## Abstract

The absolute optical thickness of a transparent plate 6-mm thick and 10 mm in diameter was measured by the excess fraction method and a wavelength-tuning Fizeau interferometer. The optical thickness, defined by the group refractive index at the central wavelength, was measured by wavelength scanning. The optical thickness deviation, defined by the ordinary refractive index, was measured using the phase-shifting technique. Two kinds of optical thicknesses, measured by discrete Fourier analysis and the phase-shifting technique, were synthesized to obtain the optical thickness with respect to the ordinary refractive index using Sellmeier equation and least-square fitting.

© 2015 Optical Society of America

## 1. Introduction

Optical thickness is an important characteristic for the design of optical devices in the semiconductor industry. As transparent plates continue to increase in size, the demand for the precise measurement of the surface shape and optical thickness grows. Although the measurement uncertainty of the surface shape by interferometry is typically on the order of *λ*/20 or 30 nm, the measurement uncertainty of the optical thickness of a transparent plate is on the order of a few microns, which is far worse than that of the surface shape. Many approaches have been developed for the optical thickness measurement of a transparent plate.

The optical thickness measurement of a transparent plate using white-light interferometers and confocal microscopy has been reported by several authors [1, 2]. In this technique, the diameter of an observing aperture is restricted to <1 cm because the accurate translation of a large reference mirror along the optical axis is difficult. When the thickness of a sample increases to more than a few millimeters, the rapid increase in the coherence length degrades the measurement resolution. Wavelength-tuning interferometry has also been used for the optical thickness measurement with respect to the group refractive index of a transparent plate [3–9]. However, these techniques are not suitable for measuring the thickness distribution, because they assume that the sample has a spatially uniform thickness. The absolute optical thickness of the mask blank glass had already been measured by discrete Fourier analysis and the phase-shifting technique [10]. However, the phase-shifting algorithm employed in Ref. 10 can only compensate for the linear calibration error of the phase shift and a sidelobe level of 1.3% [11]; therefore, there was a measurement uncertainty of 1.7 ppm.

In this study, the absolute optical thickness distribution of a transparent plate 6 mm thick and 100 mm in diameter was measured using the excess fraction method and a wavelength-tuning Fizeau interferometer. The absolute optical thickness determined using wavelength-tuning Fizeau interferometry corresponds to the group refractive index and has a large measurement uncertainty because of the synthetic wavelength. The optical thickness deviation determined using the phase-shifting technique corresponds to the ordinary refractive index. These two kinds of optical thicknesses were synthesized using the Sellmeier equation for the refractive index of fused-silica glass and least-square fitting. The absolute optical thickness and interference fringe order were finally determined using an excess fraction method to eliminate the initial uncertainty of the synthetic wavelength and refractive index.

## 2. Measurement principle

#### 2.1 Wavelength-tuning Fizeau interferometer

Figure 1 shows the optical setup for measuring the absolute optical thickness distribution of a transparent plate in a Fizeau interferometer. The temperature inside the laboratory was 20.5° C. The source is a tunable diode laser with a Littman external cavity (New Focus TLB–6300–LN) comprising a grating and a cavity mirror. The source wavelength is scanned linearly in time from 632.6 to 638.4 nm, translating the cavity mirror using a piezoelectric (PZT) transducer and a picomotor with a constant speed [12]. The beam is transmitted using an isolator and divided into two by a beam splitter: one beam goes to a wavelength meter (Anritsu MF9630A) which was calibrated using a stabilized HeNe laser with an accuracy of δ*λ*/*λ*~10^{−7} at a wavelength of 632.8 nm, and the other is incident to an interferometer. The focused output beam is reflected by a polarization beam splitter. The linearly polarized beam is then transmitted to a quarter-wave plate, becoming a circularly polarized beam. This beam is collimated to illuminate the reference surface and the measurement sample. The reflections from the multiple surfaces of the measurement sample and reference surface travel back along the same path, and then they are transmitted through the quarter-wave plate again to attain an orthogonal linear polarization. The resulting beams pass through the polarization beam splitter and combine to generate a fringe pattern on the screen, with a resolution of 640 × 480 pixels. The measurement sample is placed horizontally on a mechanical stage, with an air-gap distance of *L*.

