## Abstract

Focal plane testing methods such as the Shack–Hartmann wavefront sensor and phase-shifting deflectometry are valuable tools for optical testing. In this study, we propose a novel wavefront slope testing method that uses a scanning galvo laser, in which a single-mode Gaussian beam scans the pupils of the tested optics in the system. In addition, the ray aberration is reconstructed by the four-step phase-shifting measurement by modulating the angular domain. The measured wavefront is verified by a Fizeau interferometer in terms of Zernike polynomials.

© 2015 Optical Society of America

## 1. INTRODUCTION

Focal plane testing methods, such as the knife-edge test, the Ronchi test, the wire test, and the Hartmann test, are valuable in the field of optics and have a long history. The Hartmann method is one of the most-used methods due to its quantification capability. The Shack–Hartmann wavefront sensor (SHWS) improves upon the Hartmann testing method by using a micro-lens array coupled with a CCD camera. The SHWS is able to measure the wavefront dynamically, and thus is now frequently used in many applications.

The SWHS [1] is a robust tool with good performance, especially compared with the phase-shifting interferometer, and thus has been applied in many fields for position sensing [2] and ocular measurement [3]. The SHWS is constructed of a micro-lens array, which generates a focused spot on the CCD imaging plane. The lateral shift of each spot represents the wavefront slope of the corresponding lens. However, the measured wavefront lateral resolution is limited by the maximum possible number of lenslets given the CCD sensor size. Due to diffraction problems with the lenslet, a larger-sized lenslet is desired. As a consequence of the limited size of the micro-lens array, the SHWS does not offer a high lateral resolution of the wavefront like the interferometer. Spot assignment is critical in SHWS measurements. Any single instance of incorrect assignment of the focused spot from the lens could lead to a miscalculation or completely different measurement results.

Phase-shifting deflectometry has been shown to provide both sufficient lateral resolution and dynamic range [4,5]. Such testing methods use a sinusoidal grating object and a camera with a pinhole for modulating the aberrated beam. However, in the phase-shifting deflectometry methods, there is a trade-off between the camera aperture pinhole diffraction and the fringe visibility. In one of the deflectometry methods, the grating-slit test, the problem of the pinhole diffraction limit in one direction is improved by replacing the pinhole with a slit [6]. Therefore, this method is suitable for measuring the transverse ray aberration of an axial symmetrical lens system. Most of the time, the camera lens distortion has to be considered and carefully calibrated. The alignment of the sinusoidal fringe panel, pinhole, and tested optics is always an issue in phase-measuring deflectometry.

In this paper, we propose using a scanning-laser Gaussian beam deviated by a two-dimensional galvo mirror as an angular phase-modulating point light source. The Gaussian single-mode beam emitted from the laser diode is modulated as the sinusoidal fringes in the angular direction. In the testing process, the angular modulated Gaussian beam transmits throughout the tested optics. The aberrational fringes are then captured by a CCD camera. The regular four-step phase-shifting process is carried out to calculate the angular phase of the captured transverse ray aberration. From this, the complete ray information as a function of the CCD coordinates and pupil coordinates can be found, as well as the transverse ray aberration function. In the experiments, a commercially available laser-scanning projector is chosen to test the laboratory optics. A comparison of the measured spherical aberration with the results obtained with a Fizeau interferometer showed that this metrology method is promising.

## 2. THEORY OF MEASUREMENT

#### A. Transverse Ray Aberration Measurement

Focal plane testing methods were used to measure the wavefront slope, which was used to construct the wavefront aberration. The wavefront slope is, for the most part, measured from the quantified transverse ray aberration as a function of the exit pupil coordinate. The relation between the ray aberration and the wavefront aberration is shown in Fig. 1 and can be expressed as [7]

where $W$ is the wavefront aberration, TA is the transverse ray aberration between the positions of the actual and ideal rays on the observational plane, and $r$ is the radius curvature of the ideal wavefront.The exit pupil coordinates, $x$ and $y$, can be transformed into normalized coordinates $({\rho}_{x},{\rho}_{y})$ such that ${({\rho}_{x}^{2}+{\rho}_{y}^{2})}^{1/2}=1$ at the edge of the exit pupil. The transverse aberration is expressed in Eq. (2) below. Because $h$ is the radius of the exit pupil, the ratio of $r$ divided by $h$ is equal to twice the $F$ number:

