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Multilayer thin film thickness measurement using sensitivity separation method
Optics Communications 283 (2010) 3989–3993
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Multilayer thin film thickness measurement using sensitivity separation method Xing-zhi Gong, Liang Cheng, Fei-hong Yu ⁎ State Key Laboratory of Modern Optical Instrumentation, Optical Engineering Department, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 19 January 2010 Received in revised form 25 May 2010 Accepted 26 May 2010 Keywords: Thin films Thickness measurement Optical properties Sensitivity separation
a b s t r a c t This paper, for the first time, proposed a new sensitivity separation (SS) method for measuring thicknesses of multilayer thin-film stack with high efficiency and accuracy. Through the analysis of the relationship between the film parameters and the mean misfit error (MSE), a parameter called sensitivity is defined. With this parameter, an estimated rational thickness is assigned to a layer with lower sensitivity first, and then the layer thickness with high sensitivity is further obtained by optimization techniques. This method will greatly reduce the searching range and increase the iterating efficiency. It is a pretreatment method and it can be used with other optimization methods. Both theory and simulation results are provided in detail. The uncertainty problems are discussed and examples are given to verify the effectiveness of this method. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Optical films are widely used for light control and manipulation in various scientific and practical applications, such as optical lens manufacture, optoelectronics, sensors and so on. To achieve the desired performance, optical films need to be coated on a substrate with specific thicknesses and materials. The control and determination of optical constants of a thin film with high accuracy are key issues for optical coating technology. Many optical methods offer nondestructive measurement for optical constants rather than mechanical methods such as wavelength scanning [1–3], ellipsometry [4], and waveguide coupling [5]. Among them, wavelength scanning is fast and accurate using transmission or reflection spectrum. The determination of the film constants from a single layer has been well established with many wavelength scanning methods such as direct calculation, fringe counting [6] and envelops [7,8]. The analysis of multilayer films with optical method is still a challenge. Because of the complex expressions and the increased parameters, most of the algorithms have certain restrictions for the film structure. The FFT analysis [9] of reflective spectrum can be used to determine the optical thicknesses of multilayer thin films, but this method can only be applied to thick films, and its accuracy decreases when the thickness of different layers are close to each other. Spectrum fitting is one of the most important wavelength scanning methods, where global or local optimization algorithms [10] are employed in reflective spectrum fitting. The global minimum of the merit function is found by varying the optical parameters of the films. Unlike fringe counting and envelope methods which can only be used
⁎ Corresponding author. E-mail address: [email protected] (F. Yu). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.05.054
for single layer film, reflective spectrum fitting method can be used for both single layer and multilayer structures. In this method the thicknesses of individual layers are most interesting, while other optical properties such as refractive indices can be analyzed in other ways. Therefore, in this study the optical parameters of materials are assumed to be accurate enough so as not to bring extra errors in our analysis. Simulated Annealing (SA) algorithm is a powerful global optimization algorithm, which is widely used in multilayer film design and refining [11]. Typically the iteration starts from a set of random solution points in the search range, and then the search step are updated according to the change of the temperature until it meets the acceptance condition. Theoretically, if the decreasing speed of the temperature is slow enough, the global optimum can be found when the temperature is approaching zero. In the application of thin film thickness measurements, as the number of layers increases, the number of parameters increases and the search range becomes significantly large. In this case, the variables can be compensated with each other, which generate more local minimums instead of the global minimum. Consequently, the global search of the optimum becomes more difficult. In the fitting procedure, if the total number of iterations is fixed, as the number of parameters increase, it is more difficult to locate the correct curve fitting and the iteration is easier to be trapped into the local optimum. Take the SA algorithm as an example: if the temperature is very low and the iteration is trapped in a local optimum, the SA procedure needs to be restarted to search for the correct solution. In this paper, we proposed a novel spectrum fitting method using SS (sensitivity separation), which will significantly reduce the searching range and the number of iteration parameters. In this way the correct fitting probability will be increased and optimization
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X. Gong et al. / Optics Communications 283 (2010) 3989–3993
becomes much more efficient. SS is a pretreatment method, so in the simulation we will use SS method combined with SA to show how this method works. 2. The sensitivity analysis of thin film parameters The reflective spectrum can be calculated by conventional complex matrix after the derivation of the thicknesses and refractive indices. The merit function is formulated in Eq. (1): MF =
1 2 ∑½ðRmeasure ðλÞ−RðλÞÞ=Rmeasure ðλÞ λn −λ1 λ
1 2
