## Compressive wavefront sensing with weak values |

Optics Express, Vol. 22, Issue 16, pp. 18870-18880 (2014)

http://dx.doi.org/10.1364/OE.22.018870

Acrobat PDF (3979 KB)

### Abstract

We demonstrate a wavefront sensor that unites weak measurement and the compressive-sensing, single-pixel camera. Using a high-resolution spatial light modulator (SLM) as a variable waveplate, we weakly couple an optical field’s transverse-position and polarization degrees of freedom. By placing random, binary patterns on the SLM, polarization serves as a meter for directly measuring random projections of the wavefront’s real and imaginary components. Compressive-sensing optimization techniques can then recover the wavefront. We acquire high quality, 256 × 256 pixel images of the wavefront from only 10,000 projections. Photon-counting detectors give sub-picowatt sensitivity.

© 2014 Optical Society of America

## 1. Introduction

1. F. Roddier, *Adaptive Optics in Astronomy* (Cambridge university press, 1999). [CrossRef]

2. L. N. Thibos and X. Hong, “Clinical applications of the Shack-Hartmann aberrometer,” Optometry Vision Sci. **76**, 817–825 (1999). [CrossRef]

3. M. J. Booth, “Adaptive optics in microscopy,” Philos. T. R. Soc. A **365**, 2829–2843 (2007). [CrossRef]

4. M. Levoy, “Light fields and computational imaging,” IEEE Comput. **39**, 46–55 (2006). [CrossRef]

5. R. Tyson, *Principles of Adaptive Optics* (CRC Press, 2010). [CrossRef]

6. J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) **474**, 188–191 (2011). [CrossRef]

7. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. **17**, S573–S577 (2001). [PubMed]

8. R. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. **31**, 6902–6908 (1992). [CrossRef] [PubMed]

6. J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) **474**, 188–191 (2011). [CrossRef]

9. S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science **332**, 1170–1173 (2011). [CrossRef] [PubMed]

10. J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics **7**, 316–321 (2013). [CrossRef]

## 2. Theory

### 2.1. Weak measurement

13. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. **60**, 1351 (1988). [CrossRef] [PubMed]

14. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science **319**, 787–790 (2008). [CrossRef] [PubMed]

17. J. Lundeen and A. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of hardys paradox,” Phys. Rev. Lett. **102**, 020404 (2009). [CrossRef]

*ψ*〉. An observable of interest

*Â*is then coupled to an ancillary meter system by a weak interaction or perturbation. Finally, the system is post-selected (projected) into final state |

*f*〉. The weak measurement is read-out by a measuring device for the meter. In the limit of a very weak interaction, the measuring device’s pointer is shifted by the weak value Precision measurement applications of weak values make use of the fact that the denominator, 〈

*f*|

*ψ*〉, can be made arbitrarily small by choosing nearly orthogonal pre-selection 〈

*ψ*| and post-selection |

*f*〉.

*A*then takes on values well outside the eigenvalue range of

_{w}*Â*, a process known as weak-value amplification.

*A*—that it can have complex values. The real part of

_{w}*A*is interpreted as a shift in the measurement pointer’s position; the imaginary part of

_{w}*A*is interpreted as a shift in the measurement pointer’s momentum [18

_{w}18. J. Dressel and A. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A **85**, 012107 (2012). [CrossRef]

*A*, it can be directly measured.

_{w}19. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. **86**, 307 (2014). [CrossRef]

### 2.2. Wavefront sensing with weak values

*ψ*(

*x⃗*), where

*x⃗*= (

*x*,

*y*) are transverse spatial coordinates. In keeping with traditional presentation of weak measurement, we use a quantum formalism where

*ψ*(

*x⃗*) is treated as a probability amplitude distribution and polarization is represented on the Bloch sphere. Note that the system can still be understood classically by replacing the wavefunction with the transverse electric field amplitude (or angular spectrum) and by replacing the Bloch sphere with the Poincaré sphere. In this case, polarization states (such as |

