The term “rugate” is defined in *The American College Dictionary* as: “*adj*. wrinkled; rugose.” Rugose is defined as: “*adj*. having wrinkles; wrinkled, ridged.” The term has come into use in the optical coating community in the last three decades. Bill Southwell [1] said, in a private conversation, that the rugate concept originated with Conrad Phillippi of the Materials Laboratory at Wright-Patterson Air Force Base in 1982. Bill had been heavily involved in the rugate field and published many papers on the subject [2-7].

In this case, rugate is generally referring to **inhomogeneous** film structures versus thickness in a coating as seen in figure 1. This is the index of refraction versus thickness of the coating, which shows this wrinkled, rugate appearance. More conventional coatings are composed of **homogeneous** layers, which are composed of quarter wave optical thicknesses (QWOT) of high and low index as seen in figure 2.

*Fig. 1. Rugate filter with “perfect” envelope for antireflection (AR) in the passbands*

*Fig. 2. Conventional line filter with a QWOT stack of homogeneous layers*

The actual production of a rugate index versus thickness profiles [2, 3, and 4] is more complex and unconventional than the discrete homogeneous layers normally used in industry. To the extent that the functions of the rugate principle can be achieved with homogeneous layers, broader and more economical use can be made of the concept. A recently published paper by Li, et al. [8], extensively describes the implementation of rugate designs and production, where the index profiles are converted to many discrete homogeneous layers of **many different indices** using reactive sputtering of Si_{x}O_{y}N_{z}. They mention designs of up to 2,010 homogeneous layers. One point of this article is that, with a similar number of layers, similar results can be achieved using only two indices of refraction and using more conventional deposition processes.

The fiber Bragg grating (FBG) is also an example of a rugate filter with very small swings in index from low to high.

*Fig. 3. Reflectance spectrum of rugate line filter in Fig. 1*

Both of the index versus thickness profiles of figures 1 and 2 will generate a high reflectance or blocker band as in figure 3. Figure 3 is on a frequency (1 / wavelength) scale rather than a wavelength scale because that gives a more symmetric plot, and it better illustrates the harmonics to be discussed below. The frequency or wavelength of the reflecting band in figure 3 depends on the period or spacing of the repetitions in the index profiles of figures 1 and 2. The percent reflectance or height in the spectral (blocking) band is proportional to the difference in index from the peaks and valleys in figures 1 and 2, and the number of cycles or layer pairs in the overall thickness of the coating [6, 7, 9]. The spectral width of the reflectance band is also proportional to the difference in index from the peaks and valleys. Note that the spectrum in figure 3 happens to be the Fourier Transform of the index profile in figure 1, and figure 4 is the transform of figure 2 [Sec. 4. 6 of reference 9].

If the shape of the profile in figure 1 is perfect [7], the spectrum in figure 3 will be prefect, with reflection **only** in the blocking band, and none elsewhere from zero frequency to infinite frequency (it is an anti-reflective (AR) coating for both sides of the blocking line). The shape of the blocking band is a function of the shape of the rise and fall of the envelope in figure 1, from the start to the end of the rugate profile [7].

**The Problem**

If the index profile in figure 2 is with homogeneous layers, which are QWOTs at the frequency of 30, as in figure 3, then there will be harmonics of the reflectance band at the frequencies corresponding to 3/4 QWOT, 5/4, 7/4, etc., (or frequencies of 90, 150, 210, etc.) in the spectrum, as in figure 4. Historically, many coatings have only been concerned with a narrow spectral region around the first harmonic, such as in the visible spectrum, which is only a frequency range or bandwidth of about 2:1 (380 to 780 nm). Under these circumstances, there would be no concern for what is happening outside of that spectral band. However, if an AR were needed from 380 to 1,540 nm, this would include the range of the third harmonic (3/4 waves). If the first harmonic of the QWOT were outside the AR band at 1,600 nm, then the third harmonic would be at 1,600/3 = 533 nm. This natural harmonic would need to be overcome. The rugate index profile can, in principle, overcome this problem.

