Practical Edge Filters

The limited stop band of a multilayer edge filter often necessitates combining it with additional filters to achieve the broad rejection region typically required in practical applications. These additional filters may be multilayers themselves, as discussed in subsequent sections, or absorption filters, which provide wide rejection regions but have inflexible characteristics.

Combining Absorption and Multilayer Filters

Absorption filters can be incorporated with multilayer filters in several ways:

  • Separate Placement: The absorption filters may be placed in series with the substrates carrying the multilayers.
  • Integrated Substrates: The substrates themselves can act as absorption filters.
  • Multifunctional Materials: The multilayer materials may also function as thin-film absorption filters.

In the visible and near-ultraviolet regions, a wide range of glass filters is available to address most requirements, particularly for longwave-pass filters. However, the situation is more challenging in the infrared region. Complete filters often consist of multiple multilayers to bridge the gap between the edge of the stop band and the nearest suitable absorption filter.

Longwave-Pass Filters for Infrared Applications

Figure 7.20 illustrates an infrared longwave-pass filter. Examples of infrared absorption filters with shortwave-pass characteristics are shown in Figure 7.21. Unfortunately, not all materials listed are readily available.

For longwave-pass filters, the following materials are commonly used:

  • Silicon: Edge at 1 μm.
  • Germanium: Edge at 1.65 μm.
  • Indium Arsenide: Edge at 3.4 μm.
  • Indium Antimonide: Edge at 7.2 μm.

Indium arsenide and indium antimonide are highly absorptive and must be used as very thin slices. Typical thicknesses are approximately 0.013 cm for indium antimonide, with slightly greater thicknesses for indium arsenide. These thin slices are fragile and are typically available in circular shapes with diameters not exceeding 2.0 cm.

Example Filter Performance

Figure 7.22 presents the measured transmittance of a longwave-pass filter combining an edge filter and an absorption filter. Designed for use as a shortwave-blocking filter alongside narrowband filters at 15 μm, this filter comprises:

  1. A multilayer filter of lead telluride and zinc sulfide on a germanium substrate.
  2. An indium antimonide absorption filter in series.

The resulting filter exhibits exceptionally high rejection, as depicted in the logarithmic transmittance plot in Figure 7.22.

Extending the Rejection Zone by Interference Methods

Extending the Reflectance Zone

One straightforward method to broaden the reflectance zone is by placing a second quarter-wave stack in series with the first, ensuring that their rejection zones overlap. The second stack is typically deposited either on a separate substrate or on the opposite side of the first substrate. This approach minimizes adverse interference effects, though multiple-beam effects can still lead to performance issues.

Ideally, the combined transmittance of the filters is given by:

\[
T = T_a T_b \tag{7.46}
\]

where \( T_a \) and \( T_b \) are the individual transmittances. However, if multiple beams are reflected between surfaces and reach the receiver, the resultant transmittance will deviate from this ideal case. In the worst-case scenario, where all reflected beams are collected, the transmittance becomes:

\[
T = T_a + T_b – T_a T_b \tag{7.47}
\]

For small and equal values of \( T_a \) and \( T_b \), the transmittance is approximately half of their individual values. The net transmittance will always be less than or equal to the smaller of \( T_a \) and \( T_b \).

Avoiding Multiple-Beam Interference

To mitigate the effects of multiple beams:

  • Use a thick or slightly wedged substrate to reduce reflected beam power.
  • Insert absorbing material between the two stacks to diminish beam strength.

Stacks Deposited Together

If the second stack must be deposited directly on top of the first, special precautions are necessary to avoid transmission maxima. A transmission maximum occurs when:

\[
\phi_a + \phi_b + 2m\pi, \quad m = 0, \pm1, \pm2, \dots \tag{7.48}
\]

The height of this maximum is given by:

\[
T = \frac{(1 – \tau_a \tau_b)^2}{(1 + \rho_a \rho_b)^2} \tag{7.48}
\]

where \( \tau \) and \( \rho \) represent the transmission and reflection coefficients. To reduce transmittance at the maxima:

  • Design \( R_a \) and \( R_b \), the reflectances of the individual multilayers, to be as dissimilar as possible.
  • Ensure sufficient periods in the multilayers to increase reflectance \( R_s \) in the stop band of one stack where it overlaps the pass region of the other.

