## Abstract

The use of asymmetrical dual-cavity (ADC) filters with two transmittance peaks is proposed for a simultaneous monitoring of the thickness uniformities of high- and low-index materials. The method described works independently of the manufacturing technique used, and can detect non-uniformities of refractive indices. The properties of these ADC filters are studied using Smith’s concept of effective interfaces, and admittance diagrams. An experimental application of this method is demonstrated, which can be used for increasing the yield of manufactured optical filters.

©2003 Optical Society of America

## 1. Introduction

The ability to control precisely the thickness of deposited materials is a key factor for the successful fabrication of complex multilayer optical coatings, such as those encountered in the fields of telecommunication and ultrafast phenomena. To achieve this desired degree of precision, very stable materials and deposition rates, as well as a precise control of the deposition time or a thickness monitoring technique are required.

Equally important when manufacturing complex optical filters is the control of the thickness uniformity of the deposited materials. Thickness uniformity matters not only for maintaining the uniformity of performance over a large area (of obvious significance to commercial manufacturers), but also for fabricating complex optical coatings that are sensitive to small thickness variations under a probe beam. In both cases, not only is it important to maintain a good thickness uniformity for individual layers, but it is also critical to control the relative thickness difference between the different materials in the coatings. Figure 1 shows how slight relative differences between the optical thicknesses of the high- and low-refractive index materials in different types of coatings may affect their spectral properties. Particularly sensitive to thickness errors and non-uniformities are corrective-types of coatings, an example of which is given in column 2 of Fig. 1 (a solution to the Bow-Lake manufacturing problem [1]). For comparison, column 1 of the same figure shows that a single-cavity Fabry-Perot filter, frequently used for thickness uniformity monitoring, would not detect this type of thickness variations in the low-and high-index films. The reason is that the position of the single-cavity peak is not only a function of the cavity layer thickness, but also of the mirrors’ phase properties, which can be altered during the fabrication of the filter, and which compensate for the cavity layer non-uniformity. The purpose of this work is to demonstrate that a “class” of filters, called here the asymmetrical dual-cavity (ADC) filters (column 3 in Fig. 1), can be used for efficiently monitoring the differences in the thickness uniformity for different materials.

The method proposed in this work does not depend on the manufacturing technique used for fabricating the coatings, and can be used to optimize the uniformity of coatings, through the optimization of some deposition parameters (shadow mask corrections in our case).

## 2. Some properties of ADC filters

What we describe in this work as ACD filters corresponds to a class of optical coatings with the two following characteristics:

with cavities made of low (*L*) and high (*H*) refractive index materials, the parameters *a, b, c, d, e* being integers (the sequence of the cavities can change, and the first and/or last *H*-layer can be ommitted).

- A transmittance (or reflectance) spectrum showing two distinct maxima (or minima) around the central wavelength λ
_{center}.

A particular coating of this type (with *a*=*b*=*c*=4, *d*=*e*=1) is shown in Fig. 2(a) (with a Smith representation and its corresponding parameters that will be used in Section 3.1), and its transmittance spectrum appears in Fig. 2(b). The sensitivity of the ADC filters to thickness errors is shown in Fig. 1 (column 3) and Fig. 3 (row a) for several systematic *H* and *L* optical thickness detunings from their ideal design values. The optical thickness detunings are represented by the following parameters,

used simultaneously for all *H*-layers, or for all L-layers (*n*_{H,L}
and *d*_{H,L}
are the refractive indices and thicknesses of *H*- and *L*-layers, respectively). On Fig. 3(row a), one can see that the left-to-right transmittance peak ratio (*T*_{left}*/T*_{right}
) correlates with the ratio *mult*_{L}*/mult*_{H}
. This correlation is independent of the total thickness of the coating, as shown when comparing Figs. 3 (a2) with (a5) [or Figs. 3 (a3) with (a4)], which have similar *mult*_{L}*/mult*_{H}
ratios, but different total thicknesses. This property allows the detection of small excesses of one material over the other, and is the basis for the use of ADC filters for thickness uniformity determination (see Section 4). Figure 4(a) shows that this relation between *mult*_{L}*/mult*_{H}
and *T*_{left}*/T*_{right}
is linear around the ideal (1,1) case. In addition, Fig. 4(b) shows that the total optical thickness of the filter correlates linearly with the position of the central wavelength *λ*_{center}
. We will see in Section 4 that these correlations are useful for thickness uniformity monitoring applications.

