Abstract

A simple, low coherence, vibration insensitive, polarization Fizeau interferometer is employed in this novel optical monitoring system proposed to extract the temporal phase change of the reflection coefficient of the growing film stacks. This system can directly detect fluctuating reflection coefficient and obtain the corresponding optical admittance of the growing film in real time.

©2007 Optical Society of America

1. Introduction

The optical monitoring method is generally better than other methods to manufacture optical filters [1]. For a growing thin film stack, the refractive index of materials may change with time due to the variation of coating parameters and environment fluctuation in the chamber, so we cannot terminate the current layer at the expected point of the original design. In the conventional optical system, we usually get the normally incident transmittance or reflectance of the growing film stack. The phase extraction is difficult in such monitoring system since the operators cannot decode from the limited information.

The transmittance or reflectance will reach a local extreme value if the current layer grows to one quarter thick of the monitoring wavelength. Turning Point Method is based on such principle to produce a quarter wave stack filter, but it is lacking flexibility and sensitivity at cutting points where we should terminate the deposition for one layer. Over-shot Method has more flexibility to choose cutting points, but the termination point is short of physical meaning, and lack of the error compensation for the previous layers that will induce the shift of central wavelength [2]. Other monitoring methods, such as ellipsometry monitoring [3], use some computing algorithm to obtain optical constant by fitting the measurement, however the measurement includes too many parameters and they are hard to solve analytically.

Interferometer is usually applied on the measurement of surface profile. Recently the developed dynamic interferometers was reported to obtain several phase-shifted interferograms of two orthogonal polarization beams on a detector array at the same time and hence frozen the vibration and air turbulence [4]. A new monitoring system based on a similar interferometer structure is proposed in this article. It is an in-situ system that can instantly get the real time phase and magnitude of the reflection coefficient, as well as the optical admittance at normal incidence. It is an important reference to judge the optical behavior of the whole film stack.

2. Principle

We have employed an interferometer in our system to extract the phase change of the growing film stack. The interferometer, different from the conventional one, is applied to measure the phase instantaneously and that we use it to track one phase signal in real time, since the chamber may vibrate during the coating process. Fizeau interferometer with the advantage of common path is suitable for our case. Kimbrough and Wyant et al [5] proposed a vibration insensitive Fizeau interferometer and Twyman Green interferometer in conjunction with a low coherence source to eliminate the coherent noise due to the spurious reflection. Now let us simplify the system structure by replacing Twyman Green interferometer with a tunable retarder and set it on the coater. There are two possible types of monitoring systems. The basic layout of these two possible monitoring systems are shown in Fig. 1 and Fig. 2.

 figure: Fig. 1.

Fig. 1. Scheme of transmission optical monitoring system.

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 figure: Fig. 2.

Fig. 2. Scheme of reflection optical monitoring system.

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It is a simple interferometer added on the conventional optical monitoring system of a coater. We do not change any originally internal component of chamber, but add some optical elements near the chamber. A spatial gradient narrow-band pass filter can be inserted in front of the light source for changing the monitoring wavelength. We do not employ any quarterwave plate or half-wave plate in order to avoid the restriction of the choosing wavelength. We let the monitoring ray normally incidents into the film stack and substrate. The test surface locates at the side where the thin film grows, and the other side of substrate is the reference surface. Since what we consider is the change coming from the thin film growth instead of the whole surface profile, we only need to focus on one point of the sample just as the traditional coating monitor and do not care too much about the flatness of the reference surface. The tunable retarder was composed of two wedge x-cut uniaxial birefringent crystals. Before fixing the retarder, the optical path difference can be adjusted to match the specific thickness of the substrate by tuning the relative position of two crystals. Let p-polarization be in the direction of the fast axis of the crystal, and s-polarization be parallel to the slow axis of the crystal. We have to make the crystals induce a phase difference between two orthogonally polarized beams to match the optical phase difference induced by the reference surface and the test surface. Therefore, the interference will occur only in between either the s-polarized reference beam and the p-polarized test beam or the p-polarized reference beam and the s-polarized test beams. Other reflections from interfaces will be suppressed because of the short coherence length of light [6]. The optical path difference d that we want to cancel would be equal to double optical thickness of the substrate. Thus we have to find the length L of crystal that makes (ns-nf)L=d, where ns and nf represent the refractive indices of slow axis and fast axis, respectively. The thickness error tolerance would be ± coherence length of light. It is easy to choose a light source whose coherence length is shorter than the optical thickness of the substrate but longer than that of the multilayer film stack. The interference will occur at the polarizers whose transmission axes are at 45 and -45 degrees to the fast axis of the birefringent crystal, respectively. These two polarizers and the other two polarizers whose transmission axes are at 0 and 90 degrees to the fast axis of the birefringent crystal, respectively, were combined together to make a compact polarizer array as shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Layout of the polarizer array

