## Abstract

The Acousto-Optic Tunable Filter (AOTF) is a high-speed full-field monochromator which generates two spectrally filtered light beams with ordinary and extraordinary polarization state. The angle of diffracted light in an AOTF changes according to the scanning of wavelength, which causes an image shift on a CCD plane. An analytic design of a prism system to compensate for the angular-shift error is proposed in this paper. Analysis of light paths in a prism and experimental results verify a proposed compensation method. Experimental results agreed with simulation results based on the suggested prism model. Angular-shift errors of ordinary and extraordinary rays are simultaneously minimized at optimal conditions with the designed prism.

©2008 Optical Society of America

## 1. Introduction

Demands on nano metrology to measure nano or micro size samples in industrial fields such like semiconductor or display, biotechnology and medical science have greatly increased with the development of nano technology. White-light Scanning Interferometry (WSI) is an optical measurement method to measure images using a moving-mirror [1]. WSI produces a short coherence interferogram that measures the optical path difference between test and reference beams without 2π ambiguity. WSI is limited by the speed at which the mirror is moved. Spectral Domain Interferometry (SDI) uses the spectrum of the interferogram to measure thickness and shape information of the sample. SDI has advantages of greater sensitivity and acquisition speed, because there are no moving parts that can generate the mechanical errors including vibration and backlash. An Acousto-Optic Tunable Filter (AOTF) controls the wavelength of transmitted light using an electric signal. As such, AOTF benefits from rapid control speed. AOTF is widely used as a spectral component for spectral domain interferometry [2-7]. However, the diffraction angle between diffracted light that goes through the AOTF and a propagating axis changes according to wavelength of diffracted light. The image shift phenomenon is generated in the Charge-Coupled Device (CCD) plane due to this angular-shift error. Composing the spectral imaging system of high speed and resolution are limited by this phenomenon.

Several studies have been carried out to solve this problem. Angular-shift error was compensated by using an exit plane wedge angle and locating a prism which has the same material with an AOTF at the exit part [8-9]. However, only one ray of ordinary ray or extraordinary ray generated by AOTF was compensated. The use of an elliptical mirror was also proposed [10]. This method is difficult to apply to an actual optical system because the design of elliptical mirror is difficult, manufacturing cost is high, and the effect on angle compensation of two diffracted light is remarkably decreased when the alignment of the elliptical mirror is not good. An image processing algorithm was used to compensate for angular-shift errors [2]. The compensation of angular-shift error needs to be repetitively done whenever the system is changed. Error is generated when the compensation factor is not corrected.

Signal to noise ratio (SNR) is improved as the ordinary ray and extraordinary ray are simultaneous used, because the amplified signal is obtained by overlapping the signals of two rays. The measurement system is more efficient and has multiple modes using a characteristic of AOTF that the ordinary ray and extraordinary ray are each orthogonally polarized. A prism system is suggested to simultaneous compensate the ordinary ray and extraordinary ray. Angular-shift phenomenon is minimized using the spectral feature of the prism and information about diffraction angles of AOTF.

## 2. Method

#### 2.1 Optical characteristics of a prism

Prisms are mainly classified into dispersing prisms and reflecting prisms. Dispersing features of a prism are used to compensate angular-shift errors.

A ray entering a dispersing prism (Fig. 1,) emerges having been deflected from its original direction by an angle *δ* known as the angular deviation. At the first refraction, the ray is deviated through an angle (i
_{1}-i
_{1}’), and at the second refraction, it is further deflected through (i
_{2}’-i
_{2}). The total deviation is:

If the prism index is n and it is located in air, the final refraction angle, i
_{2}’, is expressed as:

$$={\mathrm{sin}}^{-1}\left[\left(\mathrm{sin}x\right){\left({n}^{2}-{\mathrm{sin}}^{2}{i}_{1}\right)}^{1\u20442}-\mathrm{sin}{i}_{1}\mathrm{cos}x\right]$$

Snell’s Law and trigonometry were used to derive Eq. (2). (Where, x is apex angle.)

The angular deviation is finally simplified from Eq. (1), Eq. (2) and by geometrical analysis of a prism:

The value, *δ*, is a function of a prism index, n from Eq. (3). As a result, the light which enters a prism is dispersed ay a different angle according to the wavelength, because n is a function of the wavelength of the light [11].