*h*〉) would be replaced by classical Jones vectors. For a side-by-side example of quantum and classical weak-value derivations, see [20

20. J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A **81**, 033813 (2010). [CrossRef]

*x⃗*=

*x⃗*

_{0}followed by a post-selection of momentum

*k⃗*=

*k⃗*

_{0}, where

*ψ*(

*x⃗*) and

*ψ̃*(

*k⃗*) are Fourier transform pairs. The corresponding weak value is

*A*(

_{w}*x⃗*

_{0}) is directly proportional to value of the field at position

*x⃗*=

*x⃗*

_{0}up to a linear phase.

*A*(

_{w}*x⃗*

_{0}), the field at

*x⃗*=

*x⃗*

_{0}must be weakly coupled to a meter system. The polarization degree of freedom is a convenient meter because it can be easily manipulated and measured. Let

*ψ*(

*x⃗*) be initially polarized in polarization state |

*p*〉. The full initial state is At location

_{i}*x⃗*=

*x⃗*

_{0}, the polarization is changed from |

*p*〉 to a nearby polarization |

_{i}*p*〉 (Figs. 1(a)–1(b)). At location

_{f}*x⃗*

_{0}, the state is where the transformation from |

*p*〉 to |

_{i}*p*〉 is expressed as a rotation on the Bloch sphere by angle

_{f}*θ*about unit vector

*k̂*. Note that this corresponds to a

*θ*/2 rotation in the state vector space as seen in Eq. (4). This rotation is visualized on the Bloch sphere in Fig. 1. Unit vectors

*î*,

*ĵ*, and

*k̂*form a right handed coordinate system on the Bloch sphere, where

*î*points along |

*p*〉,

_{i}*ĵ*is the orthogonal unit vector in the plane defined by |

*p*〉 and |

_{i}*p*〉, and

_{f}*k̂*=

*î*×

*ĵ*(Figs. 1(a)–1(c)). These unit vectors have corresponding Pauli operators

*σ̂*,

_{i}*σ̂*, and

_{j}*σ̂*. Note that such a coordinate system can be defined for any two polarization states |

_{k}*p*〉 and |

_{i}*p*〉.

_{f}*θ*/2 is small. A first order expansion at

*x⃗*

_{0}yields The full state is therefore Consider post-selection on a single transverse-momentum

*k⃗*=

*k⃗*

_{0}. The post-selected state |

*ψ*〉 no longer has position dependence and is given by Factoring out

_{ps}*ψ̃*(

*k⃗*

_{0}) and re-exponentiating, we find

*ψ*(

*x⃗*

_{0}) (Figs. 1(a)–1(c)). The real part of

*ψ*(

*x⃗*

_{0}) generates a rotation

*α*in the

_{R}*î*,

*ĵ*plane (Fig. 1(b)). The imaginary part of

*ψ*(

*x⃗*

_{0}) generates a rotation

*α*in the

_{I}*î*,

*k̂*plane (Fig. 1(c)).

### 2.3. Random projections of the wavefront

*ψ*(

*x⃗*=

*x⃗*

_{0}), consider instead a weak measurement of an operator

*f̂*= |

_{i}*f*〉 〈

_{i}*f*|, which takes a random, binary projection of

_{i}*ψ*(

*x⃗*) where |

*f*〉 is The filter function

_{i}*f*(

_{i}*x⃗*) consists of a pixelized, random binary pattern, where pixels in the pattern take on values of 1 or −1 with equal probability.

*f̂*, given initial state

_{i}*ψ*(

*x⃗*) and post-selected state |

*k⃗*

_{0}〉, is therefore where

*is the inner product between*

**Y**_{i}*ψ*(

*x⃗*) and

*f*(

_{i}*x⃗*) It is convenient to choose a zero-momentum post-selection

*k⃗*

_{0}= 0 to discard the linear phase factor 〈

*k⃗*

_{0}|

*f*〉 in Eq. (12).