ð1Þ
Rmeasure is the film reflectance measured with the spectrometer, and R is the reflectance calculated with the refractive indices and the iterated thicknesses. The degree of the parameters affecting the merit function is denoted as sensitivity. The sensitivity can be represented by the derivative of the merit function to the thin film thickness, as shown in Eq. (2). Si =
dMF dti
ð2Þ
where MF represents the merit function and ti represents the thickness of the layer i. Because of the complex of multilayer film's structure, it is complicate to formulate the sensitivity of film parameters with a specific analytical solution. Markku Yliammi etc [12] analyzed the sensitivity of film parameters utilizing a numerical method and the uncertainty of this method when the value of merit function is very low. The misfit is defined in Eq. (3): N
E = E0 + ∑ ðaj Δtj + bj Δnj Þ j=1
ð3Þ
E is the average misfit error, and aj, bj are the sensitivity coefficients, Δtj is the absolute error of the layer j thickness, Δnj is the absolute error of the layer j refractive index, j is the index of the multilayer thin film, N is the total layer number, and E0 represents the residual average misfit due to system error. Thus, the sensitivity of thickness corresponding to a specific layer in a multilayer thin film can be obtained from Eqs. (2) and (3). E0 and Δnj are not considered in practice in Eq. (2). Si = aj + a1
dt1 dt + a2 2 + :::::: dti dti
ð4Þ
When each layer's refractive index is known, the sensitivity of specific layer can be obtained with the following methods. Assign one layer thickness to a thickness approximately to real thickness and vary the other layer thicknesses to get the minimum average misfit. This process can be accomplished by local optimization algorithms such as CG (Conjugate Gradient). This minimum error divided by thickness variation gives the sensitivity of this layer. We assume that each layer has a similar contribution pffiffiffiffi to the overall misfit, thus each layer misfit should be divided by N, where N is the layer number. 3. Compromise calculation of sensitivity Sensitivity is a very complex parameter. It depends on the thicknesses and refractive indices of all the layers. In the measurements, if the thicknesses of all layers are unknown parameters, the exact sensitivity can't be obtained until the whole fitting procedure is completed. In this case, we can start the sensitivity calculation with an
estimated thickness rather than the exact thickness. Using a threelayer thin film stack as an example, the known layer parameters are listed in Table 1. The relationship between the sensitivities and parameters (including both thicknesses and refractive indices) are illustrated in Figs. 1 and 2. Fig. 1(a) shows Layer1's thickness varies from 0 to 250 nm while Layer2's and Layer3's thicknesses are assigned to real thicknesses. Fig. 1(b) shows Layer2's thickness varies from 0 to 250 nm while Layer1's and Layer3's thicknesses are assigned to real thicknesses. Fig. 1(c) shows Layer3's thickness varies from 0 to 250 nm while Layer1's and Layer2's thicknesses are assigned to real thicknesses. Fig. 2 is similar to Fig. 1, except the variable is refractive index. From Figs. 1 and 2, it can be found that the sensitivity is insensitive to the thickness, but very sensitive to refractive index. Thus we can use the estimated thicknesses to calculate the sensitivities to replace the sensitivities calculated with the real thicknesses. 4. Sensitivity separation Before the theoretical analysis of the sensitivity separation method, we first calculate the film sensitivity in talbe.1 using the algorithm given in Section 2. Here we list the film parameters again in Table 2, the same as those in Table 1 but add a new column called Layer Sensitivity for easy identification and comparison. The 2-D distributions of the film merit function are shown in Fig. 3 (a) and (b). In Fig. 3(a), the thickness of Layer2 is assigned to real thickness 113 nm, and Layer1 and Layer3 thicknesses are varied from 0–200 nm. The abscissa is Layer3 thickness, and ordinate is Layer1 thickness. Different colors represent different merit function values. The lowest merit function value is in dark blue band. Fig. 3(b) is almost the same as Fig. 3(a), except that Layer2 thickness is 500 nm, not the real thickness. From Fig. 3, we can find that although Layer2 thickness is different, but the lowest merit function value bands are almost in the same region. Because the merit function is small enough, we think this result is global. In Fig. 3(a), the lowest merit function points of Layer1 and Layer3 thicknesses are 52 and 91 nm respectively, and in Fig. 3(b) they are 56 and 85 nm respectively. As a result, Layer2 thickness is insensitive to merit function. If the above information is obtained in advance, the search range and the number of iteration parameters can be remarkably reduced. The fitting will be more efficient. Before calculating the thicknesses of a multilayer structure, the thickness sensitivity of each layer should be calculated first with the estimated thicknesses. Layers with low sensitivities are assigned to certain values such as the average value of the lower and upper limits while the layers with high sensitivities are calculated by global optimization algorithm first. Then the calculated results of sensitive layer thickness can be considered around the real thicknesses. At last all the thicknesses are considered as variables in the fitting procedure. As a result, thicknesses with low sensitivities are searched in a wide range while those with high sensitivities are searched in a very narrow range. This method is called sensitivity separation which can narrow the search range effectively. Thus the sensitivity separation method is summarized as follows: 1. Determine the upper and lower limits of all layers thicknesses.
Table 1 The parameters of a three-layer optical thin film on K9 glass substrate.