Designing for Overlapping High-Reflectance Bands

In regions where the high-reflectance bands overlap, ensure that:

\[
\phi_a + \phi_b \neq 2m\pi
\]

This can be achieved by inserting a layer of intermediate thickness between the two stacks. Equation (7.49) provides a guideline for determining the necessary number of periods:

\[
(1 – R_p)(1 – R_s) – R_p R_s \leq T_c \tag{7.49}
\]

where \( R_p \) is the highest reflectance in the pass region of a multilayer, \( R_s \) is the lowest reflectance in the stop band of the other multilayer, and \( T_c \) is the acceptable level of transmission in the rejection zone.

Example Composite Filters

Figures 7.23 and 7.24 illustrate two component edge filters combined into a single filter. The severe ripple in one multilayer appears in the rejection zone of the composite filter. This ripple can be minimized by adding more periods to the appropriate multilayer.

For shortwave-pass filters, the broader rejection region is often achieved using two-material structures. Figure 7.25 shows an example filter designed to block near-infrared light while transmitting visible light, such as those used to reduce infrared sensitivity in silicon receivers.

Modern Design Techniques

The modern approach to designing optical coatings heavily relies on refinement techniques:

  • A starting design, close to the requirements, is created.
  • Layer thicknesses are iteratively adjusted using refinement algorithms to meet design goals.

This process achieves the desired rejection zone and reduces ripple, eliminating the need for tedious manual adjustments. The refinement process converges on similar final designs regardless of starting conditions. Figure 7.25 demonstrates the effectiveness of this approach, which significantly simplifies the design of practical filters.

Extending the Transmission Zone

Introduction

The pass band of a shortwave-pass filter is inherently limited by the presence of higher-order stop bands. While these higher-order bands may not always pose a problem, applications such as heat-reflecting filters often require a much broader pass band. This issue was first analyzed by Epstein [14] and further studied by Thelen.

Suppression of Higher-Order Reflectance Zones

Epstein’s analysis considers the transmission coefficient \( \tau \) of a multilayer with \( s \) periods of the form:

\[
M = \begin{bmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{bmatrix}.
\]

For a single period immersed in a medium of admittance \( \eta \), the transmission coefficient is:

\[
\tau = \frac{\eta}{\eta_0} \frac{2}{M_{11} + M_{22} + M_{12}\eta + M_{21}/\eta}.
\]

Let \( \tau = |\tau|e^{i\phi} \). If the period is transparent and higher-order reflections are ignored, the total phase thickness \( \Sigma \delta \approx m\pi \), giving \( \cos\phi = \pm 1 \). A high-reflectance zone appears unless \( |\tau| = 1 \).

Suppression Conditions

For the basic stacks \( [(L/2)H(L/2)]_s \) or \( [(H/2)L(H/2)]_s \), \( |\tau| = 1 \) when:

\[
\phi = 2m\pi, \quad m = 1, 2, 3, \dots
\]

In this case, even-order high-reflectance zones are suppressed. However, even a slight deviation in layer thickness can reintroduce high reflectance.

Advanced Suppression Techniques by Thelen

Thelen extended Epstein’s method to design multilayers suppressing multiple successive orders. Using a five-layer structure \( ABCBA \), where layers \( A \) and \( B \) form an antireflection coating for \( C \), Thelen calculated conditions for suppressing specific reflectance orders.

1. Let \( \delta_A = \delta_B \) (equal optical thickness) and \( \eta_A\eta_B = \eta_C\eta_M \), where \( M \) is an artificial medium for design purposes.
2. The wavelengths for suppression are determined by:

\[
\tan^2\delta_A = \frac{\eta_A^2 – \eta_B\eta_C}{\eta_B(\eta_C – \eta_A^2/\eta_B)}. \tag{7.52}
\]

Solving for suppression wavelengths \( \lambda_1 \) and \( \lambda_2 \) gives:

\[
\delta_A = \frac{\pi}{2} \left(1 + \frac{\lambda_1}{\lambda_2}\right). \tag{7.53}
\]

The total optical thickness of the period is constrained by \( \lambda_0/2 \), where \( \lambda_0 \) is the fundamental wavelength.

Example Designs

– For a multilayer with the second- and third-order zones suppressed (\( \lambda_1 = \lambda_0/2 \), \( \lambda_2 = \lambda_0/3 \)), all layers have equal thickness \( \lambda_0/10 \). The indices \( \eta_A \), \( \eta_B \), and \( \eta_C \) are related by:

\[
\eta_B = \sqrt{\eta_A \eta_C}. \tag{7.54}
\]

– Figure 7.26 illustrates a filter with suppressed second and third orders.

