An apodized thickness design method for discrete layer notch filters is presented. The method produces error tolerant designs with low ripple in the passband regions without any additional numerical optimization. Several apodization functions including Gaussian, cosine squared, as well as quintic are considered. Theoretical and experimental results from ion beam deposited sample designs for single-notch as well as multinotch filters are presented. Good agreement is observed between the theoretical design and the experimental measurement even when the deposition process is only controlled on time. This demonstrates the low layer error sensitivity of the apodized designs.
© 2013 Optical Society of America
Multiple approaches have been used in the past for producing notch filters (also called minus filters) [1–4]. The two main approaches can be grouped into rugate and discreet layer designs. Although narrow stop band designs can be generated with both methods, rugate designs offer the advantage of low ripple (sidelobes) in the transmission regions achieved by applying an apodization function to the index amplitude variation . In addition, rugate designs do not have higher-order stop bands. However, manufacturing of rugate designs has practical process challenges, such as characterization and control of the deposited index as a function of coating thickness.
The method presented in this paper, combines the relative ease of manufacturing of discrete layer designs with the low ripple of an apodized graded index design.
2. Design Methodology
A narrow bandwidth of the reflection region of a regular quarter-wave (QW) stack [6,7] can be achieved by shifting the ratio of high index material to low index material for each half-wave pair in the stack
Perilloux  showed that a thickness-modulated design realized by applying a Gaussian thickness modulation to the ratio in Eq. (1) through the coating, can be used to design a notch filter. However, the resulting design also does not match the admittance of the substrate or media and has significant ripple in the passband regions.
Perilloux’s approach can be combined with an apodization similar to the one used in rugate designs  by applying an apodization function to the ratio in Eq. (1). For a Gaussian shape centered in the middle of the coating design, the QW optical thicknesses are given by
It is useful to consider the case of an immersed coating with the substrate and media both having the index of the low index coating material. In this case, the design naturally matches the admittance of the surrounding media on both sides. In the case of as the low index material and either fused silica or BK7 as the surrounding media, good admittance matching between the coating and the media is achieved. A separate antireflective (AR) coating can be applied for the desired transmission band region matching the top side of the coating to air. However, this can be a challenge by itself since the notch filter transmission region can be very broad.
Figure 1 shows the QW layer thicknesses for a design example with , , and using Eqs. (2) and (3). The resulting spectral transmission for this design immersed in fused silica centered at 532 nm and using and as the coating materials is shown in Fig. 2. The total thickness of the design is 27.0 μm and the FWHM of the stop band is .
The width of the stop band can be adjusted by changing . The widest stop band is achieved when . Figure 3 shows the modeled transmission and optical density (OD) spectra for a 100 layer design immersed in fused silica centered at 690 nm with and using and for the coating materials. The total coating thickness of the design is 10.7 μm. The FWHM of the stop band is .
The OD in the stop band is determined by the number of layers in the design, the ratio , the index ratio between the high index and low index coating materials, and to a lesser degree the FWHM of the apodization. Narrower stop bands (smaller ) require higher number of layers to achieve high OD.
The level of matching achieved by the apodization is adjusted by the FWHM parameter. A narrower apodization function (smaller FWHM) results in lower ripple in the transmission bands. However, lower FWHM also reduces the OD of the stop bands so a balance has to be found between matching, number of layers, and OD.
The design method also works if the high index and low index layers are interchanged and the low index material forms the thin layers. In this case, the design matches the index of the high index layers at the top and bottom of the layer stack. This makes it necessary to both have an AR coating matching the coating to the substrate as well as to air. However, the overall coating thickness is considerably thinner.
A. Truncated Apodization
It becomes apparent from Eq. (2) that reducing the ratio of the coating thicknesses starting from the middle results in a reduced reflectivity in the stop band compared to a design with the center ratio held constant throughout the coating. The coating thickness can be used more efficiently by applying the apodization to only a set number of the top and bottom layers of the coating keeping the central part of the coating stack at a fixed ratio .
Figure 4 shows the layer thicknesses for the high index layers for a design where the apodization has been applied over 101 layers with and . An additional 210 layers with a constant ratio of have been inserted in the middle for a total of 311 layers. The resulting modeled transmission and OD spectra are shown in Fig. 5 using as the coating materials when immersed in fused silica and centered at 532 nm. Note that an OD greater than 6 is achieved at 532 nm. Total design thickness is 28 μm and the FWHM of the stop band is .
B. Other Apodization Functions
Functions other than Gaussian can be used for apodization and still achieve good admittance matching. Different functions will result in a slightly different shape of the edges of the stop bands as well as different admittance matching.