Layer1 Layer2 Layer3
Material
Refractive index n
Thickness (nm)
MgF2 HfO2 Al2O3
1.37 1.82 1.62
52 113 91
X. Gong et al. / Optics Communications 283 (2010) 3989–3993
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Fig. 1. The relationship between each layer sensitivity and thickness (a) Layer1 thickness varied (b) Layer2 thickness varied (c) Layer3 thickness varied.
2. Calculate the sensitivity of each layer with the predicted thicknesses using the method mentioned in Section 2. 3. Assign low sensitive layers with certain values such as the average value of the lower and upper limits of this layer. The maximum misfit error caused by insensitive thicknesses can be calculated:
Table 2 The sensitivity of film in Table 1.
Ni
Einsensitive
∑ ðLi Δtmaxi Þ = i = 1 pffiffiffiffiffi Ni
Fig. 2. The relationship between each layer sensitivity and refractive index (a) Layer1 refractive index varied (b) Layer2 refractive index varied (c) Layer3 refractive index varied.
ð5Þ
where Einsensitive is the maximum misfit error which is caused by the insensitive thickness, Li is the sensitivity of insensitive layer i,
Layer1 Layer2 Layer3
Material
Refractive index n
Thickness (nm)
Layer sensitivity(1/nm)
MgF2 HfO2 Al2O3
1.37 1.82 1.62
52 113 91
3.83 × 10− 4 nm− 1 1.09 × 10− 4 nm− 1 1.91 × 10− 4 nm− 1
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X. Gong et al. / Optics Communications 283 (2010) 3989–3993
Fig. 3. The merit function distribution (a) Layer2 thickness is the real thickness 113 nm (b) Layer2 thickness is set to arbitrary value 500 nm in certain range.
Δtmaxi is the maximum error between estimated thickness and real thickness, Ni is the number of insensitive layers. 4. Use optimization algorithm (such as SA) to find the lowest misfit error. In this procedure the misfit of this optimization is Esensitive which is considered caused only by the misfit of sensitive thicknesses. 5. According to the misfit errors caused by sensitive and insensitive thicknesses, each sensitive layer thickness range can be calculated by Eq. (6): Δti =
Esensitive + Einsensitive pffiffiffiffiffiffiffiffiffiffiffiffiffi Si × N−Ni
Let's take the parameters in Table 2 as an example. The estimated Layer2 thickness is about 200 nm. The maximum error of Layer2 thickness can be estimated as ±100nm. The maximum misfit error of Layer2 is 0.0109 (Einsensitive). Then Layer2's thickness is assigned to 200 nm, and the SA algorithm is adopted to find the minimum average misfit. The calculated thicknesses of Layer1 and Layer2 are 60.74 nm and 91.59 nm. The average misfit is 0.0131(Esensitive). This maximum
ð6Þ
where Δti is the maximum range between calculated thickness and real thickness of sensitive layer i, Si is the sensitivity of layer i, N is the number of layers. Table 3 A three-layer structure thin film.
Layer1 Layer2 Layer3
Material
Refractive index
Thickness(nm)
Layer sensitivity(1/nm)
Al2O3 ZnS Al2O3
1.62 2.20 1.62
224 266 40
1.00 × 10− 3 1.23 × 10− 3 3.24 × 10− 4
Table 4 A four layers structure thin film.
Layer1 Layer2 Layer3 Layer4
Material
Refractive index
Thickness(nm)
Layer sensitivity(1/nm)
ZnS CaF2 ZnS Al2O3
2.20 1.4 2.20 1.62
90 236 101 143
1.21 × 10− 3 1.38 × 10− 3 1.01 × 10− 3 2.86 × 10− 4
Table 5 A five layers structure thin film.
Layer1 Layer2 Layer3 Layer4 Layer5
Material
Refractive index
Thickness(nm)
Layer sensitivity(1/nm)
ZnS CaF2 ZnS CaF2 ZnS
2.20 1.4 2.20 1.4 2.20
32 134 23 142 23
9.52 × 10− 4 4.51 × 10− 4 1.07 × 10− 3 3.86 × 10− 4 1.49 × 10− 3
Fig. 4. The results are compared between SA with SS and without SS method (a) Results calculated with the film data in Table 3 (b) Results calculated with the film data in Table 4.