The sample plate, made of synthetic fused silica (*n _{p}* ~1.45 at 632.8 nm), is 6 mm thick and 100 mm in diameter. Because the sample is aligned to be parallel with the reference surface, three dominant reflection beams from the top and rear surfaces of the sample and from the reference surface are combined to generate three different interference fringe patterns. The modulation frequency of each interference fringe is proportional to the optical path difference of each pair of interfering beams. In order to completely separate these signals in the frequency domain [13, 14], the distance

*L*( = 43.5 mm) was approximately adjusted to 5

*n*(~8.7 mm): five times the optical thickness of transparent plate

_{g}T*n*. By setting

_{g}T*L*as 5

*n*, three main signal frequencies

_{g}T*ν*

_{1}, 5

*ν*

_{1}, 6

*ν*

_{1}corresponding to optical thickness of transparent plate, surface shape and rear surface shape respectively appear in the frequency domain.

#### 2.2 Excess fraction method

First, we briefly describe the excess fraction method. When the wavelength is scanned from *λ*_{1} to *λ*_{2}, the optical thickness of the transparent plate at each wavelength is defined as

*N*

_{1, 2},

*p*

_{1, 2}, and

*n*

_{1, 2}are the interference orders, fractions, and refractive indices at each wavelength. The refractive index and optical thickness are functions of the wavelength.

The absolute lengths are measured using the conventional excess fraction method [15, 16]. This method measures the fractions *p*_{1, 2} and solves Eq. (1) to determine the interference orders *N*_{1, 2} if the wavelengths and the refractive index of air are strictly defined (typically better than 10^{−8} at the visible wavelength). However, for the optical thickness case of transparent plates, the refractive index has a much higher uncertainty. For example, fused silica has a refractive index uncertainty of 3 × 10^{−5}, which is much higher than that of air. As a result, the conventional excess fraction method is only valid for thin objects. In other words, when attempting to determine the optical thickness using excess fraction method, the dynamic range must be so small that the optical thickness is limited to tens of micrometers. Accordingly, we measured the fractions and the optical thickness simultaneously using phase-shifting and wavelength-tuning interferometry, respectively. This allowed for the exact value of interference orders *N*_{1, 2} to be determined uniquely by rounding the values of (2*n*_{1, 2}*T*/*λ*_{1, 2} – *p*_{1, 2}) to integers.

#### 2.3 Wavelength-tuning and phase measurements

From Eq. (1), the absolute optical thickness is calculated as follows:

The right-hand side of Eq. (2) is proportional to the interference order displacement *N*_{1} – *N*_{2} + *p*_{1} – *p*_{2}, which is the number of variations in the interference fringes during the wavelength scanning. The absolute optical thickness is proportional to the product of the displacement and the synthetic wavelength *λ _{s}* =

*λ*

_{1}

*λ*

_{2}/(

*λ*

_{2}–

*λ*

_{1}) [10, 16]. The coefficient on the left-hand side of Eq. (2) reduces to the group refractive index of the transparent plate at the central wavelength

*λ*= (

_{c}*λ*

_{1}+

*λ*

_{2})/2 when the dispersion of the material is small.

Using Eq. (2), the optical thickness at the central wavelength can be rewritten as

Note that the product of the synthetic wavelength and the order displacement represents not an ordinary optical thickness but one corresponding to the group refractive index. Sellmeier equation for the refractive index of fused-silica glass is given by Schott Glass, with an uncertainty of 3 × 10^{−5}, as

Using this Sellmeier equation, the ratio of the refractive indices *n*_{1}/*n*_{g} and an approximate value for the optical thickness at the wavelength *λ*_{1} can be calculated as follows:

Figure 2 indicates the variation in the wavelength with time. First, the wavelength was scanned linearly in time from *λ*_{1} - Δ*λ _{a}* to

*λ*

_{1}+ Δ

*λ*over a width of 2Δ

_{a}*λ*= 0.0917 nm, and 77 images were recorded at equal phase-shift intervals ( = π/10). During the scanning, the signal interference fringes corresponding to the optical thickness changed by four periods of 8π radians. The magnitudes of the phase shift for each step were π/10 for the optical thickness fringes. The fraction

_{a}*p*

_{1}was then calculated using phase-shifting technique with the following algorithm:

*I*is the intensity of the

_{r}*r*th recorded image. The window function

*w*[17] is defined as

_{r}The phase-shifting algorithm comprising polynomial window function defined by Eq. (8) and discrete Fourier transform term can compensate up to the second nonlinearity of the phase shift, yielding the lowest sidelobe level among the conventional phase-shifting algorithms and other window functions [17]. The fraction *p*_{1} calculated by Eqs. (7)–(8) is the fraction value of image *I*_{39}.