In the next section, we use Smith’s theory and admittance diagrams to get some insights regarding the origin of these properties.

## 3. Theory

#### 3.1. Using Smith’s concept of effective interfaces

Following Smith’s concept of effective interfaces [2], the transmittance of a non-absorbing multilayer system can be described in terms of any of its layers as a spacer (with refractive index *n*_{s}
and metric thickness *d*_{s}
), and the reflectance, transmittance and phase-changes under reflection on its effective interfaces [Fig. 2(a) gives a schematic example of this concept for an ADC design]. According to Smith, the transmittance of the whole system at a wavelength *λ*_{i}
which satisfies the condition

is given by

[*R*
_{1}, *R*
_{2}, *T*
_{1}, *T*
_{2}, *ϕ*
_{1} and *ϕ*
_{2} are defined in Fig. 2(a); *m* is an integer]. In the case of the ADC filter shown in Fig. 2(a), *R*
_{1} represents the reflectance of a quarterwave stack, *R*
_{2} is the reflectance of a single cavity Fabry-Perot, and exceptionnally the selected spacer layer has a zero thickness (*d*_{s}
=0). While Eq. (3) shows the general importance of the phase in the determination of the spectral properties of multilayer systems, one can deduce from Eq. (4) that for non-absorbing systems, with *T*
_{1}=1-*R*
_{1} and *T*
_{2}=1-*R*
_{2}, *T*_{Smith}
reaches its maximum unity values when *R*
_{1}=*R*
_{2}.

Figure 3 shows several transmittance spectra for the ADC filter represented in Fig. 2, with their corresponding *T*_{Smith}
spectra (Eq. (4)) and the sum of the phase-changes *ϕ*
_{1}+*ϕ*
_{2}. One can verify the validity of Eq. (4) at the two points where *ϕ*
_{1}+*ϕ*
_{2}=2*mπ*. The correlation of *T*_{left}*/T*_{right}
with *mult*_{L}*/mult*_{H}
can be interpreted as a consequence of the varying shift between the points where the *ϕ*
_{1}+*ϕ*
_{2}=2*mπ* condition is satisfied, and *T*_{Smith}
spectra. This varying shift is due to the fact that the values of *ϕ*
_{1} and *ϕ*
_{2} both move along the ordinate axis when small errors are introduced. This small variation in the scale of *ϕ*
_{1} and *ϕ*
_{2} has no perceptible effects on the *R*
_{1} and *R*
_{2} spectra, but affects significantly the condition given by Eq. (3), and the shape of the transmittance curve. This particular dependence on *ϕ*
_{1} and *ϕ*
_{2} explains the sensitivity of the ratio of the peaks to small variations of *mult*_{L}*/mult*_{H}
.

As seen from Fig. 3(row a), the shape of *T*_{Smith}
does not change significantly when varying *mult*_{L}*/mult*_{H}
. This observation apparently leads to a possible misinterpretation, as one would expect one of the peaks to quickly reach the minimum of the *T*_{Smith}
curve, and reverse the ratio *T*_{left}*/T*_{right}
. In fact, such an inversion of *T*_{left}*/T*_{right}
does not occur, because of an increase of the separation of the transmittance peaks, which is another consequence of the shift of *ϕ*
_{1}+*ϕ*
_{2} along the ordinate axis [see Fig. 3 (column 6)]. This behavior maintains the monotonicity of the curve *mult*_{L}*/mult*_{H}
-vs-*T*_{left}*/T*_{right}
over a wider range than shown in Fig. 4.

In addition, we can predict from Fig. 3 that any modification to the design that would make the Fabry-Perot peak *R*
_{2} narrower (i.e., increasing the number of *HL* pairs in the mirrors, or the order of the cavities in Eq. (1)) will make the transmittance peaks narrower, and bring them closer one to the other (due to a steeper variation of *ϕ*
_{1}+*ϕ*
_{2}).

It should also be noted that varying the refractive indices of the films in ADC filters has an effect on *T* that is distinct from the effect of thickness variations. Figure 5 shows that the main differences are in the peak ratio (due to a modification of *T*_{Smith}
) and their bandwidths. These observations could be used to detect non-uniformities of refractive indices (however, in the remaining part of this work, we will assume that the refractive indices are uniform).