X. Gong et al. / Optics Communications 283 (2010) 3989–3993
error is added to the misfit error caused by all the insensitive thicknesses. The final error E is 0.024. We can estimate the thickness ranges of Layer1 and Layer3 from Eq. (6). Layer1's thickness range is 60.74 ± 44 nm and Layer3 thickness range is 91.59 ± 88 nm. 5. Simulations The simulations for the other multilayer structures using SS method are shown below. We assume that the upper limit for all the thicknesses is 500 nm. We first show a three-layer structure film in Table 3. From Table 1 we find that Layer3 has the lowest sensitivity, thus Layer3 thickness is assigned to 250 nm thickness (average of the upper limit 500 nm and lower limit 0 nm). The maximum misfit error after assigning Layer3 thickness to 250 nm is 0.081. Layer1 and Layer2 thicknesses are iterated between 10 nm to 500 nm with SA algorithm. After SA Layer1 and Layer2 thicknesses are 227.5 nm and 226.4 nm, and the misfit is 0.0137. The final error is 0.0947. Then Layer1 thickness range is 227.5 ± 68 nm, and Layer2 thickness range is 226.4 ± 55 nm. A four layer structure film is shown in Table 4. Table 4 shows that Layer4 has the lowest sensitivity. We assign Layer4 thickness to 250 nm. The maximum misfit error after assigning Layer4 thickness to 250 nm is 0.0572. Layer1, Layer2 and Layer3 thicknesses are iterated between 10 nm to 500 nm with SA algorithm. After SA the first three layers thicknesses are 89.68 nm, 261.61 nm and 99.78 nm, and the misfit is 0.0221. The final error is 0.0793. Layer1 thickness range is 89.68 ± 38 nm, Layer2 thickness range is 261.61 ± 33 nm, and Layer3 thickness range is 99.78 ± 45 nm. A five layer structure film is shown in Table 5. Layer2 and Layer4 thicknesses are assigned to 250 nm first. The maximum low sensitive misfit error is 0.1480. The other layers' thicknesses are iterated by SA between 0 nm to 500 nm. After SA Layer1, Layer3 and Layer5 thicknesses are 19.83 nm, 73.45 nm and 24.50 nm, and misfit error is 0.0262. The final error is 0.1742. Layer1 thickness range is 19.83 ± 106 nm, Layer3 thickness range is 73.45 ± 94 nm, and Layer3 thickness range is 24.5 ± 68 nm. From above simulations, we can find the searching range is narrowed effectively. To show how the fitting efficiency is improved by SS, we use SA to iterate the film thicknesses in Tables 3 and 4. In the first group, the SS is not used and each layer lower and upper limit is 10 nm–500 nm. SA is used to iterate the thicknesses. In the second group, we first use SS method to narrow the lower and upper limit of sensitive layers. Then we use the narrowed limit to iterate the
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thicknesses with SA. We hope there is statistical significance in the results, so the SA is operated 500 times in each group. In Fig. 4, the abscissa is iteration times and the ordinate is the misfit error. Fig. 4(a) shows the results calculated with the film data in Table 3 and Fig. 4(b) shows the results calculated with the film data in Table 4. In Fig. 4(a), the average misfit error of SA without SS is 0.164% and average misfit with SS is 0.0056%. In Fig. 4(b), the average misfit error of SA without SS is 9.739% and average misfit with SS is 0.304%. The SS method can improve the fitting efficiency effectively. 6. Conclusion A new numerical method for determining the thickness in multilayer structure optical film stack by predetermining thickness sensitivity is proposed. The principle of sensitivity is presented, and the sensitivity proves to be very sensitive to refractive index. The estimated sensitivity can be calculated without prior knowledge of the exact thickness of each layer. The simulations demonstrate that the sensitivity separation method can significantly narrow down the searching range of the variables. Theoretically this method is general and not limited to total layer number or any specific multilayer structures. It can be used in multilayer film stack with high contrast sensitivities. The low contrast sensitivities films also have unique characteristics in merit function distribution related to thicknesses and refractive indices. Details on the relationship and the new fitting method applied for low sensitivity contrast film stack will be discussed in our future work. References [1] Y.-M. Hwang, S.-W. Yoon, J.-H. Kim, S. Kim, H.-J. Pahk, J. Opt. Lasers Eng. 46 (2008) 179 (2008). [2] K. Hibino, B.F. Oreb, P.S. Fairman, J. Burke, J. Appl. Opt. 43 (2004) 1241. [3] Y. Laaziza, A. Bennounaa, N. Chahbouna, A. Outzourhita, E.L. Amezianea, J. Thin Solid Films 372 (2000) 149. [4] C.-Y. Han, Z.-Y. Lee, Y.-F. Chao, J. Appl. Opt. 48 (2009) 3139. [5] R. Ulrich, R. Torge, J. Appl. Opt. 12 (1973) 2091. [6] R.P. Shukla, D.V. Udupa, N.C. Das, M.V. Mantravadi, J. Opt. Laser Technol. 38 (2006) 552. [7] J. Luňáček, P. Hlubina, M. Luňáčková, J. Appl. Opt. 48 (2009) 985. [8] S. Humphrey, J. Appl. Opt. 46 (2007) 4660. [9] O. Köysal, D. Önal, S. Özder, F. Necati Ecevit, J. Opt. Commun. 205 (2002) 1. [10] X. Liu, P.-f. Gu, J.-f. Tang, J. Appl. Opt. 35 (1996) 5035. [11] C.P. Chang, Y.H. Lee, J. Opt. Lett. 15 (1990) 595. [12] M. Ylilammi, T. Ranta-aho, J. Thin Solid Films 232 (1993) 56.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Multilayer thin film thickness measurement using sensitivity separation method Xing-zhi Gong, Liang Cheng, Fei-hong Yu ⁎ State Key Laboratory of Modern Optical Instrumentation, Optical Engineering Department, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 19 January 2010 Received in revised form 25 May 2010 Accepted 26 May 2010 Keywords: Thin films Thickness measurement Optical properties Sensitivity separation
