Absolute optical thickness measurement of transparent plate using excess fraction method and wavelength-tuning Fizeau interferometer

Yangjin Kim; Kenichi Hibino; Naohiko Sugita; Mamoru Mitsuishi

doi:10.1364/OE.23.004065

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 δλ/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.

Fig. 1 Wavelength-tuning Fizeau interferometer for measuring the absolute optical thickness of a transparent plate. PBS denotes the polarization beam splitter; QWP is the quarter-wave plate; HWP is the half-wave plate. The thickness of the sample and the air gap distance are T and L, respectively.

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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 5n_gT (~8.7 mm): five times the optical thickness of transparent plate n_gT. By setting L as 5n_gT, three main signal frequencies ν₁, 5ν₁, 6ν₁ 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 λ₁ to λ₂, the optical thickness of the transparent plate at each wavelength is defined as

n_{1, 2} T = \frac{λ_{1, 2}}{2} (N_{1, 2} + p_{1, 2}),

where 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 (2n_{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:

\frac{n_{1} + n_{2}}{2} (1 - \frac{λ_{1} + λ_{2}}{n_{1} + n_{2}} \cdot \frac{n_{2} - n_{1}}{λ_{2} - λ_{1}}) T = \frac{λ_{1} λ_{2}}{2 (λ_{2} - λ_{1})} (N_{1} - N_{2} + p_{1} - p_{2}) .

The right-hand side of Eq. (2) is proportional to the interference order displacement N₁ – N₂ + p₁ – p₂, 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 = λ₁λ₂/(λ₂ – λ₁) [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 = (λ₁ + λ₂)/2 when the dispersion of the material is small.

n_{g} (λ_{c}) = \frac{n_{1} + n_{2}}{2} (1 - \frac{λ_{1} + λ_{2}}{n_{1} + n_{2}} \cdot \frac{n_{2} - n_{1}}{λ_{2} - λ_{1}}) \approx n (1 - \frac{λ}{n} \cdot \frac{d n}{d λ}) .

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

{[n_{g} (λ_{c}) T]}_{m e a s} = \frac{λ_{s}}{2} (N_{1} - N_{2} + p_{1} - p_{2}) .

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

n^{2} - 1 = \frac{0.6961663 λ^{2}}{λ^{2} - {0.0684043}^{2}} + \frac{0.4079426 λ^{2}}{λ^{2} - {0.1162414}^{2}} + \frac{0.8974794 λ^{2}}{λ^{2} - {9.896161}^{2}} .

Using this Sellmeier equation, the ratio of the refractive indices n₁/n_g and an approximate value for the optical thickness at the wavelength λ₁ can be calculated as follows:

{(n_{1} T)}_{m e a s} = \frac{n_{1}}{n_{g}} \cdot {(n_{g} T)}_{m e a s} .

Figure 2 indicates the variation in the wavelength with time. First, the wavelength was scanned linearly in time from λ₁ - Δλ_a to λ₁ + Δλ_a over a width of 2Δλ_a = 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 p₁ was then calculated using phase-shifting technique with the following algorithm:

p_{1} = \frac{1}{2 π} arc \tan \frac{\sum_{r = 1}^{77} w_{r} I_{r} \sin \frac{π r}{10}}{\sum_{r = 1}^{77} w_{r} I_{r} \cos \frac{π r}{10}},

where I_r is the intensity of the rth recorded image. The window function w_r [17] is defined as

Fig. 2 Temporal variations in the source wavelength scanned by a piezoelectric transducer (fine tuning) and a picomotor (coarse tuning) attached to the end mirror of the external cavity of the laser diode (not-to-scale principle sketch).

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\begin{array}{l} w_{r} = [\frac{1}{6} r (r + 1) (r + 2)] (1 \leq r \leq 20), \\ w_{r} = [5340 + \frac{1}{2} | r - 39 | ({| r - 39 |}^{2} - 40 | r - 39 | - 1)] (21 \leq r \leq 57), \\ w_{r} = [\frac{1}{6} (80 - r) (79 - r) (78 - r)] (58 \leq r \leq 77) . \end{array}

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 p₁ calculated by Eqs. (7)–(8) is the fraction value of image I₃₉.

After recording 77 images, the wavelength was adjusted back to λ₁ by the PZT transducer of the laser cavity mirror. Using the picomotor, the wavelength was scanned linearly in time over 5.6 nm from λ₁ to λ₂, and 593 images were recorded at equal wavelength intervals. An approximate value of the interference order displacement N₁ - N₂ + p₁ - p₂ can be calculated by a discrete Fourier analysis on the 593 images. Discrete Fourier amplitude of the fringe pattern intensity is defined by

F (f) = \sqrt{{[\sum_{j = 1}^{593} I_{j} h (j) \cos 2 π f (j - 1)] / 593}^{2} + {[\sum_{j = 1}^{593} I_{j} h (j) \sin 2 π f (j - 1)] / 593}^{2}},

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

h (j) = \frac{2}{297} \cos^{2} \frac{j - 297}{593} .