_{i}*ψ*(

*x⃗*) will now receive a small polarization rotation about

*k̂*of angle

*θ*for

*f*(

_{i}*x⃗*) = 1 and −

*θ*for

*f*(

_{i}*x⃗*) = −1.

*ψ*(

*x⃗*) onto

*f*(

_{i}*x⃗*),

*. Again, taking expectation values of*

**Y**_{i}*σ̂*and

_{j}*σ̂*yields the real and imaginary parts of

_{k}*, Therefore, weak measurement allows us to directly measure random, binary projections of a transverse field*

**Y**_{i}*ψ*(

*x⃗*).

### 2.4. Compressive sensing

21. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory **52**, 1289–1306 (2006). [CrossRef]

*N*-dimensional signal

**from**

*X**M*<<

*N*measurements. The signal

**is linearly sampled by a**

*X**M*×

*N*sensing matrix

**to produce an**

*F**M*-dimensional vector of measurements

**where**

*Y***Γ**is an

*M*-dimensional noise vector. Measurement vectors (rows of F) are often random, binary vectors, so each measurement

*is a random projection of*

**Y**_{i}**[22**

*X*22. E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. **346**, 589–592 (2008). [CrossRef]

*M*<<

*N*, the system of equations in Eq. (17) is under-determined; there are many possible signals consistent with the measurements. CS posits that the correct

**is the one which is sparsest (has the fewest non-zero elements) when compressed. This**

*X***is found by solving a regularized, least squares objective function The first penalty is small when**

*X***is consistent with**

*X***. The second penalty,**

*Y**g*(

**), is small when**

*X***is compressible. A common**

*X**g*(

**) for imaging is the Total Variation (TV) of**

*X***where indices**

*X**i*and

*j*run over all pairs of adjacent pixels. TV therefore leverages compressibility in the gradient of

**. In this case, Eq. (18) is referred to as Total Variation Minimization [23**

*X*23. A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. **76**, 167–188 (1997). [CrossRef]

*k*-sparse (only

*k*significant elements when compressed)

**is possible from only**

*X**N*[24

24. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inform. Theory **52**, 5406–5425 (2006). [CrossRef]

**is imaged onto a digital micro-mirror device (DMD), an array of mirrors which can be individually oriented towards or away from a single-element detector. A series of**

*X**M*random binary patterns are placed on the DMD, each corresponding to a row of sensing matrix

**. The total optical power striking the detector for the**

*F**i*

^{th}pattern gives the projection of

**onto**

*X**, the*

**F**_{i}*i*

^{th}measurement

*. Solving Eq. (18) recovers the image.*

**Y**_{i}25. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. **58**, 1182–1195 (2007). [CrossRef] [PubMed]

26. J. Bobin, J.-L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signa. **2**, 718–726 (2008). [CrossRef]

27. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. **105**, 150401 (2010). [CrossRef]

29. E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. **25**, 21–30 (2008). [CrossRef]

30. J. Romberg, “Imaging via compressive sampling [introduction to compressive sampling and recovery via convex programming],” IEEE Signal Proc. Mag. **25**, 14–20 (2008). [CrossRef]

### 2.5. Compressive wavefront sensing

*M*random projections of the real and imaginary parts of the wavefront (as in sec 2.3), and then use compressive sensing optimization algorithms to recover the real and imaginary parts of the field.

*I*

_{SLM}(

*x⃗*

_{0}) is the SLM intensity at location

*x⃗*

_{0}and |

*d*〉 and |

*a*〉 refer to diagonal and anti-diagonal polarizations respectively. For a typical 8-bit SLM driven over a VGA video port,

*I*

_{SLM}takes on integer values of 0–255. Because SLM pixels retard |

*a*〉 and not |

*d*〉, a non-zero SLM intensity will rotate the polarization of any input state that is not purely |

*a*〉 or |

*d*〉.