a b s t r a c t This paper, for the first time, proposed a new sensitivity separation (SS) method for measuring thicknesses of multilayer thin-film stack with high efficiency and accuracy. Through the analysis of the relationship between the film parameters and the mean misfit error (MSE), a parameter called sensitivity is defined. With this parameter, an estimated rational thickness is assigned to a layer with lower sensitivity first, and then the layer thickness with high sensitivity is further obtained by optimization techniques. This method will greatly reduce the searching range and increase the iterating efficiency. It is a pretreatment method and it can be used with other optimization methods. Both theory and simulation results are provided in detail. The uncertainty problems are discussed and examples are given to verify the effectiveness of this method. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Optical films are widely used for light control and manipulation in various scientific and practical applications, such as optical lens manufacture, optoelectronics, sensors and so on. To achieve the desired performance, optical films need to be coated on a substrate with specific thicknesses and materials. The control and determination of optical constants of a thin film with high accuracy are key issues for optical coating technology. Many optical methods offer nondestructive measurement for optical constants rather than mechanical methods such as wavelength scanning [1–3], ellipsometry [4], and waveguide coupling [5]. Among them, wavelength scanning is fast and accurate using transmission or reflection spectrum. The determination of the film constants from a single layer has been well established with many wavelength scanning methods such as direct calculation, fringe counting [6] and envelops [7,8]. The analysis of multilayer films with optical method is still a challenge. Because of the complex expressions and the increased parameters, most of the algorithms have certain restrictions for the film structure. The FFT analysis [9] of reflective spectrum can be used to determine the optical thicknesses of multilayer thin films, but this method can only be applied to thick films, and its accuracy decreases when the thickness of different layers are close to each other. Spectrum fitting is one of the most important wavelength scanning methods, where global or local optimization algorithms [10] are employed in reflective spectrum fitting. The global minimum of the merit function is found by varying the optical parameters of the films. Unlike fringe counting and envelope methods which can only be used
⁎ Corresponding author. E-mail address: [email protected] (F. Yu). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.05.054
for single layer film, reflective spectrum fitting method can be used for both single layer and multilayer structures. In this method the thicknesses of individual layers are most interesting, while other optical properties such as refractive indices can be analyzed in other ways. Therefore, in this study the optical parameters of materials are assumed to be accurate enough so as not to bring extra errors in our analysis. Simulated Annealing (SA) algorithm is a powerful global optimization algorithm, which is widely used in multilayer film design and refining [11]. Typically the iteration starts from a set of random solution points in the search range, and then the search step are updated according to the change of the temperature until it meets the acceptance condition. Theoretically, if the decreasing speed of the temperature is slow enough, the global optimum can be found when the temperature is approaching zero. In the application of thin film thickness measurements, as the number of layers increases, the number of parameters increases and the search range becomes significantly large. In this case, the variables can be compensated with each other, which generate more local minimums instead of the global minimum. Consequently, the global search of the optimum becomes more difficult. In the fitting procedure, if the total number of iterations is fixed, as the number of parameters increase, it is more difficult to locate the correct curve fitting and the iteration is easier to be trapped into the local optimum. Take the SA algorithm as an example: if the temperature is very low and the iteration is trapped in a local optimum, the SA procedure needs to be restarted to search for the correct solution. In this paper, we proposed a novel spectrum fitting method using SS (sensitivity separation), which will significantly reduce the searching range and the number of iteration parameters. In this way the correct fitting probability will be increased and optimization
3990
X. Gong et al. / Optics Communications 283 (2010) 3989–3993
becomes much more efficient. SS is a pretreatment method, so in the simulation we will use SS method combined with SA to show how this method works. 2. The sensitivity analysis of thin film parameters The reflective spectrum can be calculated by conventional complex matrix after the derivation of the thicknesses and refractive indices. The merit function is formulated in Eq. (1): MF =
1 2 ∑½ðRmeasure ðλÞ−RðλÞÞ=Rmeasure ðλÞ λn −λ1 λ
1 2
ð1Þ
Rmeasure is the film reflectance measured with the spectrometer, and R is the reflectance calculated with the refractive indices and the iterated thicknesses. The degree of the parameters affecting the merit function is denoted as sensitivity. The sensitivity can be represented by the derivative of the merit function to the thin film thickness, as shown in Eq. (2). Si =