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The multiple reflections of two orthogonal polarized rays in the substrate (Fizeau cavity) for the transmission-type monitoring system are illustrated in Fig. 4. The rays are drawn spatially separate and with a slight slant in the sketch, but are all normal to the surface and collinear in the monitoring system. The numbers within the cavity indicate the number of test surface reflections of each beam has passed before it exited the cavity. The matched paths S0 and P1, and S1 and P2, etc. are drawn in Fig. 4 with the same type of lines (e.g. dashed line for S0 and P1, dotted line for S1 and P2). In Fig. 4 we assumed the transmission coefficient of the film stack (from air to films) is |τat|exp(iδ′at) and the reflection coefficient of the film stack (from substrate to films) is |ρst|exp(st). Only the paired beams that are path matched and drawn in the same type of lines will interfere with each other, since the light has short coherence length. We assumed that the reflection magnitude of the reference surface is equal to ρr, and the optical phase difference between two orthogonal polarizations induced by one pass through the substrate and the retarder are s and Γ, respectively.

 figure: Fig. 4.

Fig. 4. Multiple reflections in the substrate for transmission monitoring system

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For the 0- and 90-degree polarizers, there should be no interference taking place. The intensity thus can be represented as follows.

I0°=Is(1ρr)2τat21ρst2ρr2I90°=Ip(1ρr)2ρat21ρst2ρr2

where the subscript represents the direction of the polarizer that the intensity corresponds to.

Now take a look at the intensity of the beams after passing through the 45-degree polarizer. For the beam pair drawn as dashed lines, the phase difference induced by substrate between the p-polarized beam that has reflected once from the test surface and the s-polarized beam from the reference surface will be within the coherence range as passing through the retarder. Hence, the two beams interfere with each other and produce intensity as:

IS0P1=12(Ipρr2ρst21ρr2τat2+Isρr21ρr2τat2+2IsIp1ρr2τat2ρrρstcosθ)

where θ=δst+2s-Γ. Similarly, for the beams drawn in dotted lines, the interference intensity is as follow:

IS1P2=12ρr2ρst2(Ipρr2ρst21ρr2τat2+Isρr21ρr2τat2+2IsIp1ρr2τat2ρrρstcosθ)
=ρr2ρst2IS0P1

For the following beam pairs, we can also find the intensity produced behind 45-degree polarizer as:

IS2P3=(ρr2ρst2)2IS0P1;IS3P4=(ρr2ρst2)3IS0P1;..........ISNPN=(ρr2ρst2)NIS0P1;

It is a geometric progression. Finally, the output intensity of 45-degree polarizers can be expressed as follows:

I+45°=12(I0°+I90°)+I0°I90°ρstρrcosθ
I45°=12(I0°+I90°)+I0°I90°ρstρrcosθ

We can thereby obtain the phase of the thin films stack’s reflection coefficient:

δst=[arccos(I+45°I45°2I0°I90°ρstρr)]2s+Γ

where we can acquire 2s-Γ before the thin film was deposited. We may add a polarizer in front of the chamber to adjust the ratio of the p-polarization intensity to s-polarization intensity or coat a film on the reference surface to increase the visibility of the output. Here, we do not employ a half wave plate to adjust the intensity ratio. Therefore, we have a flexibility of the choice of monitoring wavelengths.

