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The pyramid wavefront sensor is known for its high sensitivity and dynamic range that can be tuned by mechanically altering its modulation amplitude. Here, a novel modulating digital scheme employing a reflecting phase only spatial light modulator is demonstrated. The use of the modulator allows an easy reconfigurable pyramid with digital control of the apex angle and modulation geometry without the need of any mechanically moving parts. Aberrations introduced by a 140-actuator deformable mirror were simultaneously sensed with the help of a commercial Hartmann-Shack wavefront sensor. The wavefronts reconstructed using the digital pyramid wavefront sensor matched very closely with those sensed by the Hartmann-Shack. It is noted that a tunable modulation is necessary to operate the wavefront sensor in the linear regime and to accurately sense aberrations. Through simulations, it is shown that the wavefront sensor can be extended to astronomical applications as well. This novel digital pyramid wavefront sensor has the potential to become an attractive option in both open and closed loop adaptive optics systems.
© 2013 Optical Society of America
1. Introduction
The Hartmann-Shack (HS) wavefront sensor [1] is a widely used aberration measurement device with applications that range from astronomy to vision science and nonlinear optics [2–4]. The pyramid wavefront sensor (PWS) first proposed by Ragazzoni [5] has been applied in astronomy [6,7] and vision science [8,9]. It is made up of a pyramidal prism, with the vertex angle (α) just above 0° such that the four facets deflect the light beam into four pupils as shown in Fig. 1. In order to achieve higher sensitivity and adjustable dynamic range, there is a need to modulate the pyramid. Either the refractive pyramid is physically moved to achieve modulation [5] or the beam is dynamically controlled using beam steering optics [8]. Although the PWS has advantages over HS, the need for moving parts makes this wavefront sensing device less robust. The common feature of the HS and PWS is that each of them estimate the local slopes of the wavefront and the wavefront shape is then estimated using reconstruction procedures. The number of local slopes obtained with a HS is determined by the number of microlenses. In contrast, in the case of the PWS, the local slopes of the wavefront can be obtained by an appropriate linear combination of pupil plane images [6] and hence the number of local slope measurements depends on the number of pixels that each of the four pupils occupy. In addition, the number of measurements made during one modulation has an effect on the sensing accuracy. The modulation amplitude plays a critical role in PWS. As it increases, the dynamical phase range over which the performance of the sensor can be reliably used increases [10,11]. In turn, a smaller modulation increases the sensitivity of the wavefront sensor [6, 11, 12]. One of the primary disadvantages of the PWS is that it typically needs an additional device for modulation although beam diffusers can potentially be used to avoid it [13] and the need for modulation has been questioned [14,15]. Alternatively, using a large apex angle axicon, wavefront sensing could be performed [16]. In this paper, we demonstrate that by imposing a pyramid like phase profile on an 8-bit spatial light modulator (SLM), wavefront sensing can be performed without the need for mechanically moving parts.
2. Experimental setup
The experimental setup used to test the SLM based PWS is shown in Fig. 2. A linearly-polarized 5 mW He-Ne laser beam was spatially filtered and collimated. A 140-actuator deformable mirror (DM) from Boston Micromachines™, placed in the front focal plane of a 1000 mm lens, was used to introduce aberrations in the optical system. The aberrations introduced were sensed using a commercial HS wavefront sensor placed in the conjugate plane of the DM. The SLM (Hamamatsu, LCOS SLM X10468, with a pixel pitch of 20 μm) was placed such that the lens of focal length 1000 mm focuses the beam at the center of the SLM display. The beam diameter was limited to 4 mm by an iris. This long focal length was required to avoid superposition of different diffraction orders caused by SLM pixelation. However, an SLM with a smaller pixel size could potentially reduce the overall size of the system. A beam splitter was used to redirect the light reflected from the SLM and four identical pupils were obtained by addressing the pyramid phase onto it. Using beam resizing optics, the pupils were imaged on a CCD camera. All the lenses in Fig. 2 are antireflex-coated achromatic doublets.
Fig. 2 Experimental setup to demonstrate a digital pyramid wavefront sensor: all lenses are achromatic doublets and the focal length of the lenses is shown in mm units.
