1. Introduction

Tunable Fabry-Perot (FP) interferometers fabricated from lithium niobate wafers are becoming increasingly popular for imaging spectroscopy of the solar disk [1,2]. In order to meet the exacting requirements of solar observation, these etalons need to be fabricated with physical thickness variations of 1 nm rms or better across their working aperture [3]. This in turn places very high requirements on the metrology used to measure the wafers during fabrication. This publication presents results from measurements of a fabricated wafer and demonstrates that sub-nanometer repeatability and reproducibility can be achieved from relatively simple metrology equipment [4].

2. Theory

The measurement system uses the high finesse of a FP etalon structure to enhance the resolution of the measurement, and the angular dependence of the transmission function to infer variations in optical thickness.

The transmission function of an FP etalon is given by the Airy function [5]:

where R is the reflectivity of the etalon mirrors (assumed equal), d is the physical separation between the mirrors, n is the refractive index of the medium between the mirrors, λ is the wavelength of the incident illumination, and θ_i is the angle between the normal to the surface of the etalon and the transmitted ray inside the etalon. To match the function to observable values, the internal angle is usually converted to the external angle of incidence using Snell’s law. The effects of phase change on reflection have been ignored in this instance since the effect on measured thickness over the angular range of interest (~2.7° + 0.1°) is less than 0.017 nm, determined using commercially available thin film analysis software [6].

 

Fig. 1. Normalised etalon transmission as a function of external angle of incidence for mirror reflectivities of 75%, 85%, and 95%.

Download Full Size | PDF

Figure 1 shows a typical transmission function for a 280 μm thick FP etalon with refractive index = 2.28 (matched to the refractive index of the ordinary ray of Z-cut lithium niobate) as a function of external angle of incidence. The graph shows plots for R = 75, 85, and 95% mirrors, demonstrating that the peaks in the transmission function become sharper as the mirror reflectivity, or cavity finesse, increases. Since the technique relies on accurate location of the transmission peaks with angle, this sharpening of the peaks allows the resolution of the scanning technique to be modified by varying the mirror reflectivity. Note also the symmetric nature of the response centred around normal incidence, which allows the position of normal incidence to be accurately identified, as will be discussed below.

3. Wafer metrology system

The wafers were measured using the equipment shown in Fig. 2. The components were mounted on an optical rail on a standard laboratory bench. No environmental control other than the buildings air-conditioning system (20°C ±0.5, 50% relative humidity) was required during the measurements reported in this work.

Prior to measurement, the wafer was coated on both sides with a thin layer of silver to give surfaces of approximately 95% reflectivity in order to form the necessary FP etalon structure. The wafer was then mounted on a high-resolution rotation stage between the collimating lenses. Laser light from a wavelength stabilized Helium Neon (HeNe) laser (fabricated in-house by the National Measurements Laboratory [7]) was launched via a single-mode optical fibre patch lead towards the first collimating lens, through the etalon, and then via the second lens to the digital video camera. The optical magnification and pixel count of the camera gave a spatial resolution of 0.39 mm. Note that this spatial resolution is compromised slightly as the angle of incidence increases since the multiple reflections inherent in a FP structure tend to smear out the incoming beam over a finite horizontal area. This effect is dependent on the finesse of the FP cavity and can be approximated by determining the number of reflections needed to reduce the beams intensity by an order of magnitude. For the wafer under test, this gives a spatial resolution of approximately 310 μm at an angle of incidence of 3°; therefore the true spatial resolution in this instance is closer to 0.6 mm. The precise rotation of the etalon was achieved using a micro-stepping motor with angular resolution of 0.00008° (Compumotor LE57-51-R14 micro-stepping motor coupled to a Line and Precision rotation stage). The wafer was adjusted to approximately normal incidence by observing the back-reflected spot and adjusting the tilt and yaw of the etalon so that the spot was coincident with the tip of the input fibre.