After recording 77 images, the wavelength was adjusted back to *λ*_{1} by the PZT transducer of the laser cavity mirror. Using the picomotor, the wavelength was scanned linearly in time over 5.6 nm from *λ*_{1} to *λ*_{2}, and 593 images were recorded at equal wavelength intervals. An approximate value of the interference order displacement *N*_{1} - *N*_{2} + *p*_{1} - *p*_{2} can be calculated by a discrete Fourier analysis on the 593 images. Discrete Fourier amplitude of the fringe pattern intensity is defined by

*f*= 1, 2, …, 297 and we used the Hann window [18] defined by

The amplitude *F*(*f*) has three major peaks because of three–surface interferometry. Among them, the lowest frequency *f* = *M* of the maximum gives an approximate value of the displacement as

Finally, the wavelength was scanned linearly in time by the PZT again from *λ*_{2} – Δ*λ _{b}* to

*λ*

_{2}+ Δ

*λ*over a 2Δ

_{b}*λ*

_{b}= 0.0933 nm width and another 77 images were recorded at equal phase-shift intervals. The fraction

*p*

_{2}was calculated in a manner similar to the algorithm of Eqs. (7)–(8).

#### 2.5 Synthesis of two kinds of optical thickness using least-squares fitting

The fractions *p*_{1} and *p*_{2} determined using the phase-shifting technique and frequency *N*_{1} - *N*_{2} determined using the discrete Fourier analysis yield the most suitable displacement *N*_{1} - *N*_{2} + *p*_{1} - *p*_{2}. The absolute optical thickness (*n _{g}T*)

*at the central wavelength and (*

_{meas}*n*

_{1}

*T*)

*at the initial wavelength can be calculated using Eqs. (4)–(6), respectively. However, it was shown in the experiment that the measured absolute optical thickness (*

_{meas}*n*

_{1}

*T*)

*suffers from the crosstalk noise, which must be reduced before the determination of the interference orders by the excess fraction method.*

_{meas}The crosstalk noise occurs when there is nonlinearity in the modulation during the phase-shifting, which is common in using tunable lasers. In this experiment, the PZT modulator for the fine tuning of the source wavelength has a residual nonlinearity, which causes crosstalk among the three interference signals generated from the three surfaces. The noise thus involves the fractions *p*_{1} and *p*_{2}, whose magnitudes were as high as 3 nm.

Because the absolute thickness (*n _{g}T*)

*is the product of the displacement and the synthetic wavelength, the latter of which is several tens of times higher than the source wavelength, the absolute thickness exhibits a noise level far higher than the original noise level in the fraction*

_{meas}*p*

_{1}. Here, we reduce the noise in the following manner.

Because the uncertainty of the source wavelength is 10^{−7} or less, we can regard the absolute optical thickness *n*_{1}*T* as the sum of a spatially varying component and a spatially uniform component. The deviation component is easily determined by unwrapping the fractional phase 2π*p*_{1} as follows:

*p*

_{1}denotes the unwrapped phase distribution of 2π

*p*

_{1}. The uniform component has the ambiguity of an integral multiple of a half wavelength:

The unknown integral order *A*_{1} can be determined by comparing the calculated thickness with the measured thickness using the least-squares fitting method, as follows:

*P*is the total number of image pixels.

Finally, the absolute optical thickness distribution at the initial wavelength *λ*_{1} was calculated as the sum of the uniform component *n*_{1}*T*_{0} and the deviation ingredient (*n*_{1}*T*)* _{dev}*:

#### 2.6 Interference order determination using excess fraction method

As discussed in subsection 2.2, the interference order of the absolute optical thickness can be calculated by rounding the values of (2*n*_{1}*T*/*λ*_{1} – *p*_{1}) to integer values. By substituting the optical thickness of Eq. (15) into

*λ*

_{1}can be determined. Because of the large uncertainty of 3 × 10

^{−5}in the refractive index

*n*

_{1}, the optical thickness

*n*

_{1}

*T*has an uncertainty of approximately 100 nm. However, after the interference order has been uniquely determined, the uncertainty in the optical thickness calculated by Eq. (1) is reduced to less than 10 nm. This value is mainly limited by the accuracies of the fraction measurement (accuracy of phase-shifting algorithm) and the source wavelength. Figure (3) shows the summary of the measurement procedure.

