1. Introduction

Scanning electron microscopes and scanning probe microscopes are important tools for characterizing nanoscale surface features. In order to evaluate the accuracy of these instruments, standards of periodic pitch structures were developed [1].

Pitch standards can be calibrated using a metrological atomic force microscope (MAFM) or a laser diffractometer [2]. In the MAFM, the sample is placed on a scanning stage whose displacement is measured by laser interferometers to ensure traceability. The grating pitch is finally derived from the measured positions of the sample using some special data evaluation methods [3]. The MAFM is constructed of low thermal expansion material, and is typically operated in a well controlled chamber with a temperature stability less than 0.1 °C to ensure a high measurement repeatability.

The laser diffractometer is a simple structure and provides superior measurement repeatability [2]. Nowadays, most diffractometers are based on the Littrow scheme, although various configurations have been presented to measure the grating period [4,5]. The pitch size is derived from the laser wavelength and the measured diffraction angle between the two positions of the grating, corresponding to the autocollimation condition of the positive and negative first diffracted orders. The laser diffractometer is easy to operate and the measurement time is short. Therefore, although the diffractometer cannot measure the uniformity of gratings, it is very effective for determining the mean pitch.

The minimum measurable grating period is about one half of the vacuum wavelength of the laser used if the diffractometer is operated in the atmospheric environment. For instance, a diffractometer with a 633 nm laser cannot identify a grating period of less than 317 nm. The grating period of a pitch standard has been reduced to 100 nm [1] to accommodate the rapid progress of the semiconductor and nanotechnology industries. In the near future, the grating period will be smaller than the wavelengths of all available lasers and the laser diffractometer will hit a performance wall.

A similar difficult situation also arises in photolithography. Immersion lithography has been developed to enhance the resolution without radical changes in lasers, optics and resist technology [6]. Consequently, this achievement delays the need to switch to extremely different and expensive extreme-ultraviolet lithography. Similar ideas were adopted to reduce the pitch-measuring limit of a traditional diffractometer. The refractive index of the surrounding environment is increased by placing a prism and an index-matching liquid layer on the grating’s surface. This study describes the general principles and features of the proposed scheme, and the results of feasibility testing.

2. Principles

2.1 Conventional laser diffractometry

Figure 1 illustrates a conventional laser diffractometer to measure grating periods. In the diffractometer, a laser beam is incident on a reflective grating mounted on a rotary table. When the grating is rotated, the laser beam is diffracted back upon the incident beam if the Littrow condition is satisfied. In this work, the diffracted orders of interest are +1, 0 and -1 orders. Suppose that the autocollimation angles that correspond to +1, 0 and -1 orders are γ ₊₁, 0 and γ _-1. The grating period p is related to the average angle γ=(γ ₊₁+γ _-1)/2 by

where λ_v is the vacuum wavelength of laser and n_a is the refractive index of air.

 

Fig. 1. Conventional laser diffractometer

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The minimum grating period that can be determined using a conventional laser diffractometer is approximately λ _v/2 since n_a≈1. Equation (1) reveals that the limit of the measurable pitch is a direct function of the vacuum wavelength of the laser and an inverse function of the refractive index of the surrounding environment. Hence, a diffractometer with a laser with a shorter wavelength can measure a smaller grating period, and is used in most laboratories.

2.2 Immersion diffractometry

Raising the refractive index of the environment can also improve the pitch measurement capability of a diffractometer. Accordingly, an oil film is initially spread over the grating surface and then an isosceles prism is attached to the grating, as shown in Fig. 2. The refractive indices of the prism and oil are n_p and n_o, respectively. The oil material is chosen such that n_o≈n_p to avoid total internal reflection at the oil-prism interface. If the triangular prism is replaced by a hemicylindrical prism, then the converging effect of the hemicylindrical element on the laser beam should be considered.

 

Fig. 2. Immersion grating and light paths for (a) γ<ϕ (b) γ>ϕ

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Figure 2(a) and 2(b) show the light paths for γ<ϕ and γ>ϕ, respectively, where ϕ is the prism angle and γ is the angle between the laser beam and the grating normal in air. The laser beam strikes the prism surface at an incident angle α and is refracted with an angle β. The refracted beam travels in the prism and then meets the prism-oil interface. At the interface, the incident angle is θ_p and the refracted angle is θ_o. Finally the laser beam enters the oil film and is diffracted by the grating. When the angle θ_o fulfills the Littrow condition for the first diffracted order, the grating pitch p is given by

In computing the grating period p, n _osinθ_o in Eq. (2) can be readily determined by Snell’s law as follows. At the oil-prism interface,

and at the prism-air interface,

Combining the above equations yields a single expression for p,

Note that rid="e7a">Eqs. (7a) and (7b) are in terms of n_p rather than n_o. The substitution of n_p for n_o can increase the accuracy of measurement because optical glass is a stable material and its refractive index is known precisely. Equations (7a) and (7b) are also applicable to the conventional laser diffractometry by setting n_p=n_a. Figure 3 plots the relationships between the pitch-to-wavelength ratio p/λ_v and the angle γ. A comparison of the two curves that correspond to a grating immersed in oil and air indicates that the immersion diffractometry can be used to measure a grating period of less than one-half the vacuum wavelength. The slope of the p/λ_v versus γ curve of an immersion diffractometer is a function of prism angle ϕ, as shown in Fig. 4.