**=**

*X*

*X*^{Re}+

*i*

*X*^{Im}be a one-dimensional reshaping of

*ψ*(

*x⃗*

_{0}), discretized to the SLM resolution. Let

**be a**

*F**M*×

*N*, random, binary sensing matrix whose elements take values 1 or −1 with equal probability. Each row of

**corresponds to a pattern placed on the SLM, where pixels with value 1 rotate the field’s polarization by**

*F**θ*and pixels with value −1 rotate the field’s polarization by −

*θ*.

**onto each pattern is weakly measured as in section 2.3 to produce an**

*X**M*-dimensional measurement vector The real and imaginary parts of

**are recovered by solving Eq. (18).**

*X*## 3. Experiment

### 3.1. Experimental apparatus

*a*〉 oriented polarizer and an object SLM (Cambridge Correlators SDE1024). Because the SLM retards |

*a*〉, an image placed on it will produce a nearly pure phase image with a small amount of amplitude coupling.

*h*〉) polarizer prepares the initial polarization state |

*p*〉. A sequence of

_{i}*M*random, binary patterns are placed on the SLM, executing the weak measurement. The patterns consist of randomly-permuted rows of a Hadamard matrix. This dramatically improves reconstruction algorithm speeds as repeated calculations of

**can be performed by a fast transform [31].**

*FX**μ*m pinhole performs the

*k⃗*= 0 post-selection. The Gaussian beam exiting the pinhole is collimated with a 10X objective and directed to a pair of polarization analyzers each consisting of a half-waveplate, quarter-waveplate, and a polarizing beamsplitter. The half- and quarter-waveplates are oriented to measures either 〈

*σ̂*〉 or 〈

_{j}*σ̂*〉 for the respective real and imaginary projections. The detectors are large area, photon-counting photomultiplier modules (Horiba TBX-900C).

_{k}### 3.2. SLM polarization rotation

*θ*/2 between the output polarization and horizontal.

*θ*/2 for pixels with value 1 was chosen to be 25 degrees with a corresponding

*I*

_{SLM}= 160. This angle introduces a 3 percent error in the small-angle approximation taking Eq. (4) to Eq. (6), an error we judged acceptable for our proof-of-principle experiment.

*θ*/2 rotation, two measurements must be taken for each SLM pattern

*. First, all pixels with value 1 set to*

**F**_{i}*I*

_{SLM}= 160, and all pixels with value −1 are set to

*I*

_{SLM}= 20. The latter experience a polarization rotation less than 1 degree; effectively no rotation. Then, the pattern is inverted; pixels with value −1 are set to

*I*

_{SLM}= 160 and pixels with value 1 are set to

*I*

_{SLM}= 20. Subtracting the value for the second situation from the first achieves the desired result of a positive

*θ*rotation for “1” pixels and a −

*θ*rotation for “−1” pixels.

## 4. Results

*ħ*character is given in Fig. 4. Figure 4(a) shows a camera image of the interference between the object field and reference beam. The image is predominantly a phase-only image with a small amount of amplitude coupling, particularly at the edges of

*ħ*character. The emergent edges are most likely due to the small, one inch aperture optics used in the setup, which cause a “ringing” effect when imaging sharp edges with coherent light [33

33. P. S. Considine, “Effects of coherence on imaging systems,” J. Opt. Soc. Am. **56**, 1001–1007 (1966). [CrossRef]

^{−10}are colored white. The recovered field shows has nearly uniform intensity up to the width of the illumination beam, but shows a strong variation in phase corresponding to the

*ħ*character. Only

*M*= .15

*N*= 10000 random projections were used to recover a high quality image.

*I*

_{SLM}= 30 to

*I*

_{SLM}= 240. The SLM converts this image to a phase image. These values were chosen because the polarization rotation appears roughly linear on this range (see Fig. 3). Figure 5(b) is a CCD image of the object SLM that shows that the object field has nearly uniform intensity. Figure 5(c) gives a dark port interferogram demonstrating the validity of our phase-square reconstruction, where the bottom left square has an almost negligible phase shift with the background while the top right square is nearly

*π*out of phase.

