1. Introduction

Accurate controlling of the thickness of each layer is a key technology to improve the performance of a filter and to meet the theoretical design during the coating process. To monitor the deposition of a film, the methods used usually include quartz crystal monitoring, single-wavelength optical monitoring and broadband optical monitoring. Monitoring by counting time has also been applied if the deposition rate is very stable. To produce optical filters, the optical monitoring method is better than other methods [1, 2] and broadband optical monitoring has more flexibility than single-wavelength [3].

Turning point monitoring is a well-known method since quarter-wave layers are commonly used in various designs. However, noise from the monitor system often makes it difficult for the operator to judge the terminal point of each layer. Optimally, the increment of thickness near the cutting point should produce a large change in transmittance or reflectance. The broadband monitoring system allows experienced operators to select a different monitor wavelength for each layer. Usually the wavelength that causes the termination point to be nearly halfway between the peak and valley on the transmittance curve in the runsheet will be chosen as the monitor wavelength, since transmittance or reflectance has a larger change at that point. However, very few investigations precisely pointed out the best monitor wavelength for each layer of a designed filter.

A numerical induction in this paper clarifies which monitor wavelength has the largest sensitivity on the cutting point so that the thickness error can be reduced as much as possible. A novel broadband monitoring method is described in this article. Using the selected sensitive wavelength by mathematical calculation to monitor every layer, we have pointed out the way to compensate the thickness error for the reference wavelength from a runsheet. For narrowband pass filter production, this selecting sensitive monitor wavelength (SSMW) method can achieve a better optical performance than using the turning point method. In addition, it won’t let the central wavelength peak shift.

2. Principle and method

Let us assume that we are dealing with normal incidence, zero losses and an ideally steady deposition environment so that the refractive index of material and the film growth rate are constants. For a multilayer thin film, the characteristic matrix method is used for the calculation and the layer can be represented by the following matrix [4]:

where n_E is the equivalent refractive index of all previous layers, δ is the phase thickness, n and d are the refractive index and physical thickness of the current layer, respectively, and λ is the monitor wavelength.

A typical optical filter with quarter-wave optical thickness layers is used as an example below. Air is the incident medium. The refractive index of the substrate is 1.5. The refractive indices of low and high materials are 1.46 and 2.3, respectively. The reference wavelength is 1550nm. The transmittance of this layer can be described as follows:

Note that the transmittance is not just in a simple sine or cosine form. Therefore, we won’t consequently get the largest slope (sensitivity) midway between the peak and valley of the transmittance curve in the runsheet. Fig. 1 shows the sensitivity curve of the reference wavelength of a H layer and the last layer of a quarter-wave stack, where H and L are the quarter-wave layers of high and low refractive indices, respectively [5].

Here, we define the sensitivity as the slope of the runsheet curve, that is, the change of transmittance per unit thickness increment as below:

The sensitivity at each cutting point for every layer for a different monitor wavelength can thus be found by Eq. (3).

The sensitivity of each wavelength is shown in Fig. 2. It can be seen that the reference wavelength 1550nm has a minimum sensitivity 0. Therefore, we may get the largest thickness error by monitoring with this wavelength. The best monitor wavelength is easy to see as shown in Fig. 2. For instance, 1261nm can be the best monitor wavelength for monitoring the last layer of the quarter-wave stack. It also shows that the sensitivity curve becomes sharper, has more ripples and the maximum value becomes larger as the layer number increases.

On the other hand, the sensitivity of the terminal points for the low refractive index material layers in such quarter-wave films has a similar characteristic except it has a lower value as show in Fig. 3.

 

Fig. 1. Sensitivity of reference wavelength at different high refractive index layer “H”.

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Fig. 2. Sensitivity of monitor wavelengths on cutting points in high refractive index layer “H”.

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Fig. 3. Sensitivity of monitor wavelengths on cutting points for low refractive index layer “L”.

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In the coating process, there usually exist many factors to cause an imperceptible change of material refractive index or deviation of the physical thickness and result in a change of the optical thickness. However such changes do not cause much trouble for the choice of the sensitive monitor wavelength as indicated by Fig. 4 and Fig. 5. Fig. 4 is the sensitivity vs. wavelength of the last layer of a seven-layer quarter wave stack, S/HLHLHLH/A, with a 1% standard deviation random error of optical thickness in the layers. Fig. 5 is the same except with the refractive index changed from 2.3 to 2.25. Apparently, the originally selected wavelength still works quite well in the monitoring, since the sensitivity has not changed much under a small index fluctuation.