#### 3.2. Using admittance diagrams

Additional information can be obtained when examining the properties of ADC with admittance diagrams drawn for a specific wavelength. Some basic rules for constructing and interpreting admittance diagrams are as follows (see Macleod’s book [3] for more details):

- The starting admittance
*Y*(0,λ_{0}) is given by the refractive index of the substrate (assumed to be real in this work) at the desired wavelength λ_{0}. - The admittance loci of subsequent quarterwave layers covering the substrate are given by semi-circular paths intercepting the real axis at points
*α*/${n}_{i}^{2}$ where*α*is the previous intercepting point, and*n*_{i}is the refractive index of the actual layer at the desired wavelength. - The reflectance of a coating of thickness
*d*at λ_{0}is a measure of the norm between the surface admittance*Y*(*d*,λ_{0}) and the admittance*Y*_{0}of the ambient medium (unity when air is considered):

The ADC filter described in Figs. 6 and 7 has the following structure (*HLH*)2*L*(*HL*)2*H*(*LH*)^{2}, which is simple in order to simplify the visualization of its admittance loci. Figures 6 and 7 show admittance diagrams corresponding to an ADC filter with ideal thicknesses (*mult*_{H}
=*mult*_{L}
=1) and with thickness errors (*mult*_{H}
=1/*mult*_{L}
=0.98), respectively. The admittance diagrams are divided into four parts (columns 2–5 in Figs. 6 and 7) to facilitate their interpretation and to reveal the role of some parts of the coating.

One can see from Fig. 6(column 2) that the surface admittance of the first part of the coating (a quarterwave stack) does not vary a lot with the wavelength considered (same for its reflectance, given by Eq. (5)). The part of the admittance loci that is most affected by the wavelength changes corresponds to the central quarterwave stack (column 3): The effect observed is a significant phase change of its surface admittance compared to the value at λ_{center}, reaching a summit *π*-change [compare Fig. 6(a3) with Figs. 6(b3) and (c3)]. This difference of *π* in the phase is maintained after passing through the remaining parts of the coatings (columns 4 and 5), and results in the appearance of the transmittance peaks. This interpretation is not incompatible with the Smith approach of Section 3.1, and leads to complementary observations:

- A central stack composed of more
*HL*pairs (larger parameter*b*is Eq. (1)) will lead to a smaller wavelength shift Δλ required to achieve the*π*phase change, and will result in a reduction of the separation between the transmittance peaks; - Removal of the last
*H*-layer in the design will lower the transmittance peaks and separate them slightly futher [apparent from Figs. 6(b5) and (c5)].

Figure 7 shows similar admittance diagrams, after introducing thickness “errors” in the ADC. If the admittance of the first parts of the coatings are apparently almost unaltered by this change (columns 2, 3 and 4 of Fig. 7), the admittance from the last stack is clearly affected; the diameter of the loci in Figs. 7(b5) and (c5), different from Fig. 6, lead to the variation of the transmittance peaks. Interestingly, column 5 of Fig. 7 shows that removing the last *H*-layer will reverse the ratio of the transmittance peaks (the left peak will be lower than the right peak), revealing the importance of the last layer optical thickness on the shape of the coating transmittance.

## 4. Application to thickness uniformity monitoring

Monitoring the thickness uniformity of coatings is frequently done using thick single layer coatings, or Fabry-Perot filters. In the former case, separate depositions are necessary for evaluating the uniformity of individual materials, and large thicknesses of the films are required to reduce the width of their transmittance maxima and increase the precision of thickness determinations. In the latter case, a higher precision in the total optical thickness determination can be obtained, but at the detriment of artifacts caused by the difference of uniformity of individual materials (see Fig. 1).

Although a possible solution to these problems of monitoring precision and artifacts could be the use of two Fabry-Perot filters, with *L*- and *H*-layer spacers, we have shown in this work that ADC filters are well-suited for the precise simultaneous monitoring of the uniformities of two materials. Our theoretical analysis confirmed that this method is very sensitive to small changes in thickness values, due to its sensitivity to phase terms in the Smith effective interface model. It also showed that several parameters of the ADC filter, such as the number of pairs in its quarterwave stacks (particularly the central stack) and the order of its cavities, can be used to control the width and wavelength separation of its transmittance peaks.