During the measurement, the etalon was first scanned in angle about an axis perpendicular to the optical axis of the collimating lenses, over a sufficient range to give at least two transmission fringes on either side of normal incidence. A low angular resolution was used for the initial scan to reduce overall measurement time. An image of the transmitted light was taken for each angular step in order to build up a transmission profile for each elemental section of the wafer. After the scan, the symmetry of the transmission function was used to determine the exact angular position of normal incidence for each pixel so that the effect of distortion of the wafer can be accounted for. This was done by finding the location of the mid-point between the two outer-most peaks of the transmission function for each pixel of the camera detector array - note that distortions in the horizontal plane have an increasing effect on the measured thickness as the angle of incidence increases and so must be taken into account, whereas the effect of distortions in the vertical plane remains relatively small and constant over the entire scan range. For example, the wafer used for this work had an approximately spherical distortion with maximum distortion angle of 0.06° which, in the horizontal plane creates a thickness error of approximately 5 nm at an angle of incidence of 3°, whereas the same distortion in the vertical plane introduces an error of less than 0.03 nm regardless of angle of incidence and so has been ignored in this analysis. A detailed description of distortion induced effects will be published separately. One of the inner peaks was then selected and a high resolution scan was performed over this peak only. The optical thickness of each elemental section of the wafer was then determined by re-arranging Equation 1 for thickness and using the angle at which maximum transmission occurs for each pixel of the camera detector array. At this point, the argument of the Sine term is equal to Nπ where N is the order of the transmitted fringe, and so the variation in optical thickness can be determined accurately by correctly setting the fringe order. The value of N is determined by measuring the physical thickness of the wafer independently of the angle scan measurement. This is done using an electronic dial gauge with a resolution ~±0.1 μm. In practice this means that the absolute value of substrate thickness can have quantised values equivalent to N ±1; however, since it is usually more important to critically measure only the relative variation in substrate thickness, this is acceptable in most situations. Once the optical thickness variation of the substrate is known, the physical thickness variation can be inferred using a known value of the refractive index of the material, which, assuming homogeneity, can be determined using a Sellmeier equation for the material being used.

 

Fig. 2. Wafer scanning rig.

Download Full Size | PDF

4. Wafer measurement

Using this technique, the thickness variation across the working aperture of a precision polished and optically corrected Z-cut lithium niobate wafer was measured repeatedly over a period of 4 days. The wafer was measured with the Z-axis pointing toward and away from the video camera and also after an azimuthal rotation of 90 degrees in the clockwise direction. For each orientation, a series of 4 measurements were taken and the thickness variation maps for each pair of measurements were compared. The rms repeatability between all possible pairs of each set of 4 had an average of 0.047 nm and a worst case value of 0.07 nm. An average thickness variation map was then generated from each set of four consecutive measurements, and in all cases, after suitable software-based re-orientation, rotation, and magnification of the images, the reproducibility of the measurements was ≤0.16 nm rms over the working aperture of 37.5 mm. The definitions of ‘repeatability’ and ‘reproducibility’ were taken from the ISO guide to the expression of uncertainty in measurement [8]. Figures 3, 4, and 5 show maps of the averaged thickness variation generated in all three orientations mentioned. The rms and absolute difference values between each map are given in Table 1. Figure 6 shows the difference function between Figs. 3 and 5 after flipping Fig. 5 along the vertical axis.

 

Fig. 3. Averaged measurement of the lithium niobate wafer in the ‘0° positive Z’ orientation.

Download Full Size | PDF

 

Fig. 4. Averaged measurement of the lithium niobate wafer shown in Fig. 3 in the ‘90°’ orientation.

Download Full Size | PDF

 

Fig. 5. Averaged measurement of the lithium niobate wafer shown in Fig. 3 in the ‘0° negative Z’ orientation.

Download Full Size | PDF

 

Fig. 6. Difference function between Figs. 3 and 5 after the image of Fig. 5 had been flipped along the Y-axis.

Download Full Size | PDF

Table 1. rms and absolute difference values between the maps shown in Figs. 3, Figs. 4, and Figs. 5.

View Table

5. Conclusions

A series of measurements of the thickness of a precision polished and optically corrected lithium niobate wafer has been carried out using a custom built angle scanning rig. The measurements show that the rig is capable of achieving a repeatability of ≤0.07 nm and a reproducibility of ≤0.16 nm rms without the need for specific environmental control. The enhanced resolution of this equipment is achieved by using the multiple beam interference inherent in a high finesse Fabry-Perot structure.