For the reflection monitoring system, the ray propagation through the substrate is illustrated in Fig. 5. We can conclude the intensity after passing the polarizer array as Eq. (8).

I0°=Isρr2+Is(1ρr)4ρst21ρst2ρr2I90°=Ipρr2+Ip(1ρr)4ρst21ρst2ρr2
I+45°=12(I0°+I90°)+I0°I90°ρrρst(1ρr2+(1ρr)2I0°I90°)cosθ
I45°=12(I0°+I90°)+I0°I90°ρrρst(1ρr2+(1ρr)2I0°I90°)cosθ

Then we can get the value of δ st as below.

δst={arccos[(I+45°I45°)2I0°I90°ρrρst(1ρr2+(1ρr)2I0°I90°)]}2s+Γ
 figure: Fig. 5.

Fig. 5. Multiple reflections in the substrate for reflection monitoring system

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The reflectance can be measured via the intensity change behind the 0- and 90-degree polarizers. Then the magnitude of the reflection coefficient, square root of the reflectance, can be further acquired. Since we have found both magnitude and phase of the reflection coefficient, we have the complete information about the reflection coefficient of the growing film stack.

Moreover, the admittance is related to the reflection coefficient by the following relationship [7],

ns(α+iβ)ns+(α+iβ)=ρsteiδst.

where ns is the refractive index of the substrate. We can monitor the optical admittance of the growing thin film immediately and record the real time admittance loci. We are able to calculate the optical admittance at normal incidence as follows.

α=ns(1ρst2)1+ρst2+2ρstcosδst,β=2nsρstsinδst1+ρst2+2ρstcosδst

where α and β are the real part and imaginary part of the optical admittance, respectively; δst is the phase of the reflection coefficient. This will greatly help the operator to evaluate the optical performance of the coated thin films instantly and therefore judge the cutting points more easily. Actually, some recently published papers tried to rebuild the real admittance loci from the conventional optical monitor system [2, 9, 10], but all of their proposed methods were under an assumption, the refractive index of one layer is “constant”, since they were limited by the information that conventional monitor system can provide.

Figure 6 demonstrates the simulation of the monitoring diagrams for homogenous and inhomogenous thin films. The values change as the film thickness grows and form the admittance loci. We can see that there is an obvious difference between the admittance loci which cannot be clearly inspected on the conventional monitoring diagrams, since the traditional monitoring systems detect no phase information. Unlike a homogenous thin film, the admittance locus of the inhomogenous thin film is not a perfect circle. The refractive index and thickness of each moment can be calculated analytically from following relation [7],

α+iβ=insinδ+(α+iβ)cosδcosδ+iαβnsinδ

where δ is the optical phase of newly deposited thin film, n is the corresponding refractive index, and α + and α+ are the admittance of the previously deposited film stack and the new equivalent admittance, respectively. The solutions of Eq. (12) will be:

δ=arctan[±(α2ααα2+αβ2αβ2αβ+αβ)]n=±(αα)(α2ααα2+αβ2αβ2)(αα)

We should always choose the set of answer whose n and δ are positive. Therefore, inhomogenous phenomenon as well as the actual thickness of thin film would be obviously inspected in this system. As we have mentioned above, the monitoring diagram may never pass the design point due to the change of refractive index or the thickness error of the previous layers. The rules of the determinations of the termination points for a non-quarter wave stack have little physical meaning in the traditional monitoring method. However, in our monitoring method, the error can be compensated through stopping the monitoring admittance locus at the cross point of the locus of the next layer [8].

 figure: Fig. 6.

Fig. 6. Admittance loci simulation of homogenous and inhomogenous thin films

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3. Conclusion

The extraction of optical phase information for a growing film stack is very difficult, since no one, to our knowledge, can build up an interferometer on a vibrating coater machine. Now we have extended the advantage of application of the interferometer for surface profile measurement, and used it as a tool to detect the temporal phase change on one spot of a coating sample. Both the magnitude and phase of the reflection coefficient can be found and give us access to the complete information about reflection coefficient, optical admittance, refractive index and actual thickness of the growing film stack at every moment.