dMF dti
ð2Þ
where MF represents the merit function and ti represents the thickness of the layer i. Because of the complex of multilayer film's structure, it is complicate to formulate the sensitivity of film parameters with a specific analytical solution. Markku Yliammi etc [12] analyzed the sensitivity of film parameters utilizing a numerical method and the uncertainty of this method when the value of merit function is very low. The misfit is defined in Eq. (3): N
E = E0 + ∑ ðaj Δtj + bj Δnj Þ j=1
ð3Þ
E is the average misfit error, and aj, bj are the sensitivity coefficients, Δtj is the absolute error of the layer j thickness, Δnj is the absolute error of the layer j refractive index, j is the index of the multilayer thin film, N is the total layer number, and E0 represents the residual average misfit due to system error. Thus, the sensitivity of thickness corresponding to a specific layer in a multilayer thin film can be obtained from Eqs. (2) and (3). E0 and Δnj are not considered in practice in Eq. (2). Si = aj + a1
dt1 dt + a2 2 + :::::: dti dti
ð4Þ
When each layer's refractive index is known, the sensitivity of specific layer can be obtained with the following methods. Assign one layer thickness to a thickness approximately to real thickness and vary the other layer thicknesses to get the minimum average misfit. This process can be accomplished by local optimization algorithms such as CG (Conjugate Gradient). This minimum error divided by thickness variation gives the sensitivity of this layer. We assume that each layer has a similar contribution pffiffiffiffi to the overall misfit, thus each layer misfit should be divided by N, where N is the layer number. 3. Compromise calculation of sensitivity Sensitivity is a very complex parameter. It depends on the thicknesses and refractive indices of all the layers. In the measurements, if the thicknesses of all layers are unknown parameters, the exact sensitivity can't be obtained until the whole fitting procedure is completed. In this case, we can start the sensitivity calculation with an
estimated thickness rather than the exact thickness. Using a threelayer thin film stack as an example, the known layer parameters are listed in Table 1. The relationship between the sensitivities and parameters (including both thicknesses and refractive indices) are illustrated in Figs. 1 and 2. Fig. 1(a) shows Layer1's thickness varies from 0 to 250 nm while Layer2's and Layer3's thicknesses are assigned to real thicknesses. Fig. 1(b) shows Layer2's thickness varies from 0 to 250 nm while Layer1's and Layer3's thicknesses are assigned to real thicknesses. Fig. 1(c) shows Layer3's thickness varies from 0 to 250 nm while Layer1's and Layer2's thicknesses are assigned to real thicknesses. Fig. 2 is similar to Fig. 1, except the variable is refractive index. From Figs. 1 and 2, it can be found that the sensitivity is insensitive to the thickness, but very sensitive to refractive index. Thus we can use the estimated thicknesses to calculate the sensitivities to replace the sensitivities calculated with the real thicknesses. 4. Sensitivity separation Before the theoretical analysis of the sensitivity separation method, we first calculate the film sensitivity in talbe.1 using the algorithm given in Section 2. Here we list the film parameters again in Table 2, the same as those in Table 1 but add a new column called Layer Sensitivity for easy identification and comparison. The 2-D distributions of the film merit function are shown in Fig. 3 (a) and (b). In Fig. 3(a), the thickness of Layer2 is assigned to real thickness 113 nm, and Layer1 and Layer3 thicknesses are varied from 0–200 nm. The abscissa is Layer3 thickness, and ordinate is Layer1 thickness. Different colors represent different merit function values. The lowest merit function value is in dark blue band. Fig. 3(b) is almost the same as Fig. 3(a), except that Layer2 thickness is 500 nm, not the real thickness. From Fig. 3, we can find that although Layer2 thickness is different, but the lowest merit function value bands are almost in the same region. Because the merit function is small enough, we think this result is global. In Fig. 3(a), the lowest merit function points of Layer1 and Layer3 thicknesses are 52 and 91 nm respectively, and in Fig. 3(b) they are 56 and 85 nm respectively. As a result, Layer2 thickness is insensitive to merit function. If the above information is obtained in advance, the search range and the number of iteration parameters can be remarkably reduced. The fitting will be more efficient. Before calculating the thicknesses of a multilayer structure, the thickness sensitivity of each layer should be calculated first with the estimated thicknesses. Layers with low sensitivities are assigned to certain values such as the average value of the lower and upper limits while the layers with high sensitivities are calculated by global optimization algorithm first. Then the calculated results of sensitive layer thickness can be considered around the real thicknesses. At last all the thicknesses are considered as variables in the fitting procedure. As a result, thicknesses with low sensitivities are searched in a wide range while those with high sensitivities are searched in a very narrow range. This method is called sensitivity separation which can narrow the search range effectively. Thus the sensitivity separation method is summarized as follows: 1. Determine the upper and lower limits of all layers thicknesses.
Table 1 The parameters of a three-layer optical thin film on K9 glass substrate.