*Fig. 4. Schematic %R spectrum of a QWOT stack line filter in Fig. 2*

**Higher Harmonic Reflection Bands**

The fundamental high reflection band generated by a QWOT stack is also illustrated in figure 5. This figure is a “real world” example showing the sidebands and shape of the blocking band. One can consider the implications of higher harmonic replications of this fundamental reflection, where the peak reflections occur at 1/4, 3/4, 5/4 QWOTs. The extension of the effects is seen in figures 4 and 5, where there are reflection peaks at every odd harmonic of the fundamental QWOT stack which are shown on an optical density scale and in wavenumbers (frequency).

Optical density (OD) is defined as:

*OD** = log _{10}(1/T)*

Here, T is the fractional transmittance, not the percent T.

As will be seen in more detail later, different bands can be suppressed by usinga unity ratio between the thicknesses of the high and low index layers. The QWOT stack has equal optical thickness in the high and low index layers (the ratio between the **overall thickness** of the layer pair to the thinnest layer is 2:1 in this case.)

*Fig. 5. Real reflectance spectrum in OD of a QWOT stack like Fig. 4*

*Fig. 6. Spectrum like Fig. 5, but with 3:1 ratio of layers*

The second, fourth, sixth, etc., harmonic reflections seen in figures 4 and 5 are suppressed or missing. For example, a 3:1 ratio between the **overall thickness** of the pair to the thinnest layer will add the second and fourth harmonics but suppress the third, sixth, ninth, etc., as seen in figure 6. A 4:1 ratio will add the second but not the fourth, eighth, twelfth, etc.

The width of each of these block bands is the same in wavenumbers. This then leads to the fact that the third harmonic frequency will have one-third the relative width (Δω/ω) at that frequency, or in (Δλ/λ) *wavelength* as at the fundamental QWOT wavelength. Similarly, the fifth harmonic will be one-fifth as wide. This then allows the creation of narrower bands when needed, but it is at the expense of three or five times as thick a stack for a given block band wavelength. Using high and low index materials that are closer to each other in index can accomplish the same thing [9], if the appropriate materials are available. The free width between block bands also needs to be considered.

These properties can be useful in designing blockers for laser lines when the laser frequency is doubled, tripled, quadrupled, etc. Other applications using these harmonics may appear in the future, in areas such as fiber optics communications filtering.

Gradual transitions in index of refraction from layer to layer are illustrated in figure 1. It can be shown that the first order reflection band is not significantly changed from when the transition between layers is “square,” as in figure 2, even to the case where it is sinusoidal. Figure 7 shows the way one might simulate sinusoidal rugate transitions from one index to the next as was recently done in reference 8.

*Fig. 7. Index versus thickness approximating a rugate** structure*

Figures 3 and 4 show that such a “rugate” index profile produces essentially the same fundamental reflection at the wavenumber/wavelength where the peak to valley spacing in the index values are one QWOT apart. However, the higher harmonics of this are not present with the rugate.

**Bandwidth Limitations On Short Wave Pass Filters**

The rugate’s removal of harmonics is useful when very wide passbands (ARs) are needed next to blocking edges. There are differences in the behavior of wide passband edge filters with short wavelength passbands (SWP) and of those with long wavelength passbands (LWP). The **bandwidth** of the passband here is **defined** by the longest wavelength in the band divided by the shortest wavelength, or the highest frequency divided by the lowest. This is virtually unlimited in the case of a LWP filter, as seen in figures 4 and 5. However, it is significantly limited in the case of the usual approach of using quarter wavelength layer thickness stacks for SWP filters. This limitation is seen in figures 4 and 5 because of the third and higher harmonics that exist when QWOTs are used. Bandwidths of over two start to have increasingly higher reflection losses, and bandwidths of two and one-half become virtually impractical for QWOT stacks.