 

Fig. 3. Relationships between the ratio p/λ_v and angle γ corresponding to a grating immersed in oil and air

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Fig. 4. Different ϕ changes the slope of the p/λ_v versus γ curve of an immersion diffractometer

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3. Experiments

A one-dimensional holographic grating (Moxtek MXS-301BE) with a period of 288 nm was measured under three different conditions to verify the feasibility of the proposed method. First, the grating was measured in the atmosphere and the light source was a 543 nm He-Ne laser. Next, the grating was immersed in oil and the same laser was used. Third, the grating remained in oil but the light source was changed to a 633 nm He-Ne laser.

3.1 Grating immersed in air

Figure 1 displays the laser diffractometer used to measure the period of the grating in air. Half portion of the laser beam passes through a beam splitter and strikes the grating. The grating is fixed on a rotary table (Aerotech ADR240) with an accuracy of one arc second. First, the zero-order diffracted beam is aligned to travel along the incident beam. The beam splitter reflects half part of the return beam to a position sensitive detector (Duma Optronics) with a resolution of one micrometer. The distance between the grating and the detector is 34 cm. The position of the center of the detector is adjusted to coincide with the laser spot. The center position of the detector is now the reference point for determining whether other diffracted beams meet the Littrow condition. Then, the table is rotated to find the autocollimation angles of the +1 and -1 diffracted orders. Averaging these two measured angles yields the angle γ.

3.2 Grating immersed in oil

Figure 5 presents the experimental apparatus for measuring the groove spacing of a immersion grating. A small drop of index-matching oil (Nikon 50 type A, n_o≈1.515) is placed upon the grating surface, and the hypotenuse face of the right angle prism with angles of 45°-90°-45° is brought into contact with the grating.

 

Fig. 5. Immersion diffractometer

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Fig. 6. Adjustments for (a) γ<ϕ (b) γ>ϕ

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The measurement procedures are similar to those described in Sec. 3.1. Figure 6 illustrates the adjustment that must be made before the pitch measurement is performed. This adjustment ensures that the same part of the grating is measured when the laser beam illuminates the grating at two different angles, γ ₊₁ and γ _-1. The immersed grating is moved outward and the axis of rotation is located outside the prism for γ<ϕ, while the immersed grating is moved inward and the axis of rotation is located inside the prism for γ>ϕ. The offset adjustment for the grating is confirmed experimentally that the incident beam hits the same spot on the grating at two angles. If the beam strikes two different spots, no measurement error is involved for a uniform-pitch grating. Nevertheless, it causes a problem for measuring a small grating while the distance between the two spots exceeds the grating dimension.

4. Results

According to Eqs. (7a) and (7b), the grating period p can be determined from the measured γ if λ_v, n_p, n_a and ϕ are known. The vacuum wavelengths λ_v of the green and red HeNe lasers were checked using a wavelength meter with a relative accuracy of 1×10^-7. The experiments were conducted in a laboratory whose temperature and relative humidity were maintained at 20±1°C and 50±10%, respectively. Substituting the mean atmospheric pressure, air temperature, and relative humidity into the revised Edlén equation yields the refractive index of air n_a≈1.00027. The prism is made of BK7 whose refractive index n_p is calculated by the Sellmeir equation with dispersion constants from the manufacture. The prism angle ϕ was measured using the autocollimation method and the apparatus shown in Fig.1. The prism angle ϕ is 45.00002°. Table 1 lists the measurements of the 288 nm grating under three different conditions. Comparing the grating pitches obtained by immersion diffractometry and conventional laser diffractometry yields a maximum deviation of about 0.06 nm.

Table 1. Results of 288 nm grating measured under different conditions

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5. Summary

This work demonstrated that the immersion diffractometry can measure a grating period of under one-half of the laser wavelength. A prism and an index-matching film are introduced to a traditional laser diffractometer without significantly changing the configuration. The prism raises the refractive index of the environment. Therefore, the new method can measure a smaller pitch. In this study, the minimum measurable grating pitch in theory is reduced to approximately one third of the laser wavelength. Future investigation will implement this technology with an excimer laser to measure grating standards with a pitch of less than 100 nm.