Using experimentally determined optical constants for any pair of non-absorbing *L*- and *H*-materials, one can calculate curves such as those appearing in Fig. 4 and use them as calibration curves for the non-ambiguous inverse determination of *mult*_{L}
/*mult*_{H}
and the total *OT* from measured values of λ
_{center}
and *T _{left}/T_{right}* for a particular ADC filter design. The variation of these parameters at different locations on a sample will indicate the total thickness non-uniformity, and the amount of material (

*L*or

*H*) that is in excess of the other. The movie in Fig. 8 gives an example of the variation of

*T*measured along the surface of a typical ADC filter made in our Dual Ion-Beam Sputtering deposition system (Spector, Veeco-IonTech) using TiO

_{2}and SiO

_{2}on a 2 inches×4 inches B270 glass substrate. The substrate was mounted on a 12 inches diam. rotating substrate-holder, in an orientation that is clearly apparent from Fig. 8.

From every transmittance spectrum measured on the sample, values of *T*_{left}*/T*_{right}
and λ
_{center}
can be estimated, and used with the calibration curves to extract position-dependent *mult*_{L}*/mult*_{H}
and total *OT* values. Figure 9 shows typical experimental results obtained at different positions along the *Y*-axis of an ADC filter (different from Fig. 8). The data were normalized to values found at the monitoring position, around *Y*=6.6 cm, to remove the effect of deposition thickness-errors on the shape of the ADC filter and uniformity calculations. The total *OT* values, over (or under) the unity line, indicates any global excess (or lack) of material, compared to the monitoring position. The *mult*_{L}*/mult*_{H}
ratio, over (or under) the unity line, indicates that less *L*- (or *H*-) material should be deposited. The combination of these two sets of data offers a few interpretations, which are summarized by the regions identified in Fig. 9, and described in Table 1. A *mult* ratio close to unity (for example, at the boundary between regions A and B in Fig. 9) indicates that the requested corrections to the deposition rates at this position will have to be performed on both materials simultaneously. In addition, when the *mult* ratio and the normalized *OT* values fall on different sides of the unity value, corrections can be made on one material, but no conclusions can be drawn on the second material, which means that iterations are still needed for correcting both materials.

Figure 10 shows two uniformity measurements done before and after such an iteration. In our case, the iteration meant a variation in the profiles of the shadow masks used in front of SiO_{2} and TiO_{2} targets during the deposition. It can be seen from Figs. 10(a1) and (b1) that a correction done on one extremity of the shadow masks can easily improve the total *OT* uniformity of the coating, but that more iterations are required to improve the *mult* ratio. Once again, it should be noted that single-cavity Fabry-Perot filters alone would not lead to this last conclusion, giving instead a false impression of good uniformity after the masks iteration.

## 5. Conclusion

We demonstrated experimentally the use of this uniformity monitoring method with our dual ion-beam sputtered films, and showed its usefulness for the correction of the uniformity of optical coatings through the optimization of experimental parameters (in our case, shadow mask corrections).

Part of this work was first presented at the Optical Society of America’s 8
^{th}
Topical Meeting on Optical Interference Coatings (OIC) held in Banff, Canada July 15–20, 2001. [4] During the writing of the present manuscript, it was brought to my attention that the method as described at the OIC meeting is now used in the optical thin films industry, for example in the production of wavelength division multiplexing filters.[5]

## Acknowledgments

The author wishes to thank Tom Cassidy and Steve Guétré, and Dr. Pierre Verly, for having critically tested the method described in this work. He also thanks Drs. J. A. Dobrowolski and Dan Dalacu for their proof reading of the manuscript.

## References and links

**1. **J. A. Dobrowolski, S. Browning, M. Jacobson, and M. Nadal, “2001 Optical Society of America Topical Meeting on Optical Coatings: Manufacturing Problem,” Appl. Opt. **41**, 1–14 (2002). [CrossRef]

**2. **S. D. Smith, “Design of Multilayer Filters by Considering Two Effective Interfaces,” J. Opt. Soc. Am. **48**, 43–50 (1958). [CrossRef]

**3. **H. A. Macleod, *Thin-film optical filters* (Institute of Physics, Bristol, 2001). [CrossRef]

**4. **D. Poitras, T. Cassidy, and S. Guétré, “Asymmetrical Dual-Cavity Filters: Theory and Application,” in *Optical Interference Coatings, OSA Technical Digest*, pp. MD3-1-3 (Optical Society of America, Washington D.C., 2001).

**5. **D. I. Pearson, P. Luff, M. Davis, and A. T. Howe, “Use of asymmetric dual-cavity filters to track small H and L variations as a tool to increase yield of DWDM filters,” in *Photonic Integrated Systems*, L. A. Eldada, A. R. Pirich, P. L. Repak, R. T. Chen, and J. C. Chon, eds., Proc. SPIE4998, 178–185 (2003). [CrossRef]