}^2 + M_{12}^2)}.
\]

Approximation

Neglecting terms of second and higher order in \(\epsilon\) and terms involving \((\eta_L / \eta_H)^x\), the phase shift for the system \((LH \ldots LHLH|\eta_{\text{sub}})\) simplifies to:
\[
\tan \phi = \frac{2\eta_H \eta_L(\eta_L \epsilon_H – \eta_H \epsilon_L)}{\eta_H^2 – \eta_L^2}.
\]

Final Expression

The phase shift \(\phi\) is thus given by:
\[
\tan \phi = \frac{2\eta_H \eta_L (\eta_L \epsilon_H – \eta_H \epsilon_L)}{\eta_H^2 – \eta_L^2}.
\]

This result shows the dependency of the phase shift on the indices of refraction of the high-index and low-index materials (\(\eta_H\) and \(\eta_L\)) and the layer perturbations (\(\epsilon_H\) and \(\epsilon_L\)).

Phase Shift: Case II

In this case, the reflection coefficient, \(\rho\), is derived in a manner similar to Equation 7.19, with \(M\) replaced by \(N\). Following the same procedure as for Case I, the expression for the phase shift becomes:

\[
\tan \phi = \frac{2\eta_0(\eta_L \epsilon_L – \eta_H \epsilon_H)}{\eta_H^2 – \eta_L^2}
\]
(for \(\text{HLH…LHLH}|\eta_{\text{sub}}\)). (Equation 8.22)

Generalized Expression for Phase Shift

Equations 8.21 and 8.22 provide a general form for the phase shift. For simplification, we note:
\[
\delta = \frac{2\pi nd}{\lambda} = 2\pi \nu \cdot \frac{nd}{\nu_0} = 2\pi g
\]
where \(g = \nu/\nu_0\). This simplifies:
\[
\epsilon_H = \frac{\pi}{2}(g – 1) \quad \text{and} \quad \epsilon_L = \frac{\pi}{2}(g – 1).
\]

For Fabry–Perot filters, the incident medium for Case I is typically a high-index spacer layer, and for Case II, it is a low-index spacer layer. This results in the simplified expression:
\[
\tan \phi = -\pi \frac{n_H – n_L}{n_H n_L} (g – 1),
\]
valid for both cases.

Resolving Power and Bandwidth

Using this phase shift, the derivative of \(\phi\) with respect to \(g\) becomes:
\[
\frac{d\phi}{dg} = \pi \frac{\eta_H – \eta_L}{\eta_H \eta_L}.
\]
This result, derived by Seeley, is inserted into Equation 8.18, leading to:

High-index cavity:
\[
\frac{\Delta \nu_h}{\nu_0} = \frac{\Delta \lambda_h}{\lambda_0} = \frac{4}{mF} \frac{\eta_H \eta_L}{\eta_H – \eta_L}.
\]
(Equation 8.23)

Low-index cavity:
\[
\frac{\Delta \nu_h}{\nu_0} = \frac{\Delta \lambda_h}{\lambda_0} = \frac{4}{mF} \frac{\eta_H \eta_L}{\eta_L – \eta_H}.
\]
(Equation 8.24)

These results apply to first-order reflecting stacks and \(m\)-order cavities.

Observations

– The phase shift has a greater effect when the indices \(\eta_H\) and \(\eta_L\) are closer in value and the cavity order \(m\) is lower.
– For materials commonly used in the visible and near-infrared spectrum, such as zinc sulfide (\(\eta_H = 2.35\)) and cryolite (\(\eta_L = 1.35\)), the factor \((\eta_H – \eta_L)/(\eta_H + \eta_L/m)\) is approximately 0.43 for first-order cavities.
– In the infrared region, materials like zinc sulfide (\(\eta_H\)) and lead telluride (\(\eta_L\)) have a higher factor, around 0.57.

Figures 8.8 and 8.9 illustrate the characteristics of typical all-dielectric narrowband single-cavity filters.

Transmission Sideband Suppression

Since the reflectors in all-dielectric multilayers are effective over a limited range, transmission sidebands appear on either side of the peak and must often be suppressed. The solutions include:

1. Shortwave Sidebands:
– Removed by adding a longwave-pass absorption filter, commonly available as polished glass disks.

2. Longwave Sidebands:
– Absorption filters are less effective due to their shallow edges, which can significantly reduce peak transmittance.
– Preferred solution: Use a metal–dielectric blocking filter, especially of the multiple-cavity type, which can suppress sidebands without reducing transmittance.

Effects of Absorption Losses

Absorption losses in the layers are analyzed similarly to quarter-wave stacks. The loss in a weakly absorbing multilayer is:
\[
A = (1 – R)\Sigma A,
\]
where \(\Sigma A\) is a sum of absorption terms across the layers.

For low-index cavities:
\[
A = \pi^2 \frac{k_H}{\lambda_0} \left[\frac{1}{n_H^2} + \frac{n_H^2}{n_L^2} \right] + \ldots,
\]
(Equation 8.25)

For high-index cavities:
\[
A = \pi^2 \frac{k_L}{\lambda_0} \left[\frac{1}{n_L^2} + \frac{n_L^2}{n_H^2} \right] + \ldots,
\]
(Equation 8.26)

Manufacturing Challenges

– Achieving high uniformity is critical, particularly for filters with half-widths below 0.1 nm.
– Variations in peak wavelength should not exceed one-third of the half-width across the surface to maintain performance.

High-index cavities are generally preferred for their better performance under tilt, energy grasp, and manufacturing feasibility, particularly in the visible and near-infrared regions.