 

Fig. 4. Comparison of the sensitivity of the cutting point in the seventh layer: dark line, each layer in quarter-wave stack; grey line, each layer not in exact quarter-wave stack.

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Fig. 5. Comparison of the sensitivity on the cutting point in the seventh layer: dark line, n_H=2.3; grey line, n_H=2.25.

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The traditional turning point method has an advantage of self-compensation in thickness error [6]. This is particularly useful during the fabrication of a narrow-band pass filter. When a very sensitive monitor wavelength is chosen to monitor the cutting point, a layer thickness error still can’t be avoided although it is very tiny. Such a small layer thickness error will affect the deposition of the following layers and may result in the shift of the central wavelength. However, the error can be compensated by the following layer based on the admittance loci theory of Macleod to keep the center wavelength from shifting as described in the following.

Let us select 939nm, 1194nm and 1150nm to be the monitor wavelength of the first, second, and the third layers respectively to deposit quarter-wave layers. As a result of noise, the first layer is assumed terminated too early and it changes the optical thickness from 0.25λ0 to 0.24λ0, where λ0 is the central wavelength 1550nm. The shortfall in the thickness can’t be calculated directly from the runsheet. However the problem can be solved by analyzing the admittance loci pattern of the first two layers as shown in Fig. 6. The admittance loci theory points out that the points of intersection of the admittance circle and the real axis correspond to the turning points of the monitoring curve. Admittance values Y at these points are real numbers and can be calculated from following formula:

where R is reflectance and can be calculated by R=1-T, where T is transmittance shown in the runsheet. Therefore, the admittance value can be figured out when the transmittance curve passes through a turning point in the second layer using 1194nm to monitor as shown in Fig. 7. Note that if the monitor wavelength is too short, the points of valley and peak might be reversed and Eq. (4) would give a wrong admittance value. Hence, we suggest that the monitor wavelength be chosen at the first peak of the sensitivity diagram that is closest to the reference wavelength to avoid the ambiguity.

 

Fig. 6. Admittance loci of 1550nm of the first two layers.

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Fig. 7. Admittance loci of 1194nm of the first two layers.

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The growth of these two layers can be described by the following matrix [7]:

The imaginary part of Y is required to be zero, as illustrated in Fig. 7, and we have:

Substituting the refractive index of the substrate ns for n in Eq. (5) to Eq. (7) and noting that the real part of the admittance value at the turning point Re{Y}=(1-√R)/(1+√R), we can find out the optical phase δ_A and δ_B, namely, the phase thickness of the first layer and the phase thickness between the starting point and the turning point of the second layer, respectively, from Eqs. (6) and (7).

Then, we convert δ_A to optical thickness and use it to find the terminal point of the first layer in the admittance loci of the central wavelength 1550nm. Suppose the optical phase of the first layer for 1550nm is δ_A’, and the value of the first terminal point is α′+iβ′, then we insert this in the thin film matrix as follows:

$\left[\begin{array}{c}B\prime \\ C\prime \end{array}\right]=\left[\begin{array}{cc}\cos {\delta }_c& i\frac{\sin {\delta }_c}{n}\\ \mathit{in}\sin {\delta }_c& \cos {\delta }_c\end{array}\right]\left[\begin{array}{c}1\\ \alpha \prime +i\beta \prime \end{array}\right]$

Let the imaginary part of C’/B’ be zero:

Now, we get the phase thickness δ_c for the central wavelength. The explicit solutions for Eq. (8) are expressed as below:

One of the solutions will be different from the other by π/2, and the operator shall choose the reasonable one.