The Zernike polynomial and least squares fitting technique are used in order to reconstruct the wavefront aberration [8]. In Eq. (2), $W$ is expanded by the Zernike polynomial [7], and can now be rewritten in matrix form as Eq. (3), where $d$ is the differential symbol, and $C$ is the coefficient of terms for the Zernike polynomial. This equation is in the form $Ax=B$, so the least squares fitting technique can be used to find the coefficient of each term, and then the wavefront aberration can be reconstructed by using the coefficients and the Zernike polynomial, as follows:

#### B. Laser-Scanning Pico Projector

The laser-scanning pico projector [9] is composed of three laser diodes of different wavelengths and a micro-electromechanical system (MEMS) mirror. The three available wavelengths in the pico projector are 440, 537, and 632 nm. A schematic diagram of the laser-scanning pico projector is shown in Fig. 2.

When the laser beam is reflected by the galvo mirror, the direction of propagation of the laser is changed two times by the rotational angle of the galvo mirror, but the light still comes from the same laser light source point. Given this property, the light from the projector can be regarded as a point light source with the modulation of the intensity in the angular direction. The modulation of the angular frequency is set according to both the size of the numerical aperture of the tested optics entrance pupil and the minimum angular resolution of the galvo mirror. We used an off-the-shelf laser pico projector with a spatial resolution of $848\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{step pixels}\times 480\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{step pixels}$, which corresponds to $43.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{angular deg}\times 24.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{angular deg}$. This means that each angular pixel has an extended angle of 0.051 deg or 3 arcmin.

#### C. Galvo Mirror Angular Stability Test

The galvo mirror system inside the laser projector plays an important role in accurately referencing the angular coordinates of the tested optics, so the angular stability is critical for the precise mapping of the transverse ray aberration. We therefore conducted a preliminary experiment to measure the angular stability of the galvo mirror system to ensure that it fulfilled the testing requirements.

The preliminary experimental setup was rather simple. We shined an arbitrary bright pixel onto a distant CCD camera 40 cm away. The CCD recorded the Gaussian beam over time, as shown in Fig. 3. The weighted centroid coordinates of the Gaussian beam were calculated over a 1 min period, with 10 samples measured.

According to the results, the root-mean-square stability error for a MEMS mirror 40 cm away is about 35 μm, which corresponds to about 18 arcsec, which is equivalent to 1/10 of an angular pixel. The stability error will determine the angular resolution of the entrance pupil of the tested optics. That means the system will not yield a measurement result with an angular pupil resolution better than 18 arcsec. This resolution is sufficient for most applications as long as the tested optics are not collimated in the object space with the object space’s numerical aperture near zero.

#### D. Gaussian Beam Propagation

Since a single-mode Gaussian beam propagates throughout the testing system, it is necessary to know the properties of the Gaussian beam during the testing process, especially in the CCD sensor plane, and the tested lens entrance pupil coordinates. The complete complex amplitude function of the Gaussian beam can be expressed as follows [10]:

When a Gaussian beam propagates through a lens with a focal length of $f$, its complex amplitude is multiplied by this phase factor. As a result, the transmitted wavefront curvature is changed, but the beam width is not altered.

The phase of the Gaussian beam before entering the lens is $kz+k{\rho}^{2}/2R(z)-\xi (z)$, while the phase of the transmitted wave is altered to $kz+k{\rho}^{2}/2R(z)-\xi (z)-k{\rho}^{2}/2f$. This can be rewritten as $kz+k{\rho}^{2}/2{R}^{\prime}(z)-\xi (z)$, where ${R}^{\prime}$ satisfies the imaging equation $1/R-1/{R}^{\prime}=1/f$. As a result, when the beam is transmitted through a lens, the beam waist radius can be expressed as a function of the magnification of the lens $M(z,f)$ [10], which depends on the focal length $f$, the position of the lens $z$, and the width emitted from the laser diode ${W}_{0}$. It can be expressed as follows:

After the beam propagates to the CCD, the new beam waist is located at a distance of ${z}^{\prime}$. The Gaussian beam width at the CCD is determined with the following equation [10]:

It is worth noting the width of the Gaussian beam on the CCD sensor plane has to be as small as possible to observe a better fringe contrast on the CCD plane. This guarantees better precision of the later phase-shift measurement. However, according to Eq. (5), a smaller beam width means a higher divergence angle, which corresponds to a larger beam area on the tested lens. A larger beam width on the tested lens will average out the transverse ray aberration at the location of interest. This will reduce the lateral resolution of the tested optics. Thus, one has to design the beam delivery system wisely so as to optimize the beam width over both the tested optics and the CCD coordinates. We are presently working on trying to find the best optimization process to obtain the best measurement results.