An example of apodization using a quintic function  is given by2) and (3), but is now related to the FWHM by
Similarly a cosine square function can be used6 using Gaussian, quintic, and cosine-squared apodization. The modeled transmission and OD are shown in Fig. 7 for these three designs using a 532 nm reference wavelength and and as the coating materials. All three apodization functions give very similar filter performance. The quintic apodization has slightly steeper side edges on the blocking region while the Gaussian apodization has slightly higher OD in the blocking region.
The similar performance for the different apodizing functions shows that the admittance matching and thereby the ripple in the transmission band is not sensitive to the functional shape of the apodization. Admittance matching is achieved as long as the thickness apodization follows a smooth function gradually varying the thickness ratio from the center of the coating design.
C. Multinotch Designs
The apodized designs will have stop bands at higher orders of the design wavelength (). Multinotch filters with the desired stop bands spaced at the wavelengths of these higher orders are therefore easily designed with the method described above. A filter designed for 1064 nm will, for example, also have stop bands centered close to 532 and 355 nm. The dispersion of the coating material indices will shift the centering of the higher orders, which might make it hard to align the higher order stop bands to the desired wavelengths for narrow stop band designs.
Another approach for multinotch designs is to expand the single-notch apodization to cover more complex basic periods. An example of such a design is given by the design formula2) and (3) for the Gaussian apodization. This design matches the admittance of the high index material on top and bottom of the coating stack. Separate AR coatings need to be applied to match the design to air and the substrate. Since the transmission bands often cover a relatively limited wavelength range for this type of multinotch filter, the design of an AR is easier than, for example, the broad transmission range shown in Fig. 3.
The modeled transmission and OD spectra of a 85 layer design using Eqs. (11) and (12) is shown in Fig. 8 with and and a reference wavelength of 464 nm. The coating materials are and the coating is modeled as immersed in . Total coating thickness is .
Basic periods other than the example given by Eq. (11) can also be used in the multinotch designs as long as either all the high index layers or all the low index layers have a thickness in the middle that is equal to or less than a QW at the design wavelength. The thickness of this material can then be apodized from the middle of the coating keeping the sum of the QW thicknesses of each pair of high index and low index layers constant. This gives considerable flexibility in the designs, making it possible to achieve different number of, widths of, and OD of the stop bands while maintaining low ripple in the transmission regions.
3. Experimental Results
Several notch designs have been coated using ion beam deposition (IBD) in a Veeco SPECTOR IBD system. The coating materials were and .
The measured transmission spectra of a deposited single-notch filter coating centered at 532 nm is shown in Fig. 9. The design shown in Figs. 1 and 2 was used. The deposition was controlled by time only. The immersed spectrum was measured by optically contacting the coated fused silica sample to an uncoated fused silica substrate. The scan includes the reflection losses from the outer two uncoated surfaces (). The nonimmersed sample has reflection losses from the backside and from the coated side resulting from the fact that the coating is matched to . As expected, the loss from the coating to air side is similar to the uncoated fused silica reflection. The dip in transmission at is due to the fused silica substrate water absorption line.
Figure 10 shows the measured transmission together with the modeled transmission of a 100 layer deposited notch filter centered at 690 nm using the design shown in Fig. 3. The design was coated on standard microscope slides. The coated slide was optically contacted to an uncoated slide so that the immersed coating performance could be measured. The measurement and model include the reflection losses from the two outside uncoated surfaces. The deposition was controlled by time only. Good agreement is seen between the model and the measured transmission.
The measured and modeled transmission spectra of a 92 layer multinotch filter is shown in Fig. 11. The spectra include the reflection loss from the uncoated back side of the substrate. The design is the same shown in Fig. 8 except that an AR coating for the transmission regions is applied on the substrate side matching it from the admittance to the substrate and on the air side to match it to air. The center wavelength of the modeled design was shifted to 460 nm to match the centering of the deposited coating.
Good agreement is observed between the design and the measured transmission spectra when it comes to the shape of the stop bands. The ripple in the transmission region is partially due to imperfect performance of the AR coatings in addition to random layer thickness errors. Some reduction of transmission is also observed at the shorter wavelength passbands. This reduction is mainly thought to be due to residual absorption in the thick layers and can be further minimized by adjusting the deposition process to increase the oxidation during deposition.
All of the experimental results show relatively good agreement with the modeled design performance. Since the deposition of each layer is deposited based on time only, which introduces layer errors of the order of 1%, it can be concluded that the generated designs have relatively low sensitivity to layer errors.
The apodized thickness modulated design method presented in this work generates discrete layer notch filter designs with very low ripple in the transmission regions without the need for numerical optimization. The designs are tolerant to layer errors and functional shape of the thickness apodization. Several sample designs have been coated on time control using IBD deposition, with performance close to the theoretical designs.
The authors would like to thank Professor Angus Macleod at the Thin Film Center for helpful comments on the presented design method.
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