A diffraction equation for AOTFs is expressed as:

(Where Δ θ d: deflection angle, θ i: incident angle, Δ n_{b}: refractive index, θ _{a}: angle of an acoustic wave)

The diffraction angle of white-light going through an AOTF increases as the wavelength of diffracted light becomes shorter because a refractive index, Δ n_{b} is inversely proportional to wavelength. AOTFs have uniaxial birefringent crystal, so they display two distinct principal indices of refraction. The diffraction angles of ordinary ray and extraordinary ray are differed as extraordinary ray has much larger refractive index than ordinary ray.

The dispersion angle of a prism increases as the wavelength of refracted light gets shorter (Eq. (3)) because the wavelength is approximately inversely proportional to the refraction index of a prism. Thus, the dispersion phenomenon according to the wavelength is compensated as the prism system is located at the back of an AOTF.

## 2.2 A prism system

A simple prism system, (Fig. 2,) is suggested in this paper to simultaneous compensate the angular-shift errors of ordinary and extraordinary rays of an AOTF. A prism is symmetrically located about the propagating axis of an AOTF, and both bottom side angles of a prism, A and B, are designed to have different values from each other because the diffraction angles of ordinary ray and extraordinary ray do not exactly correspond according to difference of refractive indices between ordinary ray and extraordinary ray.

The final refraction angles of two rays going through a prism and the AOTF have almost uniform values without regard to wavelength, as both A and B are optimized. Thus, the image shift phenomenon is minimized in CCD plane at these values.

The direction of extraordinary ray, (Fig. 3) is changed when the extraordinary ray enters with diffraction angle of θ about the propagating axis to a suggested prism. The incident angle about a prism system is defined as:

The final refraction angle of a prism is obtained by the light trace that is based on Snell’s law and geometrical analysis. The final refraction angle is given by:

This value, i_{2}’, is given by function of a prism index and the bottom side angle of a prism from Eq. (6). The final refraction angle of ordinary ray is derived using a similar method as:

The dispersion angle is minimized at whole wavelength ranges as the both bottom side angles of a prism are optimally designed.

The direction of extraordinary rays undergoes a change when the extraordinary rays enter with angle of θ about the propagating axis and the tilting angle of prism is α (Fig. 4).

The incident angle about a prism is defined as:

The final refraction angle of a prism is given by:

The value, i_{2}’, is given by function of a tilting angle, α where the prism index and the bottom side angle are fixed. The final refraction angle of ordinary rays is derived as:

The dispersion angle is minimized at full wavelength range where the tilting angle is appropriately selected.

## 2.3 Diffraction angles

The diffraction angles according to wavelength of ordinary and extraordinary rays are measured preferentially to compensate the variation of diffraction angles for the two rays that go through the AOTF. The diffraction angles were measured using a simple optical system setup (Fig. 5).

A tungsten halogen lamp (60W) was used as a light source and along with an AOTF of Brimrose Corporation (TEAF10-40-65). A mirror was located in front of the AOTF to align the axis of the AOTF with the propagting axis. A polarizer was used to separate ordinary and extraordinary rays and modulate the intensity of radiation.

The distance between diffracted rays and undiffracted rays on the screen and the distance from the exit plane of the AOTF to the screen were measured. The diffraction angles of ordinary and extra ordinary rays were calculated using distance information.

## 3. Results

The wavelength of diffracted light was changed from 614.4230nm to 456.1140nm when the driving frequency of the AOTF changed from 120MHz to 180MHz. Diffraction angles according to driving frequency were obtained (Fig. 6). The diffraction angles were linearly increased in proportion to driving frequency. The diffraction angle was 6.021° at 120MHz (614.723nm) and 6.684° at 180MHz (456.114nm) for ordinary rays. The diffraction angles of extraordinary rays were 0.055° larger than that of ordinary rays at full spectral range because extraordinary ray has much larger refractive index than ordinary ray. The spectral image of extraordinary ray on the CCD was shifted as the diffraction angles of two rays were changed (Fig. 7). The sample that was filled with SiO2 thin film above patterns was used to check the image shifting phenomenon in the CCD. The image of extraordinary ray was shifted 244 pixels in the direction of one axis on the CCD and the divergence of diffraction angles was about 0.56° when the driving frequency of AOTF was changed from 120MHz to 170MHz. The image of ordinary ray was also shifted 244 pixels in the direction of one axis on the CCD at same spectral range.