3. Background
The pyramid transmittance function can be written as [17],
where ‘c’ is a constant factor that can be used to maximise the phase at the center of the wrapped pyramidal phase, T(X, Y). ‘β’ is a constant factor that defines the apex angle of the phase only pyramid. In order that the four pupil images fit within the CCD camera (see Fig. 2), the value of β was chosen such that the divergence angle equals 0.41°. H represents the Heaviside function that takes a value 1 when both arguments are either greater than or equal to zero. In other cases, the Heaviside function takes a value 0.Although a convolution method could be used to simulate the pupil plane images [17] using the defined pyramid transmittance function (Eq. 1), we adopted the Fast Fourier Transform technique [10]. Consequently, the pupil plane images can be calculated,
where ‘ϕ(x, y)’ is the wavefront aberration in the system, FT represents the Fourier Transform function and P(x, y) is the defined pupil function. A simulated intensity image corresponding to a defocus aberration and simulated pyramid phase are shown in Fig. 3.Fig. 3 (a) Simulated aberrations (in μm units): (top to bottom) No aberrations, defocus ( ), astigmatism ( ), coma ( ), secondary astigmatism ( ) and a combination of defocus and astigmatism aberrations ( ) and corresponding (b) simulated pupil plane images, (c) simulated x-slope matrix, (d) simulated y-slope matrix and (e) reconstructed wavefronts (in μm units). RMS wavefront reconstruction error for sensing these reconstructed aberrations (top to bottom): 0.01, 17.06, 22.12, 15.10, 20.93 and 30.43 nm. The generated aberrations have a peak-to-valley of 0.50 μm. The modulation amplitude is 0.80 mm. The reconstructed peak-to-valley values are shown above the plots. The slope values are in × 10−3.
3.1. Wavefront slope
The ‘x’ and ‘y’ slopes of the wavefront ϕ(x, y), Sx(x, y) and Sy(x, y) can be approximated by using the following relations defined from the recorded individual pupil intensities for the four pupils (see Fig. 1).
Here, ‘f’ is the distance between the lens aperture and the SLM and ξ is the magnitude of circular modulation. The wavefront was reconstructed from the calculated slopes using the slope geometry of Southwell [18].3.2. Wavefront estimation
The wavefront was estimated using a zonal reconstruction method from the calculated slope values (see Eqs. 3, 4). Following the slope geometry of Southwell [18], the phase is related to the slopes in the following manner for N2 phase points,
where i = 1, 2,..., N−1 and j = 1, 2,..., N and h is the ratio of the width of the aperture to the square root of the number of phase points. where i = 1, 2,..., N and j = 1, 2,..., N−1. Using matrix formulation, the relation between the slope measurements and the phase values defined in Eqs. 5 and 6 can be written, DS = Aϕ̂, where D and A are sparse matrices. S is a vector containing ‘x’ slopes followed by ‘y’ slope values. ϕ̂ is a vector containing the phase values of the estimated wavefront. Since ‘A’ is not a square matrix by definition, it is not possible to calculate its inverse to estimate ϕ̂. Hence, AT is multiplied on either side, i.e. AT DS = AT Aϕ̂. To overcome singularity problems with AT A, a singular value decomposition technique is used to write AT A = UΛTV. Finally, the estimated phase can be written as:3.3. Experimental methodology
The center of the pyramidal phase is moved along a circular path on the SLM to achieve modulation. The intensity at the plane of the four pupils, Ipyr(x, y) was thereby obtained by superposing 36 pupil plane images, captured at equi-spaced modulation angles. For small modulation amplitude, the discretization in the pixels causes a step-like modulation along the boundary of a circle. Finally, the wavefronts were back reconstructed using Eq. 7.
Primary defocus aberration ( ), primary astigmatism ( ), primary coma ( ), secondary astigmatism ( ) and a combination of primary defocus and primary astigmatism were introduced using the deformable mirror. The introduced aberrations were sensed simultaneously using the HS and the PWS. To achieve the desired beam divergence angle of 0.41°, the apex angle of the pyramid was fixed at 179° for all the measurements presented here. This required modulo 3π phase wrapping for every 5 pixels on the SLM as shown in Fig. 4. It can be noted that the use of the SLM allows flexibility in the choice of the apex angle, which controls the spacing between the pupil images.
Fig. 4 A portion of the phase map addressed on the SLM to simulate the pyramid phase. The five grayscale steps used in this map are 255, 191, 128, 64 and 0; starting from the center, moving outward.
3.4. Simulations
Simulations were performed in order to compare the obtained experimental results with theoretical estimates. Also, the trade-offs between sensitivity and dynamic range can be studied by varying the radius of circular modulation and phase magnitude of the aberrations. Wavefront aberrations defined in terms of Zernike polynomials ( , n is the radial index and m is the azimuthal index) were used:
Here, α defines the magnitude of the aberration. In the simulations, using Eq. 2, the pupil plane intensity was obtained by adopting circular modulation and averaging 36 pupil plane images for a given modulation amplitude. A pyramidal phase identical to that employed in the experiments was used. The wavefront error was calculated in terms of the root mean square (RMS) value between the estimated wavefront (Eq. 7) and the actual wavefront (Eq. 8), where ϕ is the initially simulated aberration defined in Eq. 8 and ϕ̂ is the estimated wavefront, reconstructed from the pyramid wavefront sensor. To quantify linearity of the wavefront sensor, peak-to-valley (PV) measurements were employed, while sensing introduced defocus aberrations of different magnitude.4. Results
Wavefronts reconstructed using the PWS are comparable with those obtained using the HS. Figure 5 shows a comparison of the simulated aberrations that were introduced by the deformable mirror (in closed-loop with the HS), corresponding HS measured wavefronts and the wavefronts reconstructed using the PWS. The comparison shown here is for the case of a modulation radius equal to 0.4 mm. It is noted that there is a small tilt detected by the PWS, more significantly visible in the cases of primary defocus and primary astigmatism. This small tilt is attributed to the slight offset of the pyramid apex with respect to the aberrated point spread function. The tilt aberration arising from this positional offset can be reduced by adjusting mirror M7 such that all four pupils assume equal intensities when no aberration is addressed on the deformable mirror. However, even a sub-pixel offset of the pyramid phase could lead to a significant tilt in the detector plane of the pupils. Although this tilt aberration can be removed using high precision adjustable mounts, it was unavoidable in our optical setup. More discussion on this tilt is presented in the later part of this section.