Layer1 Layer2 Layer3
Material
Refractive index n
Thickness (nm)
MgF2 HfO2 Al2O3
1.37 1.82 1.62
52 113 91
X. Gong et al. / Optics Communications 283 (2010) 3989–3993
3991
Fig. 1. The relationship between each layer sensitivity and thickness (a) Layer1 thickness varied (b) Layer2 thickness varied (c) Layer3 thickness varied.
2. Calculate the sensitivity of each layer with the predicted thicknesses using the method mentioned in Section 2. 3. Assign low sensitive layers with certain values such as the average value of the lower and upper limits of this layer. The maximum misfit error caused by insensitive thicknesses can be calculated:
Table 2 The sensitivity of film in Table 1.
Ni
Einsensitive
∑ ðLi Δtmaxi Þ = i = 1 pffiffiffiffiffi Ni
Fig. 2. The relationship between each layer sensitivity and refractive index (a) Layer1 refractive index varied (b) Layer2 refractive index varied (c) Layer3 refractive index varied.
ð5Þ
where Einsensitive is the maximum misfit error which is caused by the insensitive thickness, Li is the sensitivity of insensitive layer i,
Layer1 Layer2 Layer3
Material
Refractive index n
Thickness (nm)
Layer sensitivity(1/nm)
MgF2 HfO2 Al2O3
1.37 1.82 1.62
52 113 91
3.83 × 10− 4 nm− 1 1.09 × 10− 4 nm− 1 1.91 × 10− 4 nm− 1
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X. Gong et al. / Optics Communications 283 (2010) 3989–3993
Fig. 3. The merit function distribution (a) Layer2 thickness is the real thickness 113 nm (b) Layer2 thickness is set to arbitrary value 500 nm in certain range.
Δtmaxi is the maximum error between estimated thickness and real thickness, Ni is the number of insensitive layers. 4. Use optimization algorithm (such as SA) to find the lowest misfit error. In this procedure the misfit of this optimization is Esensitive which is considered caused only by the misfit of sensitive thicknesses. 5. According to the misfit errors caused by sensitive and insensitive thicknesses, each sensitive layer thickness range can be calculated by Eq. (6): Δti =
Esensitive + Einsensitive pffiffiffiffiffiffiffiffiffiffiffiffiffi Si × N−Ni
Let's take the parameters in Table 2 as an example. The estimated Layer2 thickness is about 200 nm. The maximum error of Layer2 thickness can be estimated as ±100nm. The maximum misfit error of Layer2 is 0.0109 (Einsensitive). Then Layer2's thickness is assigned to 200 nm, and the SA algorithm is adopted to find the minimum average misfit. The calculated thicknesses of Layer1 and Layer2 are 60.74 nm and 91.59 nm. The average misfit is 0.0131(Esensitive). This maximum
ð6Þ
where Δti is the maximum range between calculated thickness and real thickness of sensitive layer i, Si is the sensitivity of layer i, N is the number of layers. Table 3 A three-layer structure thin film.
Layer1 Layer2 Layer3
Material
Refractive index
Thickness(nm)
Layer sensitivity(1/nm)
Al2O3 ZnS Al2O3
1.62 2.20 1.62
224 266 40
1.00 × 10− 3 1.23 × 10− 3 3.24 × 10− 4
Table 4 A four layers structure thin film.
Layer1 Layer2 Layer3 Layer4
Material
Refractive index
Thickness(nm)
Layer sensitivity(1/nm)
ZnS CaF2 ZnS Al2O3
2.20 1.4 2.20 1.62
90 236 101 143
1.21 × 10− 3 1.38 × 10− 3 1.01 × 10− 3 2.86 × 10− 4
Table 5 A five layers structure thin film.
Layer1 Layer2 Layer3 Layer4 Layer5
Material
Refractive index
Thickness(nm)
Layer sensitivity(1/nm)
ZnS CaF2 ZnS CaF2 ZnS
2.20 1.4 2.20 1.4 2.20
32 134 23 142 23
9.52 × 10− 4 4.51 × 10− 4 1.07 × 10− 3 3.86 × 10− 4 1.49 × 10− 3
Fig. 4. The results are compared between SA with SS and without SS method (a) Results calculated with the film data in Table 3 (b) Results calculated with the film data in Table 4.
