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Digital pyramid wavefront sensor Vyas Akondi, Sara Castillo, Md. Atikur Rahman Jewel and Brian Vohnsen Advanced Optical Imaging Group, School of Physics, University College Dublin, Dublin 4, Ireland
[email protected]
Abstract: We demonstrate the possibility of sensing defocus and a combination of defocus and astigmatism aberrations using a digital pyramid wavefront sensor based on a spatial light modulator which allows easy modulation of the pyramid without the need for moving parts. © 2013 Optical Society of America OCIS codes: 010.7350, 080.4298, 120.5050, 010.1080, 280.4788
1.
Introduction
First proposed by Ragazzoni [1], the pyramid wavefront sensor has high sensitivity and adjustable dynamic range with applications in astronomy [2, 3] and vision science [4, 5]. Largely, either the refractive pyramid is physically moved to achieve modulation [1] or the beam is dynamically controlled using beam steering optics [4]. It is made up of a pyramidal prism, with the vertex angle slightly less than 1800 such that the four facets of the prism split the light beam into four pupils (see Fig. 1). The local slopes of the wavefront can be obtained by an appropriate linear combination of the pupil plane images [2]. As the modulation magnitude increases, the phase range over which the performance of the sensor is linear increases [6, 7]. Also, a smaller modulation radius increases the sensitivity of the wavefront sensor [2, 7, 8]. One of the primary disadvantages of the pyramid sensor is that it needs an additional device for modulation. Beam diffusers can be used to avoid modulation [9]. Also, the need for modulation itself was questioned [10, 11]. Alternatively, using a large apex angle axicon, wavefront sensing could be performed [12]. In this paper, we demonstrate that by imposing a pyramid like phase profile on a spatial light modulator (SLM), wavefront sensing can be performed without the need for moving parts. 2.
Theory
The pyramid transmittance function can be written as [13], 1
T (X,Y ) =
1
n
m y−c]
∑ ∑ H((−1)n X, (−1)mY )e−iα[(−1) x−c+(−1)
(1)
n=0 n=0
where ‘c’ is the half diagonal distance of the pyramid base. ‘α’ is the tangent of the divergence angle of the pyramid and H is the Heaviside function. Although a convolution method could be used to simulate the pupil plane images [13] using the above defined pyramid transmittance function, we adopted the Fast Fourier Transform technique [6] and consequently, the pupil plane images can be simulated, 2 I pyr (x, y) = FT (FT (P(x, y).eiφ (x,y) ).T )
(2)
where FT represents the Fourier Transform function and P(x, y) is the defined pupil function. The ‘x’ and ‘y’ slopes of the wavefront φ (x, y) can be obtained by using the following relations defined from the recorded individual pupil intensities for the four pupils (see Fig. 1). Sx (x, y) =
I1 (x, y) − I2 (x, y) − I3 (x, y) + I4 (x, y) I1 (x, y) + I2 (x, y) + I3 (x, y) + I4 (x, y)
(3)
Sy (x, y) =
I1 (x, y) + I2 (x, y) − I3 (x, y) − I4 (x, y) I1 (x, y) + I2 (x, y) + I3 (x, y) + I4 (x, y)
(4)
The above expressions are valid when [7],
OM2A.3.pdf
Imaging and Applied Optics © OSA 2013
Fig. 1. Principle of the pyramid wavefront sensor
Fig. 2. Comparison of the wavefronts reconstructed using pyramid wavefront sensor and HS sensor
∂φ f < d ∂x
(5)
where ‘φ (x, y)’ is the wavefront aberration in the system, ‘ f ’ is the distance between the lens aperture and the SLM where the pyramid is addressed. ‘d’ is the aperture size as on the pyramid. The wavefront was reconstructed from the calculated slopes using the slope geometry of Southwell [14]. 3.