When band-passes broader than about two are needed for edge filters with a SWP, recourse to rugate-like designs is needed. Such designs can be achieved with only two **homogeneous** materials by employing the concept of the Herpin [10] approximation, although many layers may be required. Herpin showed that a single layer of any thickness could be simulated by a three-layer combination of layers of two indices which are higher and lower than the one simulated. Generally, it has been found that this can also be done with only two layers.

Edge filters have a certain slope at the edge which is usually specified, some degree of blocking which is also usually specified by wavelength band and optical density, and a transmission or passband AR which is specified. If the passband is on the short wavelength side of the edge, it is referred to as a SWP filter. If the passband is on the long wavelength side of the edge, it is a LWP filter.

The passband of a LWP filter is essentially like an AR coating except that further attention is paid to the filter edge, which is normally outside the edge of the AR band (on the short wavelength side). In such a case, design targets would be added to the optimization process in addition to those already there for the AR band, which would apply pressure for a steeply rising reflection edge at the desired wavelength. The steepness of the edge is a nearly linear function of the number of layers. The LWP bandwidth has no inherent limitation, because there are no harmonic bands at longer wavelengths than the fundamental blocking band generated by a QWOT stack at the center wavelength of the blocking bands, as illustrated in figure 5 (at the left end).

When a SWP filter is designed with a basic QWOT stack, the passband width is limited to the region between the first and third harmonic (1,000 and 3,000 cm^{-1}), as seen in figure 5. Small adjustments in the thicknesses of what started as QWOT layers in order to improve the average transmittance in the passband cannot overcome the above limitation. We will discuss the effects of this limitation and a **non-rugate** approach, which can overcome this limitation.

**Design Limitations Of QWOT Stack SWP Filters**

It has been found [9] that average reflectance (Rave) in the AR band was primarily a function of the bandwidth (B) and almost no function of the number of layers. The slope of the edge as measured in the change in %T or %R per nm was almost a linear function of the number of layers. Therefore the designer can use whatever number of layers is needed to obtain the required slope or steepness.

**When Wider Passbands Are Needed For SWP Edge Filters**

Macleod [11] discusses this problem and some solutions, and he refers to the earlier work of Epstein [12] and Thelen [13]. Thelen also has a chapter in his book that deals with the subject [14]. As will be seen below, the solutions are of a **rugate**** nature** with respect to the profile of the index of refraction with thickness. In the practical case, the inhomogeneous index versus thickness profile can be replaced by non-QWOT layer structures of just two indices along the lines introduced by Herpin [10] and Epstein [15]. These substitutions of Herpin equivalent layers or Epstein periods (homogeneous layers) as **surrogates** for **rugate** (inhomogeneous) structures will be here called **“surrugate”** designs.

*Fig. 8. Blocking- and pass-bands produced by the index** versus thickness profile of a QWOT stack like that of Fig. 2*

Figure 8 shows the reflectance spectrum, which is produced by a 40 layer or 20 cycle design of QWOTs with the index profile like that of figure 2. The first step is to replace the two layers per cycle of figure 1 with six layers per cycle as in figure 9. These have been optimized with some of the layers linked to others to maintain symmetry, while reducing the third harmonic band to zero, as seen in figure 10. This gives blocking bands at harmonics one, five, seven, etc., but **NOT** at the third harmonic, as seen in figure 8.

*Fig. 9. Index versus thickness profile of symmetric layer stack with six layers per cycle*

*Fig. 10. Blocking**- and pass-bands produced by the index** versus thickness profile of Fig. 9*

When the above approach is extended to 10 layers per cycle as in figure 11, this gives blocking bands at harmonics one, seven, etc., but **NOT** at the third and fifth harmonics, as seen in figure 12.