The angular-shift error was minimized when the angle deviation between the final refraction angles of light at the shortest wavelength and the longest wavelength was zero. The wavelength bandwidth of light source ranged from 473.795nm to 614.723nm. The final refraction angles of light at 473.795nm and 614.723nm were obtained from Eqs. (8)-(9) and diffraction angles were measured according to wavelength. The angle deviation between refraction angles of two wavelength was simulated and the optimal tilting angle was determined when a BK-7 prism, having the bottom side angles of 45°, was used (Fig. 8).

Angle deviations of extraordinary and ordinary rays were zero where each tilting angle of a prism was +3.337° and -3.274°, respectively from simulation results. The optical system was set to compare simulation results with actual experimental results (Fig. 9). A BK-7 prism having bottom side angles of 45° was located at the back of AOTF and the tiling stage was used to minutely tilt the prism. The experimentally determined image shifting phenomena in the CCD were minimized where each tilting angle of a prism was about +3.21° and -3.13° for extraordinary and ordinary rays, respectively. These tilting angles were very similar to optimal values obtained from simulation results.

Each bottom side angle of a prism was designed using these results. The angular deviations at optimal angles of α obtained from experiment were selected in simulation results. The optimal bottom side angle was fitted to these angle deviations from Eqs. (6) and (7). Each bottom side angles were 46.57° and 46.62° for extraordinary and ordinary rays, respectively. A specific prism was designed using these values (Fig. 10).

We noticed that dimensions of a prism don’t affect on compensation of angular-shift error from Eqs. (6) and (7), so each dimension is selected to minimize a prism size without any loss of lights in a prism and at the interface between air and prism.

Image shifting phenomena were measured for the designed prism that was perpendicularly located with a propagating axis at the back of the AOTF. Each image of extraordinary and ordinary rays was shifted by only 6 pixels and 5 pixels and the divergence of diffraction angles by 0.015° and 0.012° respectively, which the driving frequency was changed from 120MHz (643.723nm) to 170MHz (473.795nm) (Fig. 11).

Our results are compared with other methods that use no shifting compensation hence produces shifting in the images of extraordinary and ordinary rays by 244 pixels in the CCD and divergence of diffraction angles by 0.562. Also, our resulting angular divergences were still smaller than 0.04°obtained by Yano and Watanabe [8] and 0.03° compensated by Wachman et al [9].

## 4. Discussion

Final refraction angles of ordinary and extraordinary rays going through a prism are function of a bottom side angle, A(or B) and tilting angle, α of a prism from Eqs. (9) and (10), so final refraction angles are varied when bottom side angles of a prism are changed or a prism was tilted about the propagating axis. One prism with a specific bottom side angle was used instead of using many prism sets with various bottom side angles to compare experimental results with simulation results.

Image shifting phenomena of the AOTF were compensated using a designed prism. Each image of extraordinary and ordinary rays was shifted 6 pixels and 5 pixels, respectively, when bottom side angles of each prism were optimized. The angular-shift error was not perfectly compensated because of error factors, such as the misalignment error of the optical system used to find optimal angles of a prism, inaccuracy of used diffraction angles, manufacturing errors of prisms and a medium inhomogeneity of designed prisms. However, total 6 pixel and 5 pixel errors are not significant values comparing to 244 pixels obtained without a prism. Total 6 pixel errors in the CCD are approximately equivalent to the divergence of diffraction angles within 0.02°.

Intensities of ordinary and extraordinary rays might be reduced by reflection effects generated at the interface between the air and a prism. Transmittances of ordinary and extraordinary rays going through the designed prism were 0.94 and 0.82, respectively from simulation results using the Fresnel Equation [12]. Therefore, the closer these transmittance values become 1, less important they become in consideration for angular-shift error compensation by the proposed prism design.

Ordinary and extraordinary rays are used at the same time to obtain information about a measured target, because angular-shift phenomena of the two rays are simultaneously compensated by the designed prism. This suggested method can be widely used for SDI and optical coherence tomography (OCT) systems using AOTFs [13].

In conclusion, our optimized prism system was designed to compensate for angular-shift errors in the AOTF. Analytic simulations of light paths and experimental results prove the validity of proposed compensation method. The angular-shift errors of ordinary and extraordinary rays were minimized by the designed prism.

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