– Thelen also devised a multilayer suppressing the second, third, and fourth orders, with layer thicknesses:

\[
A : B : C = \lambda_0/12 : \lambda_0/12 : \lambda_0/6.
\]

Figure 7.27 shows the transmittance of such a filter.

Half-Wave Hole Problem

The half-wave hole is a narrow dip in transmittance around half the fundamental reference wavelength. It results when quarter-waves in a structure become half-waves, rendering the layers absentee. This issue becomes pronounced when tilting a filter to oblique angles, as shown in Figure 7.30.

Suppressing the Half-Wave Hole

To address the half-wave hole:

  • Apply an antireflection coating between high- and low-index materials effective only at the second-order wavelength.
  • Use the same materials as the filter for simplicity. The optical thicknesses of the two layers in the antireflection coating are equal and derived from the admittance diagram (Figure 7.31).

Modified Design

The modified design includes antireflection coatings with adjusted thicknesses to suppress the half-wave hole. The starting structure is:

\[
(0.3368L\, 0.1632H\, 0.1632L\, 0.6737H\, 0.1632L\, 0.1632H\, 0.3368L)_{25}.
\]

After refinement, the performance improves significantly, as shown in Figures 7.29 and 7.32. The antireflection coatings reduce the half-wave hole, but adjustments for incidence angle may still be needed.

By refining layer thicknesses and using antireflection coatings, both higher-order reflectance suppression and elimination of the half-wave hole can be achieved. These advanced techniques enable the design of practical filters with extended transmission zones and minimized reflectance artifacts.

Reducing the Transmission Zone

The absence of even-order high-reflectance bands in a standard quarter-wave multilayer design is often desirable. However, in certain cases, it is useful to reintroduce these high-reflectance bands. The method used to suppress higher orders can be reversed to enhance reflectance at the even orders.

Basic Concept

The even-order peaks in a standard quarter-wave stack are suppressed because each layer’s optical thickness equals an integral multiple of half-wavelengths, making them absentee layers. To restore reflectance peaks at even orders:

  • Modify the layer thicknesses such that one layer’s thickness increases while the other decreases, maintaining the overall optical thickness constant.

Example Design

If reflectance bands are desired at \( \lambda_0 \), \( \lambda_0/2 \), and \( \lambda_0/3 \), but not \( \lambda_0/4 \), the following conditions can be applied:
\[
n_A d_A = n_B d_B/3, \quad n_A d_A = \lambda_0/8
\]
The resulting multilayer designs could be:
\[
H \, LH \, LH \, L \dots \quad \text{or} \quad L \, HL \, HL \, H \dots
\]
Here, \( H \) and \( L \) represent high- and low-refractive-index layers. The peak at \( \lambda_0/4 \) is suppressed because layers at that wavelength retain integral half-wave thicknesses.

Trade-Offs

  • Increasing deviations from the ideal quarter-wave condition narrows the first-order reflectance band.
  • The method allows customization of reflectance zones as needed.

Edge Steepness

The edge steepness in longwave- and shortwave-pass filters is usually adequate given the number of layers needed for rejection in the stop band. However, when exceptionally steep edges are required, the following methods can be employed.

Increasing the Number of Layers

  • Adding more layers to the design increases edge steepness.
  • This also brings the first pass band minimum closer to the edge, potentially increasing ripple in the pass band.
  • Advanced ripple-reduction techniques may be necessary if the layer count becomes excessive.

Using Higher-Order Stacks

  • Higher-order stacks improve edge steepness for a fixed number of layers.
  • Edge steepness increases in proportion to the order of the stack.

Challenges with Higher-Order Stacks

1. Reduced Rejection Zone Width: The width of the rejection zone decreases inversely with the order number. This can be mitigated by adding a first-order stack to extend the rejection zone.

2. Tighter Tolerances:

  • Performance depends on the sine and cosine of the layer thickness. For higher-order stacks (e.g., fifth-order), the phase thickness exceeds \( 2\pi \).
  • Tolerable random errors in layer thickness decrease inversely with the order number. For example, if 5% errors are acceptable for a first-order filter, only 1% errors are tolerable for a fifth-order filter.
  • Optical monitoring, rather than crystal monitoring, is preferred to accommodate tighter tolerances.

3. Material Requirements:

  • Higher-order filters require significantly more material for each layer, increasing production costs and time.

Restoring even-order reflectance bands requires strategic modifications to layer thicknesses, maintaining overall optical thickness. For sharper edge steepness, increasing the number of layers or using higher-order stacks are viable options, though they introduce challenges such as narrower rejection zones, tighter tolerances, and increased material usage. These considerations must be balanced to meet specific design requirements.

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