Furthermore, the method to obtain the optical admittance during coating process was demonstrated. This system is simple, compact, and will be useful in the fabrication of valuable coated elements.

Acknowledgement

The authors thank the National Science Council of Taiwan for financially supporting under Contract No. NSC95-2221-E-008-115-MY3 and NSC96-2221-E-008-052-MY3.

References

1. H. A. Macleod, “Monitor of optical coatings,” Appl. Opt. 20, 82–89 (1981). [CrossRef]   [PubMed]  

2. C. C. Lee and K. Wu, “In situ sensitive optical monitoring with error compensation,” Opt. Lett. 32, 2118–2120 (2007). [CrossRef]   [PubMed]  

3. S. Dligatch, R. Netterfield, and B. Martin, “Application of in-situ ellipsometry to the fabrication of multi-layered coatings with sub-nanometre accuracy,” Thin Solid Films 455456, 376–379 (2004). [CrossRef]  

4. N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005). [CrossRef]  

5. B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low Coherence Vibration Insensitive Fizeau Interferometer,” Proc. SPIE6292, 62920F (2006).

6. J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005) [CrossRef]  

7. H. A. Macleod, Thin Film Optical Filters, 3nd ed (Inst. of Physics Publishing2001).

8. Y. R. Chen and C. C. Lee, “Monitoring of Multilayer by Admittance Diagram”, in Conference on Optical Interference Coatings, Technical Digest (CD) (Optical Society of America, 2007) , paper PWC7.

9. B. Chun, C. K. Hwangbo, and J. S. Kim, “Optical monitoring of nonquarter-wave layers of dielectric multilayer filters using optical admittance,” Opt. Express 14, 2473–2480 (2006) [CrossRef]   [PubMed]  

10. A. Tikhonravov and M. Trubetskov, “Eliminating of cumulative effect of thickness errors in monochromatic monitoring of optical coating production: theory,” Appl. Opt. 46, 2084 (2007). [CrossRef]   [PubMed]  

References

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  1. H. A. Macleod, “Monitor of optical coatings,” Appl. Opt. 20, 82–89 (1981).
    [Crossref] [PubMed]
  2. C. C. Lee and K. Wu, “In situ sensitive optical monitoring with error compensation,” Opt. Lett. 32, 2118–2120 (2007).
    [Crossref] [PubMed]
  3. S. Dligatch, R. Netterfield, and B. Martin, “Application of in-situ ellipsometry to the fabrication of multi-layered coatings with sub-nanometre accuracy,” Thin Solid Films 455– 456, 376–379 (2004).
    [Crossref]
  4. N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005).
    [Crossref]
  5. B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low Coherence Vibration Insensitive Fizeau Interferometer,” Proc. SPIE6292, 62920F (2006).
  6. J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
    [Crossref]
  7. H. A. Macleod, Thin Film Optical Filters, 3nd ed (Inst. of Physics Publishing2001).
  8. Y. R. Chen and C. C. Lee, “Monitoring of Multilayer by Admittance Diagram”, in Conference on Optical Interference Coatings, Technical Digest (CD) (Optical Society of America, 2007) , paper PWC7.
  9. B. Chun, C. K. Hwangbo, and J. S. Kim, “Optical monitoring of nonquarter-wave layers of dielectric multilayer filters using optical admittance,” Opt. Express 14, 2473–2480 (2006)
    [Crossref] [PubMed]
  10. A. Tikhonravov and M. Trubetskov, “Eliminating of cumulative effect of thickness errors in monochromatic monitoring of optical coating production: theory,” Appl. Opt. 46, 2084 (2007).
    [Crossref] [PubMed]

2007 (2)

2006 (1)

2005 (1)

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
[Crossref]

2004 (1)

S. Dligatch, R. Netterfield, and B. Martin, “Application of in-situ ellipsometry to the fabrication of multi-layered coatings with sub-nanometre accuracy,” Thin Solid Films 455– 456, 376–379 (2004).
[Crossref]

1981 (1)

Brock, N.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
[Crossref]

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005).
[Crossref]

Chen, Y. R.