#### E. Angular Phase-Modulated Gaussian Beam

As shown in Fig. 4, an angular phase-modulated Gaussian beam is generated by the laser projector. The Gaussian beam is transmitted through the entrance pupil of the tested lens. A CCD camera is placed near the paraxial focus of the lens to record the distorted fringes. The lens magnification of the laser diode source to the CCD camera plane determines the width of the Gaussian beam on the CCD plane, as in Eq. (7). On the other hand, the tested optical aberration determines the location where the aberrated ray is incident on the CCD; therefore, the distortion of the fringes on the CCD plane is determined by the aberration of the system.

Since the Gaussian beam is modulated in the angular domain $(\alpha ,\beta )$, which expresses the angles between the beams and the optical axis, the exact angular position at the entrance pupil can be found using the standard four-step phase-shifting algorithm [11]. The phase measured on the CCD detector is therefore the angular position at the entrance pupil of the tested optics. The sinusoidal fringe signal is sent into the pico projector.

As can be seen in Fig. 5, when the Gaussian beam is modulated by the galvo mirror, it is modulated only in one direction. The intensity of the projected sinusoidal fringe ${I}_{A}$ in the angular domain can be expressed as in Eq. (8):

Assume that the lens magnification between the laser diode and the paraxial image conjugate magnification is $M$, and the CCD camera is placed near the paraxial image conjugate plane. By the energy conservation law, the intensity at the CCD coordinates $({\epsilon}_{x},{\epsilon}_{y})$ can be expressed as in Eq. (10):

With the presence of the optical aberration of the lens transfer function, there are transverse ray aberrations ${\mathrm{TA}}_{x}({\rho}_{x},{\rho}_{y})$. In addition, the CCD camera plane is not at the paraxial image plane. Therefore, the coordinates at the CCD plane $({\epsilon}_{x},{\epsilon}_{y})$ are a function of the transverse ray aberration and the amount of defocusing $\delta $, and can be obtained using

The measured transverse ray aberration is coordinated in the CCD pixel coordinates while the defocusing is the wavefront compensator, which needs to be removed.

Equations for the other orthogonal direction take the same form as Eqs. (8)–(11). From Eq. (10), we know that any phase-shifting process can lead to the solutions we are looking for [11], i.e., normalized pupil coordinate maps, which are also called angular maps, as a function of the transverse ray aberration. This also means that, since the fringes are captured on the CCD plane, the acquired quantiles are the ray data $\psi ({\rho}_{x},{\rho}_{y},{\epsilon}_{x},{\epsilon}_{y})$. The number of rays measured equals the number of pixels of the CCD camera illuminated by the scanning-laser fringes. Since the measured pupil coordinates $({\rho}_{x},{\rho}_{y})$ in the ray data $\psi $ are not uniform, we have to interpolate uniform mesh data out of the ray data $\psi ({\rho}_{x},{\rho}_{y},{\epsilon}_{x},{\epsilon}_{y})$ to find the transverse ray aberration as a function of the normalized pupil coordinates ${\epsilon}_{x}({\rho}_{x},{\rho}_{y})$ and ${\epsilon}_{y}({\rho}_{x},{\rho}_{y})$.

Finally, after removing the defocused terms and other alignment-induced aberrations, the transverse ray aberration map ${\mathrm{TA}}_{x}({\rho}_{x},{\rho}_{y})$ as a function of the pupil coordinates can be found. The standard path-dependent or modal methods can be used to construct the wavefront from the wavefront slope data in both orthogonal directions, as shown in Eq. (2).

## 3. EXPERIMENTAL SETUP

#### A. System Setup and Tested Samples

The experimental setup for the optical testing experiments is shown in Fig. 6.

We chose three different laboratory lenses to test. Their descriptions are shown in Table 1. Each tested lens was set up on the stage with enough degrees of freedom for alignment to minimize any potential off-axis aberration due to alignment errors.