Experimental setup
A spatially filtered He-Ne laser (λ =632.8 nm) was used as source. A 1000 mm achromat was used to focus the beam on the apex of the SLM pyramid. The airy disk occupied 17.5 pixels on the SLM when no aberrations were present in the system. Defocus, astigmatism, coma and a combination of these aberrations were generated with a deformable mirror (DM) placed in the back Fourier plane of the 1000 mm achromat. The pyramid on the SLM splits the beam into four pupils (see Fig. 1) that were recorded with a CCD camera. The aberration was sensed with a Hartmann-Shack (HS) wavefront sensor for comparison. The long focal length lens was required in order to avoid the overlapping of copies of the pupils that arise due to the diffraction caused by pixelated SLM. A phase magnitude of 331λ is needed to produce a pyramid with vertex angle of 1780 , which could be used for sensing ocular aberrations [15]. 4.
Results and Conclusions
The shape of the circular modulation applied to the pyramid is limited by SLM pixelation and strongly depends on the radius of modulation. The wavefronts reconstructed with the pyramid wavefront sensor were compared with those reconstructed with the HS sensor (see Fig. 2). Fig. 2(a) shows simulated wavefront aberrations that were intended to be generated using the DM. The calculated ‘x’ and ‘y’ slopes obtained from Sx and Sy (see Eqs. 3 and 4) are shown in Figs. 2(b) and 2(c) respectively. It can be seen that the wavefronts reconstructed with the digital pyramid wavefront sensor (Fig. 2d) are in good agreement with those reconstructed using the HS (Fig. 2e). Eliminating the pupil images arising due to the higher diffraction orders is critical to improve the accuracy of wavefront sensing. In conclusion, a novel digital pyramid wavefront sensor was demonstrated in sensing primary defocus and a combination of defocus and astigmatism. In the future, we aim to test the sensitivity of the digital pyramid wavefront sensor, the limits of its operation and its performance with more complex wavefront aberrations.
OM2A.3.pdf
Imaging and Applied Optics © OSA 2013
Acknowledgement Financial support from Science Foundation Ireland (grants: 07/SK/B1239a and 08/In.1/B2053) is gratefully acknowledged. References 1. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” Journal of modern optics 43, 289-293 (1996). 2. Esposito, Simone, and Armando Riccardi. “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astronomy and Astrophysics 369, 9-12 (2001). 3. C. Verinaud, M. Le Louarn, V. Korkiakoski, and M. Carbillet, “Adaptive optics for high-contrast imaging: pyramid sensor versus spatially filtered ShackHartmann sensor,” Monthly Notices of the Royal Astronomical Society: Letters, 357: L26L30 (2005). 4. I. Iglesias, R. Ragazzoni, Y. Julien, and P. Artal, “Extended source pyramid wave-front sensor for the human eye,” Opt. Express 10, 419-428 (2002). 5. S. Chamot, C. Dainty, and S. Esposito, “Adaptive optics for ophthalmic applications using a pyramid wavefront sensor,” Opt. Express 14, 518-526 (2006). 6. C. Verinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Optics Communications 233, 27-38 (2004). 7. A. Burvall, E. Daly, S. Chamot, and C. Dainty, “Linearity of the pyramid wavefront sensor,” Opt. Express 14, 11925-11934 (2006). 8. J. LeDue, L. Jolissaint, J. Vran, and C. Bradley, “Calibration and testing with real turbulence of a pyramid sensor employing static modulation,” Opt. Express 17, 7186-7195 (2009). 9. R. Ragazzoni, D. Emiliano and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Optics communications 208: 51-60 (2002). 10. J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” Proc. SPIE 4839, Adaptive Optical System Technologies II, 288 (2003). 11. J. Costa, “Modulation effect of the atmosphere in a pyramid wave-frontsensor,” Appl. Opt. 44, 60-66 (2005). 12. B. Vohnsen, S. Castillo, and D. Rativa, ”Wavefront sensing with an axicon,” Opt. Lett. 36, 846-848 (2011). 13. V. Korkiakoski, C. Verinaud, ` M. Le Louarn, and R. Conan, “Comparison between a model-based and a conventional pyramid sensor reconstructor,” Appl. Opt. 46, 6176-6184 (2007). 14. W. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998-1006 (1980). 15. E. M. Daly, C. J. Dainty, “Measuring phase aberrations using a pyramid wave front sensor,” Proc. SPIE 7726, Optical Sensing and Detection, 77260W (2010).