Y. R. Chen and C. C. Lee, “Monitoring of Multilayer by Admittance Diagram”, in Conference on Optical Interference Coatings, Technical Digest (CD) (Optical Society of America, 2007) , paper PWC7.

Chun, B.

Dligatch, S.

S. Dligatch, R. Netterfield, and B. Martin, “Application of in-situ ellipsometry to the fabrication of multi-layered coatings with sub-nanometre accuracy,” Thin Solid Films 455– 456, 376–379 (2004).
[Crossref]

Hayes, J.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
[Crossref]

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005).
[Crossref]

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low Coherence Vibration Insensitive Fizeau Interferometer,” Proc. SPIE6292, 62920F (2006).

Hwangbo, C. K.

Kim, J. S.

Kimbrough, B.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
[Crossref]

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low Coherence Vibration Insensitive Fizeau Interferometer,” Proc. SPIE6292, 62920F (2006).

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005).
[Crossref]

Lee, C. C.

C. C. Lee and K. Wu, “In situ sensitive optical monitoring with error compensation,” Opt. Lett. 32, 2118–2120 (2007).
[Crossref] [PubMed]

Y. R. Chen and C. C. Lee, “Monitoring of Multilayer by Admittance Diagram”, in Conference on Optical Interference Coatings, Technical Digest (CD) (Optical Society of America, 2007) , paper PWC7.

Macleod, H. A.

H. A. Macleod, “Monitor of optical coatings,” Appl. Opt. 20, 82–89 (1981).
[Crossref] [PubMed]

H. A. Macleod, Thin Film Optical Filters, 3nd ed (Inst. of Physics Publishing2001).

Martin, B.

S. Dligatch, R. Netterfield, and B. Martin, “Application of in-situ ellipsometry to the fabrication of multi-layered coatings with sub-nanometre accuracy,” Thin Solid Films 455– 456, 376–379 (2004).
[Crossref]

Millerd, J.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
[Crossref]

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low Coherence Vibration Insensitive Fizeau Interferometer,” Proc. SPIE6292, 62920F (2006).

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005).
[Crossref]

Netterfield, R.

S. Dligatch, R. Netterfield, and B. Martin, “Application of in-situ ellipsometry to the fabrication of multi-layered coatings with sub-nanometre accuracy,” Thin Solid Films 455– 456, 376–379 (2004).
[Crossref]

North-Morris, M.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
[Crossref]

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005).
[Crossref]

Novak, M.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
[Crossref]

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005).
[Crossref]

Tikhonravov, A.

Trubetskov, M.

Wu, K.

Wyant, J.

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low Coherence Vibration Insensitive Fizeau Interferometer,” Proc. SPIE6292, 62920F (2006).

Wyant, J. C.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
[Crossref]

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005).
[Crossref]

Appl. Opt. (2)

Opt. Express (1)

Opt. Lett. (1)

Proc. of SPIE (1)

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern Approaches in Phase Measuring Metrology,” Proc. of SPIE 5856, 14–22 (2005)
[Crossref]

Thin Solid Films (1)

S. Dligatch, R. Netterfield, and B. Martin, “Application of in-situ ellipsometry to the fabrication of multi-layered coatings with sub-nanometre accuracy,” Thin Solid Films 455– 456, 376–379 (2004).
[Crossref]

Other (4)

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE5875, 58750F (2005).
[Crossref]

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low Coherence Vibration Insensitive Fizeau Interferometer,” Proc. SPIE6292, 62920F (2006).

H. A. Macleod, Thin Film Optical Filters, 3nd ed (Inst. of Physics Publishing2001).

Y. R. Chen and C. C. Lee, “Monitoring of Multilayer by Admittance Diagram”, in Conference on Optical Interference Coatings, Technical Digest (CD) (Optical Society of America, 2007) , paper PWC7.