35. M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. **1**, 586–597 (2007). [CrossRef]

36. G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express **21**, 23822–23837 (2013). [CrossRef] [PubMed]

## 5. Conclusion

*M*<<

*N*) with only single-element detectors and without scanning. We anticipate that our technique will be a valuable and useful addition to the field of wavefront sensing and adaptive optics.

## Acknowledgments

37. J. D. Hunter, “Matplotlib: A 2D graphics environment,” IEEE Comput. Sci. Eng. **9**, 90–95 (2007). [CrossRef]

## References and links

1. | F. Roddier, |

2. | L. N. Thibos and X. Hong, “Clinical applications of the Shack-Hartmann aberrometer,” Optometry Vision Sci. |

3. | M. J. Booth, “Adaptive optics in microscopy,” Philos. T. R. Soc. A |

4. | M. Levoy, “Light fields and computational imaging,” IEEE Comput. |

5. | R. Tyson, |

6. | J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) |

7. | B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. |

8. | R. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. |

9. | S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science |

10. | J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics |

11. | D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “ |

12. | R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. |

13. | Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. |

14. | O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science |

15. | P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. |

16. | A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: When less is more,” Phys. Rev. X |

17. | J. Lundeen and A. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of hardys paradox,” Phys. Rev. Lett. |

18. | J. Dressel and A. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A |

19. | J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. |

20. | J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A |

21. | D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory |

22. | E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. |

23. | A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. |

24. | E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inform. Theory |

25. | M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. |

26. | J. Bobin, J.-L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signa. |

27. | D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. |

28. | G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X |

29. | E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. |

30. | J. Romberg, “Imaging via compressive sampling [introduction to compressive sampling and recovery via convex programming],” IEEE Signal Proc. Mag. |

31. | C. Li, “Compressive sensing for 3D data processing tasks: applications, models and algorithms,” Ph.D. thesis, Rice University (2011). |

32. | C. Li, W. Yin, and Y. Zhang, “Users guide for TVAL3: TV minimization by augmented lagrangian and alternating direction algorithms,” CAAM Report (2009). |

33. | P. S. Considine, “Effects of coherence on imaging systems,” J. Opt. Soc. Am. |

34. | J. Goodman, |

35. | M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. |

36. | G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express |

37. | J. D. Hunter, “Matplotlib: A 2D graphics environment,” IEEE Comput. Sci. Eng. |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(270.0270) Quantum optics : Quantum optics

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: May 14, 2014

Revised Manuscript: July 15, 2014

Manuscript Accepted: July 20, 2014

Published: July 28, 2014

**Citation**

Gregory A. Howland, Daniel J. Lum, and John C. Howell, "Compressive wavefront sensing with weak values," Opt. Express **22**, 18870-18880 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-18870

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### References

- F. Roddier, Adaptive Optics in Astronomy (Cambridge university press, 1999). [CrossRef]
- L. N. Thibos and X. Hong, “Clinical applications of the Shack-Hartmann aberrometer,” Optometry Vision Sci.76, 817–825 (1999). [CrossRef]
- M. J. Booth, “Adaptive optics in microscopy,” Philos. T. R. Soc. A365, 2829–2843 (2007). [CrossRef]
- M. Levoy, “Light fields and computational imaging,” IEEE Comput.39, 46–55 (2006). [CrossRef]
- R. Tyson, Principles of Adaptive Optics (CRC Press, 2010). [CrossRef]
- J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London)474, 188–191 (2011). [CrossRef]
- B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg.17, S573–S577 (2001). [PubMed]
- R. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt.31, 6902–6908 (1992). [CrossRef] [PubMed]
- S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science332, 1170–1173 (2011). [CrossRef] [PubMed]
- J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics7, 316–321 (2013). [CrossRef]
- D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.
- R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag.83, 914730 (2008).
- Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett.60, 1351 (1988). [CrossRef] [PubMed]
- O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science319, 787–790 (2008). [CrossRef] [PubMed]
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