#### B. Measurement Process

The measurement steps are shown in Fig. 7. A screen of parallel sinusoidal fringes is generated and sent to the projector. The camera records the distorted fringes as shown in Fig. 8. After the fringes are shifted by a 1/4 angular period for four steps, the process is then repeated for the other orthogonal direction. A standard four-step phase-shifting equation is applied to calculate the fringe phase recorded on the CCD plane. After the phase is unwrapped, we can obtain the four-dimensional ray data $\psi ({\rho}_{x},{\rho}_{y},{\epsilon}_{x},{\epsilon}_{y})$. Since the measured pupil coordinates $({\rho}_{x},{\rho}_{y})$ in the ray data $\psi $ are not uniform, we have to interpolate to find the transverse ray aberration as a function of the uniform and normalized pupil coordinates ${\epsilon}_{x}({\rho}_{x},{\rho}_{y})$ and ${\epsilon}_{y}({\rho}_{x},{\rho}_{y})$. With two orthogonal transverse ray aberration maps, the wavefront aberration without tilts and defocusing can be reconstructed by Eq. (3).

The angular maps for ${\rho}_{x}({\epsilon}_{x},{\epsilon}_{y})$ and ${\rho}_{y}({\epsilon}_{x},{\epsilon}_{y})$ are then solved by the four-step phase-shifting algorithm, as shown in Fig. 9(a). After the phase unwrapping process, it becomes a smooth and continuous function. The result is shown in Fig. 9(b). Note that in Fig. 9(b), the scale is divided by $2\pi $ so that the fringes are shown across the pupil peak to valley. The four-dimensional ray data $\psi ({\rho}_{x},{\rho}_{y},{\epsilon}_{x},{\epsilon}_{y})$ is then the combination of the two orthogonal unwrapped angular phase maps.

The positions of the Gaussian beams on the pupil and on the CCD are modulated in the ray data. Therefore, the transverse ray aberration as a function of the normalized pupil coordinates can be solved with the ray data. However, the normalized pupil coordinates do not uniformly mesh in the ray data, so it is necessary to interpolate for the transverse ray aberration as a function of the uniform, normalized pupil coordinates. The transverse ray aberration maps are shown in Fig. 10.

Finally, the wavefront aberration can be reconstructed by using Eq. (3) with the two orthogonal transverse ray aberrations. The wavefront aberration, with the tilt and defocusing terms removed, is shown in Fig. 11, prevailing the dominating spherical aberration.

## 4. EXPERIMENTAL RESULTS AND DISCUSSION

To verify the measurement accuracy and precision, we set up a Fizeau interferometer and used the retro-reflective method at the same finite conjugate position. A schematic diagram of the Fizeau interferometer setup is shown in Fig. 12.

In the Fizeau interferometer, the reference lens is set up to generate converging test beams. The tested wavefront converges to a point coincident with the conjugate, as we tested the lens with the scanning galvo laser. Therefore, the distance between the focal point and the lens to be tested is the same as the distance between the projector and lens undergoing testing in the experiments. The light propagates to the retro-reflective ball and returns to the light source along the same path. This makes the wavefront measured double the proposed scanning galvo laser measured.

Therefore, after the reconstruction of the wavefront, the spherical aberration term of the Zernike polynomial is compared with the one measured by the Fizeau interferometer. The results are shown in Table 2.

As can be seen in Table 2, the Fizeau and experimental results are close, but do not match each other. One of the problems found during the testing is the proper definition of the edge of the aberrated beam footprint on the CCD plane. Therefore, the measurement fiducial mask cannot be defined without a sharp pupil edge. Any slight differences in the definitions of the radius of the fiducial mask could have a phenomenal impact on the spherical aberration of the Zernike polynomials.

## 5. SUMMARY

In summary, we demonstrated a novel wavefront measurement device produced with an off-the-shelf scanning-laser pico projector. With the scanning galvo laser, the Gaussian beam is projected into the entrance pupil of the lens undergoing testing, forming the distorted fringe pattern in the image space. Complete ray information as a function of the exit pupil and the image plane coordinates is acquired. Moreover, we can calculate the transverse ray aberration, from which the wavefront aberration can be easily reconstructed by the least squares fitting technique. The measurement performance of the system, which is verified by a comparison with results obtained with the Fizeau interferometer, showed that this metrology method is promising.

## Funding

Ministry of Science and Technology, Taiwan (MOST) (103-2221-E-008-050).

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