## 3. Experiment

A transparent plate made of fused silica (*n* = 1.45 and *n*_{g} = 1.47) was measured using a Fizeau interferometer and the optical setup shown in Fig. 1. Figure 4(a) is a laboratory photo of the transparent plate in the wavelength-tuning Fizeau interferometer, and Fig. 4(b) is an observed raw interferogram of the transparent plate at the wavelength of 632.8 nm.

First, the wavelength was finely scanned from 632.8041 to 632.8957 nm, and 77 interference images were recorded at equal wavelength intervals. Because there was a nonlinearity of approximately 3% in the PZT response (fine-scanning mode) of the source-laser cavity, a quadratic voltage increment was applied to the PZT to make the resultant wavelength scanning linear. As a result, the nonlinearity decreased to 1% of the total phase shift. The initial phase was calculated using the algorithm of Eqs. (7)–(8). Secondly, the wavelength was scanned back to *λ*_{1} = 632.8507 nm and then scanned coarsely to *λ*_{2} = 638.4011 nm over a 5.56 nm width at a rate of 0.01 nm/s. During the scanning, 593 interference images on the screen were recorded by CCD camera. The order displacement *M* of the interference orders of the image was estimated using Fourier analysis. Finally, the wavelength was finely scanned from 638.3544 to 638.4475 nm, and an additional 77 interference images were recorded at equal wavelength intervals. The fractional phase *p*_{2} was calculated similarly using these images. The absolute optical thickness (*n*_{g}*T*)* _{meas}* at the central wavelength was calculated using Eq. (4). The absolute optical thickness (

*n*

_{1}

*T*)

*at the initial wavelength was then calculated using Eq. (6) along with the Sellmeier equation.*

_{meas}Figure 5(a) indicates the measured fraction *p*_{1}, and Fig. 5(b) indicates the absolute optical thickness at the initial wavelength, which was calculated using Eqs. (4)–(6). The repeatability of the fraction measurement was 1 nm rms. The absolute optical thickness exhibits noise of approximately 1 μm PV, in a pattern following the interference fringes. These noise pixels were distributed mainly on the dark interference fringes of the optical thickness. This noise was caused by the crosstalk between the different frequency components in the phase-shifting calculations. It is common in using tunable lasers and was originally involved in the fractions *p*_{1} and *p*_{2}. The crosstalk occurred because there was a residual nonlinearity in the phase shift during the recording of the 77.

The optical thickness deviation calculated by unwrapping the fraction *λ*_{1}*p*_{1}/2 was fitted to the measured optical thickness (*n*_{1}*T*)* _{meas}*, as shown in Fig. 5(b), using the least-squares fitting method, which was discussed in subsection 2.5. The resultant uniform component of the optical thickness and the mean square error of the fitting were given by

*n*

_{1}

*T*

_{0}= 8839.975 μm (

*A*

_{1}= 27937; see Eq. (13)). Finally, substituting this uniform thickness and the optical thickness deviation into Eq. (15), the interference orders

*N*

_{1}at the initial wavelength were determined. Figure 6(a) indicates the final absolute optical thickness at the initial wavelength

*λ*

_{1}that was calculated using Eq. (1). Figure 6(b) indicates the calculated interference orders.

The accuracy of the wavelength meter was 10^{−7} after the calibration. The stability of the source wavelength during the observation time of 5 min was better than 10^{−7}. The measured phase contains a crosstalk noise of 2 nm PV, which was caused by the nonlinearity in the fine tuning of the wavelength. There was also a random motion of the PZT, which yielded a phase measurement error of 1 nm rms.

The uncertainty of the absolute optical thickness distribution at the initial wavelength can be expressed as

Because the uncertainty of the source wavelength is less than or equal to 10^{−7}, the uncertainty of the absolute optical thickness, which is estimated as 3.3 nm, is dominant in the phase-shifting technique. The relative uncertainty of the optical thickness is therefore limited to 0.47 ppm, being converted to 4.16 nm of absolute optical thickness.

## 4. Conclusion

The absolute optical thickness distribution of a fused silica parallel plate 6-mm thick and 100 mm in diameter was measured using a wavelength-tuning Fizeau interferometer and the excess fraction method. Two kinds of optical thicknesses, measured by discrete Fourier analysis and the phase-shifting technique, were synthesized to obtain the optical thickness with respect to the ordinary refractive index *n*_{1} at the initial wavelength *λ*_{1}. The absolute optical thickness and interference fringe order were finally determined using an excess fraction method to eliminate the initial uncertainty of the synthetic wavelength and refractive index.

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