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

M = round (N_{1} - N_{2} + p_{1} - p_{2})

where function round gives the nearest integer. The discrete Fourier analysis causes systematic errors in the observed fractional phases that yield the wrong interference order. These systematic errors were corrected by determining a correlation integral analysis between the observed interference signal and the theoretically predicted interference signal [10, 16].

Finally, the wavelength was scanned linearly in time by the PZT again from λ₂ – Δλ_b to λ₂ + Δλ_b over a 2Δλ_b = 0.0933 nm width and another 77 images were recorded at equal phase-shift intervals. The fraction p₂ 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 p₁ and p₂ determined using the phase-shifting technique and frequency N₁ - N₂ determined using the discrete Fourier analysis yield the most suitable displacement N₁ - N₂ + p₁ - p₂. The absolute optical thickness (n_gT)_meas at the central wavelength and (n₁T)_meas 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 (n₁T)_meas suffers from the crosstalk noise, which must be reduced before the determination of the interference orders by the excess fraction method.

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 p₁ and p₂, whose magnitudes were as high as 3 nm.

Because the absolute thickness (n_gT)_meas 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 p₁. 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 n₁T as the sum of a spatially varying component and a spatially uniform component. The deviation component is easily determined by unwrapping the fractional phase 2πp₁ as follows:

{(n_{1} T)}_{d e v} = \frac{λ_{1}}{4 π} \cdot unwrap [p_{1} (x, y)],

where unwrap p₁ denotes the unwrapped phase distribution of 2πp₁. The uniform component has the ambiguity of an integral multiple of a half wavelength:

n_{1} T_{0} = \frac{λ_{1}}{2} \cdot A_{1} .

The unknown integral order A₁ can be determined by comparing the calculated thickness with the measured thickness using the least-squares fitting method, as follows:

\frac{1}{P} {\sum_{i = 1}^{P} {{[{(n_{1} T)}_{m e a s}]}_{i} - {[{(n_{1} T)}_{d e v}]}_{i} - n_{1} T_{0}}}^{2} = \min,

where P is the total number of image pixels.

Finally, the absolute optical thickness distribution at the initial wavelength λ₁ was calculated as the sum of the uniform component n₁T₀ and the deviation ingredient (n₁T)_dev:

n_{1} T = n_{1} T_{0} + {(n_{1} T)}_{d e v} .

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 (2n₁T/λ₁ – p₁) to integer values. By substituting the optical thickness of Eq. (15) into

N_{1} = round (\frac{2 n_{1} T}{λ_{1}} - p_{1}),

the interference order at the initial wavelength λ₁ can be determined. Because of the large uncertainty of 3 × 10⁻⁵ in the refractive index n₁, the optical thickness n₁T 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.

Fig. 3 Summary of the measurement procedure.

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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.

Fig. 4 (a) Laboratory photo of transparent plate in wavelength-tuning Fizeau interferometer, (b) raw interferogram at wavelength of 632.8 nm.

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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 λ₁ = 632.8507 nm and then scanned coarsely to λ₂ = 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 p₂ was calculated similarly using these images. The absolute optical thickness (n_gT)_meas at the central wavelength was calculated using Eq. (4). The absolute optical thickness (n₁T)_meas at the initial wavelength was then calculated using Eq. (6) along with the Sellmeier equation.

Figure 5(a) indicates the measured fraction p₁, 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 p₁ and p₂. The crosstalk occurred because there was a residual nonlinearity in the phase shift during the recording of the 77.

Fig. 5 (a) Fraction p₁, (b) absolute optical thickness distribution at initial wavelength.

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The optical thickness deviation calculated by unwrapping the fraction λ₁p₁/2 was fitted to the measured optical thickness (n₁T)_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 n₁T₀ = 8839.975 μm (A₁ = 27937; see Eq. (13)). Finally, substituting this uniform thickness and the optical thickness deviation into Eq. (15), the interference orders N₁ at the initial wavelength were determined. Figure 6(a) indicates the final absolute optical thickness at the initial wavelength λ₁ that was calculated using Eq. (1). Figure 6(b) indicates the calculated interference orders.

Fig. 6 (a) Final absolute optical thickness distribution at initial wavelength, (b) interference order at initial wavelength.

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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

\frac{\partial (n_{1} T)}{n_{1} T} = \frac{\partial λ_{1}}{λ_{1}} + \frac{\partial (N_{1} + p_{1})}{N_{1} + p_{1}} \approx \frac{\partial λ_{1}}{λ_{1}} + \frac{\partial p_{1}}{N_{1}} .

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 n₁ at the initial wavelength λ₁. 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.

References and links

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Absolute optical thickness measurement of transparent plate using excess fraction method and wavelength-tuning Fizeau interferometer

Abstract

1. Introduction

2. Measurement principle

2.1 Wavelength-tuning Fizeau interferometer

2.2 Excess fraction method

2.3 Wavelength-tuning and phase measurements

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

2.6 Interference order determination using excess fraction method

3. Experiment

4. Conclusion

References and links

Cited By

Figures (6)

Equations (17)

Optics Express