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Figures (6)

Fig. 1.
Fig. 1. Scheme of transmission optical monitoring system.
Fig. 2.
Fig. 2. Scheme of reflection optical monitoring system.
Fig. 3.
Fig. 3. Layout of the polarizer array
Fig. 4.
Fig. 4. Multiple reflections in the substrate for transmission monitoring system
Fig. 5.
Fig. 5. Multiple reflections in the substrate for reflection monitoring system
Fig. 6.
Fig. 6. Admittance loci simulation of homogenous and inhomogenous thin films

Equations (16)

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I 0 ° = I s ( 1 ρ r ) 2 τ at 2 1 ρ st 2 ρ r 2 I 90 ° = I p ( 1 ρ r ) 2 ρ at 2 1 ρ st 2 ρ r 2
I S 0 P 1 = 1 2 ( I p ρ r 2 ρ st 2 1 ρ r 2 τ at 2 + I s ρ r 2 1 ρ r 2 τ at 2 + 2 I s I p 1 ρ r 2 τ at 2 ρ r ρ st cos θ )
I S 1 P 2 = 1 2 ρ r 2 ρ st 2 ( I p ρ r 2 ρ st 2 1 ρ r 2 τ at 2 + I s ρ r 2 1 ρ r 2 τ at 2 + 2 I s I p 1 ρ r 2 τ at 2 ρ r ρ st cos θ )
= ρ r 2 ρ st 2 I S 0 P 1
I S 2 P 3 = ( ρ r 2 ρ st 2 ) 2 I S 0 P 1 ; I S 3 P 4 = ( ρ r 2 ρ st 2 ) 3 I S 0 P 1 ; . . . . . . . . . . I SNPN = ( ρ r 2 ρ st 2 ) N I S 0 P 1 ;
I + 45 ° = 1 2 ( I 0 ° + I 90 ° ) + I 0 ° I 90 ° ρ st ρ r cos θ
I 45 ° = 1 2 ( I 0 ° + I 90 ° ) + I 0 ° I 90 ° ρ st ρ r cos θ
δ st = [ arccos ( I + 45 ° I 45 ° 2 I 0 ° I 90 ° ρ st ρ r ) ] 2 s + Γ
I 0 ° = I s ρ r 2 + I s ( 1 ρ r ) 4 ρ st 2 1 ρ st 2 ρ r 2 I 90 ° = I p ρ r 2 + I p ( 1 ρ r ) 4 ρ st 2 1 ρ st 2 ρ r 2
I + 45 ° = 1 2 ( I 0 ° + I 90 ° ) + I 0 ° I 90 ° ρ r ρ st ( 1 ρ r 2 + ( 1 ρ r ) 2 I 0 ° I 90 ° ) cos θ
I 45 ° = 1 2 ( I 0 ° + I 90 ° ) + I 0 ° I 90 ° ρ r ρ st ( 1 ρ r 2 + ( 1 ρ r ) 2 I 0 ° I 90 ° ) cos θ
δ st = { arccos [ ( I + 45 ° I 45 ° ) 2 I 0 ° I 90 ° ρ r ρ st ( 1 ρ r 2 + ( 1 ρ r ) 2 I 0 ° I 90 ° ) ] } 2 s + Γ
n s ( α + i β ) n s + ( α + i β ) = ρ st e i δ s t .
α = n s ( 1 ρ st 2 ) 1 + ρ st 2 + 2 ρ st cos δ st , β = 2 n s ρ st sin δ st 1 + ρ st 2 + 2 ρ st cos δ st
α + i β = in sin δ + ( α + i β ) cos δ cos δ + i α β n sin δ
δ = arctan [ ± ( α 2 α α α 2 + α β 2 α β 2 α β + α β ) ] n = ± ( α α ) ( α 2 α α α 2 + α β 2 α β 2 ) ( α α )

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