In case the admittance of the second layer does not stop at the real axis, δ_C can be recalculated together with the third layer by using Eqs. (4) to (9) as in the previous procedure except that the monitor wavelength of the third layer is 1150 nm. During the calculation of the optical phase between the beginning point and the turning point of the third layer, we have substituted the admittance value of the first layer terminal point as a new substrate. In this case, the admittance value of the equivalent substrate is α+iβ and the explicit solutions of δ_B will be:

where Y’ is the value of the point which intersects the real axis of the admittance loci of the third layer. Y’ can be found when the runsheet curve passes through the turning point. The sign in Eq. (10) is selected as that causing the value of δ _B to lie in the range 0~π/2 if we always pick the first turning point value as Y’. The corresponding solution for δ_A is too complicated to present here.

The cutting point with error compensation for the central wavelength of the following layers can be calculated by repeating the above procedure. Thus we can terminate the layer more precisely than the turning point method without losing the advantage of error compensation. Although some of these equations might be too complicated to calculate by hand, it can be solved by a computer program instantly.

3.Simulation

The advantage of the proposed selecting sensitive monitor wavelength (SSMW) method as compared with the turning point method and the overshoot method is demonstrated below. First, let’s look at a simple quarter-wave reflector S/(HL)^4 H/A with reference wavelength 1550nm. The refractive indices of high and low materials and substrate were 2.3, 1.46 and 1.5, respectively. Let the termination of each layer have a 1% standard deviation error in transmittance. We get the optical performance resulting from each monitoring method as shown in Fig. 8.

 

Fig. 8. Spectrum of reflectors monitored by different methods with a 1 % standard deviation error.

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It can be seen that the performance achieved using the SSMW method is much better than that obtained using the turning point method. The full width at half maximum of the reflection band and the maximum reflectance produced by the SSMW method is 17 nm wider and 0.82% higher, respectively, than that produced by the turning point method and almost behaves the same as the theoretical design.

Secondly, let’s use the same materials to simulate single-cavity, two-cavity and three-cavity narrow-band pass filters. The selected sensitive monitor wavelength for each layer of the single cavity is given in the last column of Table 1, and the resulting optical thickness of each layer is listed in the third column in which the optical thickness is in terms of a central wavelength 1550nm. The optical thickness of each layer monitored by turning point is also list in the second column for comparing. The comparison of the optical performance of the single-cavity, two-cavity and three-cavity narrow-band pass filters monitored by the SSMW method and the turning point method are shown in Fig. 9, Fig. 10 and Fig. 11, respectively. During the simulation, a 0.3 % standard deviation transmittance error in the cutting points of layers was assumed. It can be seen obviously that the filters monitored by the SSMW method not only achieve better performance, but also keep the transmittance peak at the central wavelength after the calculation of the error compensation for it. On the other hand, the filters monitored by the turning point method have wider half-maximum bandwidth and each layer thickness is farther away from the quarter wave due to the low sensitivity at the cutting points. The wider half-maximum bandwidth of the single-cavity narrow-band pass filter monitored by the turning point method was due to the low reflectance as indicated in Fig. 8. The shift to shorter wavelength of the center of the single-cavity narrow-band pass filter monitored by the turning point method was caused by the last layer being less than quarter wave, or 0.2063 waves. The fallen shoulder of the 2-cavity filter and shifted peak of the 3-cavity filter as monitored by the turning point method were due to the peaks of each cavity not being at the same wavelength as shown in Fig. 12. Therefore it is hard to make a good filter with more than two cavities by using the turning point method, unless some assistant tools such as a quartz monitor or time counting for each deposited layer were used also.

4.Conclusion

By a series of mathematical analyses, the selecting sensitive monitor wavelength (SSMW) method was introduced. Simulations demonstrate that the performance of filters deposited using the SSMW method is much superior to those using the turning point monitoring method. This new procedure combines the advantage of the sensitive monitor wavelength with the error compensation for the reference wavelength to reach an excellent result. It is believed that the SSMW method is also very useful to produce various other types of optical thin film filters.

 

Fig. 9. Comparison of 1-cavity narrow-band pass filter performance between two monitoring methods.

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Fig. 10. Comparison of 2-cavity narrow-band pass filter performance between two monitoring methods.

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Fig. 11. Comparison of 3-cavity narrow-band pass filter performance between two monitoring methods.

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Fig. 12. The first, second and third cavities of the filters monitored by turning point method shown in Fig. 10 and Fig. 11.

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Table 1. Result optical thickness of the filters monitored by two methods under a 0.3% standard deviation of transmittance error and the monitor wavelength used by the SSMW method

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