When extended further to 14 layers per cycle, as in figure 13, the third, fifth, and seventh harmonics can be suppressed, as in figure 14. This process can be extended by increments of four layers to suppress higher and higher harmonics. As this is done, the Herpin equivalents would come closer and closer to approximating the sinusoidal rugate index profile that would have only the first harmonic band. This, in turn, would allow a pass-bandwidth as wide as desired. However, there are limitations to how much the reflection in the pass band can be reduced [9].

*Fig. 11. Index versus thickness profile of symmetric layer stack with 10 layers per cycle*

*Fig. 12. Blocking- and pass-bands produced by the index** versus thickness profile of Fig. 11*

*Fig. 13. Index versus thickness profile of symmetric layer stack with 14 layers per cycle. Sine index profile as in a rugate is superimposed over the Herpin-like surrugate index profile to show the effective similarity. This gives blocking*** bands at harmonics one, nine, etc., but NOT at the third, fifth, and seventh harmonics, as seen in Fig. 14**.

*Fig. 14. Blocking- and pass-bands produced by the index** versus thickness profile of Fig. 13. Note: third, fifth, and seventh harmonics are suppressed.*

*Fig. 15. Average reflectance in a passband versus bandwidth for designs of six, 10, and 14 layers per cycle, which suppress the third, fifth, and seventh harmonics respectively*

The points in figure 15 show the Rave in the passband which was achieved for a given bandwidth by optimization of two cycles of layers on each side of a fixed set of 16 cycles of layers. Therefore, the total number of cycles was kept to a total of 20, which would be 20 layer pairs or 40 layers in the QWOT case of two layers per cycle. It can be seen that the Rave becomes progressively larger with increased bandwidth in a linear fashion until it approaches the next harmonic band. At that point, the Rave rises, indicating the need to suppress yet another harmonic.

**Conclusions**

When band-passes of bandwidths broader than about two are needed for edge filters with a short-wave passband, or for ARs, recourse to rugate-like designs is needed. By employing the concept of the Herpin approximation, two materials **only** can be used as a surrogate for the rugate principle, or a **“surrugate”** design. The number of layers per cycle required to eliminate the third, fifth, seventh, etc., harmonic blocking bands is six, ten, 14, etc. The design approach to broaden the pass bandwidth with rugate and surrugate designs has been described, and it has been demonstrated how rugates can be replaced by **surrugates**.

**References**

- W. H. Southwell, Table Mountain Optics, 509 Marin St, Thousand Oaks, CA 91360.
- R. L. Hall, W. H. Southwell, and W. J. Gunning, "The Fabrication of Rugate Filters Using the Digital Technique,"
*Optical Interference Coatings*, 1988 Technical Digest Series 6, 154-157. (1988). - W. H. Southwell and W. J. Gunning, "High-speed ellipsometer for thin-film deposition monitoring," SPIE 1019, 84-89 (1988).
- William J. Gunning, Randolph L. Hall, Frank J. Woodberry, W. H. Southwell, and Natalie S. Gluck, "Codeposition of continuous composition rugate filters," Applied Optics 28, 2945-2948 (1989).
- W. H. Southwell, "Spectral response calculations of rugate filters using coupled-wave theory," J. Optical Soc. Amer. A 5, 1558-1564 (1988).
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- R. R. Willey,
*Practical Design of Optical Thin Films*, 4^{th}, Ed., Willey Optical (2014). - M. A. Herpin: "Calcul du pouvoir réflecteur d'un systèm estratifié quelconque,"
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*Thin Film Optical Filters, Fourth Edition,*Sec. 7.2.3.9, CRC Press, Boca Raton, FL (2010). - L. I. Epstein, “Improvements in Heat-Reflecting Filters,”
*J.O.S.A.***45**, 360-362 (1955). - A. Thelen, “Multilayer Filters with Wide Transmittance Bands,”
*J.O.S.A.***53**, 1266-1270 (1963). - A. Thelen,
*Design of Optical Interference**Coatings*, Chap. 6, McGraw-Hill, New York (1988). - L. I. Epstein: "The Design of Optical Filters,"
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