X. Gong et al. / Optics Communications 283 (2010) 3989–3993
error is added to the misfit error caused by all the insensitive thicknesses. The final error E is 0.024. We can estimate the thickness ranges of Layer1 and Layer3 from Eq. (6). Layer1's thickness range is 60.74 ± 44 nm and Layer3 thickness range is 91.59 ± 88 nm. 5. Simulations The simulations for the other multilayer structures using SS method are shown below. We assume that the upper limit for all the thicknesses is 500 nm. We first show a three-layer structure film in Table 3. From Table 1 we find that Layer3 has the lowest sensitivity, thus Layer3 thickness is assigned to 250 nm thickness (average of the upper limit 500 nm and lower limit 0 nm). The maximum misfit error after assigning Layer3 thickness to 250 nm is 0.081. Layer1 and Layer2 thicknesses are iterated between 10 nm to 500 nm with SA algorithm. After SA Layer1 and Layer2 thicknesses are 227.5 nm and 226.4 nm, and the misfit is 0.0137. The final error is 0.0947. Then Layer1 thickness range is 227.5 ± 68 nm, and Layer2 thickness range is 226.4 ± 55 nm. A four layer structure film is shown in Table 4. Table 4 shows that Layer4 has the lowest sensitivity. We assign Layer4 thickness to 250 nm. The maximum misfit error after assigning Layer4 thickness to 250 nm is 0.0572. Layer1, Layer2 and Layer3 thicknesses are iterated between 10 nm to 500 nm with SA algorithm. After SA the first three layers thicknesses are 89.68 nm, 261.61 nm and 99.78 nm, and the misfit is 0.0221. The final error is 0.0793. Layer1 thickness range is 89.68 ± 38 nm, Layer2 thickness range is 261.61 ± 33 nm, and Layer3 thickness range is 99.78 ± 45 nm. A five layer structure film is shown in Table 5. Layer2 and Layer4 thicknesses are assigned to 250 nm first. The maximum low sensitive misfit error is 0.1480. The other layers' thicknesses are iterated by SA between 0 nm to 500 nm. After SA Layer1, Layer3 and Layer5 thicknesses are 19.83 nm, 73.45 nm and 24.50 nm, and misfit error is 0.0262. The final error is 0.1742. Layer1 thickness range is 19.83 ± 106 nm, Layer3 thickness range is 73.45 ± 94 nm, and Layer3 thickness range is 24.5 ± 68 nm. From above simulations, we can find the searching range is narrowed effectively. To show how the fitting efficiency is improved by SS, we use SA to iterate the film thicknesses in Tables 3 and 4. In the first group, the SS is not used and each layer lower and upper limit is 10 nm–500 nm. SA is used to iterate the thicknesses. In the second group, we first use SS method to narrow the lower and upper limit of sensitive layers. Then we use the narrowed limit to iterate the
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thicknesses with SA. We hope there is statistical significance in the results, so the SA is operated 500 times in each group. In Fig. 4, the abscissa is iteration times and the ordinate is the misfit error. Fig. 4(a) shows the results calculated with the film data in Table 3 and Fig. 4(b) shows the results calculated with the film data in Table 4. In Fig. 4(a), the average misfit error of SA without SS is 0.164% and average misfit with SS is 0.0056%. In Fig. 4(b), the average misfit error of SA without SS is 9.739% and average misfit with SS is 0.304%. The SS method can improve the fitting efficiency effectively. 6. Conclusion A new numerical method for determining the thickness in multilayer structure optical film stack by predetermining thickness sensitivity is proposed. The principle of sensitivity is presented, and the sensitivity proves to be very sensitive to refractive index. The estimated sensitivity can be calculated without prior knowledge of the exact thickness of each layer. The simulations demonstrate that the sensitivity separation method can significantly narrow down the searching range of the variables. Theoretically this method is general and not limited to total layer number or any specific multilayer structures. It can be used in multilayer film stack with high contrast sensitivities. The low contrast sensitivities films also have unique characteristics in merit function distribution related to thicknesses and refractive indices. Details on the relationship and the new fitting method applied for low sensitivity contrast film stack will be discussed in our future work. References [1] Y.-M. Hwang, S.-W. Yoon, J.-H. Kim, S. Kim, H.-J. Pahk, J. Opt. Lasers Eng. 46 (2008) 179 (2008). [2] K. Hibino, B.F. Oreb, P.S. Fairman, J. Burke, J. Appl. Opt. 43 (2004) 1241. [3] Y. Laaziza, A. Bennounaa, N. Chahbouna, A. Outzourhita, E.L. Amezianea, J. Thin Solid Films 372 (2000) 149. [4] C.-Y. Han, Z.-Y. Lee, Y.-F. Chao, J. Appl. Opt. 48 (2009) 3139. [5] R. Ulrich, R. Torge, J. Appl. Opt. 12 (1973) 2091. [6] R.P. Shukla, D.V. Udupa, N.C. Das, M.V. Mantravadi, J. Opt. Laser Technol. 38 (2006) 552. [7] J. Luňáček, P. Hlubina, M. Luňáčková, J. Appl. Opt. 48 (2009) 985. [8] S. Humphrey, J. Appl. Opt. 46 (2007) 4660. [9] O. Köysal, D. Önal, S. Özder, F. Necati Ecevit, J. Opt. Commun. 205 (2002) 1. [10] X. Liu, P.-f. Gu, J.-f. Tang, J. Appl. Opt. 35 (1996) 5035. [11] C.P. Chang, Y.H. Lee, J. Opt. Lett. 15 (1990) 595. [12] M. Ylilammi, T. Ranta-aho, J. Thin Solid Films 232 (1993) 56.