Fig. 5 A comparison of the wavefronts reconstructed using the novel digital pyramid wavefront sensor (modulation radius = 0.4 mm) with that measured by the HS.
Modulation radius affects the accuracy of wavefront sensing with a PWS. Figure 6 shows the variation of RMS wavefront error, predicted from simulations, as a function of modulation radius for different magnitudes of primary defocus ( ) aberration. As the magnitude of the aberration increases, the smallest modulation radius for which the RMS wavefront error is minimum, drifts towards higher values. This evidence necessitates modulation in order to achieve good accuracy in open loop adaptive optical wavefront sensing and also assists in faster convergence in closed loop adaptive optical systems [19]. Figure 6 illustrates that a smaller modulation radius has an increased sensitivity to low magnitude aberrations. An example of closed loop correction with the PWS is shown in Fig. 7.
Fig. 6 Simulations: Influence of the modulation amplitude on the RMS wavefront error while sensing primary defocus aberration of different peak-to-valley (PV).
Fig. 7 Simulation: RMS wavefront error in a closed loop adaptive optics with a pyramid wavefront sensor.
Figure 8(a) shows a plot of the experimentally observed RMS wavefront error as a function of modulation radius for primary defocus, primary astigmatism and a combination of these two. Figure 8(b) shows its theoretical counterpart. It was observed that the measured RMS wavefront error is higher than the theoretically predicted value. This could be due to imperfections in optical alignment. In addition, any extraneous amplitude modulation caused by the SLM and print-through effects were neglected in the simulations.
Fig. 8 (a) Experimentally observed RMS wavefront error and (b) theoretically predicted RMS wavefront error for the introduced aberrations: primary astigmatism, primary defocus and a combination of these aberrations. The corresponding HS measured peak-to-valley values of the introduced aberrations are shown in the legend.
There is another noticeable difference between the experimental and theoretical plots in Figs. 8(a) and 8(b). Theoretically, the RMS wavefront error saturates with increasing modulation radius. However, the experimental RMS wavefront error does not saturate. Although the error decreases with increasing modulation radius, beyond a certain minimum, the RMS wavefront error starts to raise again as the modulation amplitude is increased. This increase is attributed to the tilt aberration caused by the offset of the pyramid apex with respect to the aberrated point spread function. To illustrate this, simulations were performed by introducing a positional offset of the apex of the pyramid on the SLM. Figure 9 shows that for a primary defocus aberration, with a peak-to-valley of 1.5 μm, an offset of the pyramid apex on the SLM even by a mere one pixel has a similar impact as seen experimentally. Care has to be taken to reduce this offset in order to obtain good wavefront sensing accuracy.
Fig. 9 Simulation: Testing the effect of offset of the pyramid apex with respect to the aberrated point spread function. One pixel equals an offset of 20 μm.
Linearity becomes an important parameter when using the pyramid wavefront sensor. As a test of linearity, the peak-to-valley of the sensed aberration is plotted against the HS determined values as shown in Fig. 10. This demonstrates that the linearity of the wavefront sensor improves with increasing modulation radius. However, it is noted that even in the cases of large modulation, the slope of the curve is slightly different from unity. We suspect that errors due to optical alignment inaccuracies, non-negligible SLM amplitude modulation and DM print-through effects cause an overestimate of the wavefront error while sensing a low magnitude defocus with a large modulation radius.
Fig. 10 Test of linearity of the digital pyramid wavefront sensor with defocus aberration. (a) Experimental results (b) Theoretical prediction for different modulation radii.
The pyramid wavefront sensor could also be employed in astronomy for sensing aberrations due to atmospheric turbulence. Based on the Kolmogorov turbulence model and a Fast Fourier Transform (FFT) technique [20], an atmospheric turbulence phase screen for a 1 m telescope was randomly simulated (see Fig. 11) on a 40 × 40 grid, with Fried parameter, r0 = 12.5 cm and the largest scale of turbulence, L0 = 100 m. The magnitude of the simulated aberration was scaled to 2 μm within 4 mm pupil. The corresponding pupil plane images were calculated using Eq. 2 and the wavefront slopes were estimated using Eq. 5 and Eq. 6. The reconstructed wavefront estimated using Eq. 7 was compared with the initially simulated aberration in terms of the RMS wavefront error. Assuming that a high resolution wavefront corrector, with 40 × 40 degrees of freedom was used, the residual wavefront error was calculated. The residual wavefront was again reconstructed using the above described procedure for the pyramid wavefront sensor. This was repeated another three times so as to complete five closed loop operations and reduce the RMS wavefront error significantly as shown in Fig. 11(j). It can be observed that four closed loop operations make the pupil images in Fig. 11(g) uniform compared to those in Fig. 11(b), corresponding to the randomly simulated wavefront in Fig. 11(a).
Fig. 11 Simulations: closed-loop high resolution wavefront correction of an atmospheric phase screen using the digital pyramid wavefront sensor. (a) Simulated atmospheric turbulence phase screen; (b) pupil plane image corresponding to (a); (c) X and (d) Y slopes (in × 10−3) corresponding to (a); (e) reconstructed wavefront corresponding to (a). (f) Residual wavefront after four closed loop operations; (g) pupil plane image corresponding to (f); (h) X and (i) Y slopes (in × 10−3) corresponding to (f); (i) reconstructed wavefront after five closed loop operations.
5. Discussion and conclusions
An SLM with a smaller pixel size will offer a greater advantage in terms of being able to use a shorter focal length lens to focus the beam onto the SLM pyramid without leading to overlapping diffraction orders. This would make the optical setup more compact. A smaller pixel size will not only help lessen the complexity of the optical system, but also help in reducing the tilt aberration induced by the pyramid apex offset. Also, a greater phase modulation capability could reduce aliasing effects due to the wrapped pyramid phase. The accuracy of wavefront sensing is lower in Fig. 5 when compared to the simulation results in Fig. 3. This is due to a lower modulation amplitude and larger aberration magnitude in the laboratory situation. A zero order diffraction spot that occurs due to a limited 95% fill factor of the SLM pixels and imperfect anti-reflection coating leading to a fractional unmodulated beam observed in the center of the four pupils. This unwanted light could be suppressed by making an in-plane translation of the zero order beam. Although no significant improvement in the accuracy of wavefront sensing was observed while doing so with a blazed phase grating, this could be critical in cases where the light intensity is lower. To avoid boundary/edge errors in wavefront sensing with the HS, a reduced beam was used for calibration with the deformable mirror. The digital PWS imaged the full aperture of the DM in the detector plane where the pupils were recorded. As discussed earlier, working with the offset error leads to inaccurate wavefront sensing and considering that the RMS wavefront error increases beyond a certain modulation (see Figs. 8(a), 9), it is recommended that the modulation radius be kept below this value and apply wavefront correction in a closed loop mode to reduce the RMS wavefront error. The difficulty in correcting the pyramid offset is also attributed to the complications that arise when equalizing the intensity in the four pupils due to the unavoidable DM print-through effects. This high frequency noise from the DM could be eliminated by applying appropriate filtering operations in the Fourier plane of the DM. However, care has to be taken not to remove the required spatial frequencies. The finite number of degrees of freedom for the DM limits our capability in testing the PWS with higher order Zernike aberrations. A closed loop operation with a digital pyramid wavefront sensor is limited by the finite response time of the SLM. The wavelength dependence of the SLM limits the application of this novel pyramid wavefront sensor with white light sources.
From simulations it is noted that as the number of measurements during a single modulation increases the RMS wavefront error falls exponentially. Beyond 20 measurements it saturates for the case of lower-order Zernike aberrations with peak-to-valley of 2 μm when modulation radius is 1.6 mm. A similar fall-off is found for other aberration modes. To eliminate the limitations due to fewer measurements during a single modulation, 36 measurements were used in this paper. The saturation of the RMS wavefront error is due to the constraint on the number of slope measurements used for wavefront estimation and SLM pixelation.
The large active area (12×16 mm2) of the SLM allows us to retain the high dynamic range property, which is as good as that attained while using a glass pyramid and a shorter focal length lens.
In conclusion, we demonstrated the working of a PWS using a spatial light modulator. It is shown that the digital version of the sensor allows a tunable modulation of the pyramid, making it feasible to improve the dynamic range and sensitivity, and a very good choice in both open and closed loop adaptive optical systems.
Acknowledgments
We thank the support of Md. Atikur Rahman Jewel during the course of this work. Financial support from Science Foundation Ireland (grants: 07/SK/B1239a and 08/IN.1/B2053) is also gratefully acknowledged.
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