## Measurement limitations in knife-edge tomographic phase retrieval of focused IR laser beams |

Optics Express, Vol. 20, Issue 21, pp. 23875-23886 (2012)

http://dx.doi.org/10.1364/OE.20.023875

Acrobat PDF (884 KB)

### Abstract

An experimental setup to measure the three-dimensional phase-intensity distribution of an infrared laser beam in the focal region has been presented. It is based on the knife-edge method to perform a tomographic reconstruction and on a transport of intensity equation-based numerical method to obtain the propagating wavefront. This experimental approach allows us to characterize a focalized laser beam when the use of image or interferometer arrangements is not possible. Thus, we have recovered intensity and phase of an aberrated beam dominated by astigmatism. The phase evolution is fully consistent with that of the beam intensity along the optical axis. Moreover, this method is based on an expansion on both the irradiance and the phase information in a series of Zernike polynomials. We have described guidelines to choose a proper set of these polynomials depending on the experimental conditions and showed that, by abiding these criteria, numerical errors can be reduced.

© 2012 OSA

## 1. Introduction

1. P. S. Carney, B. Deutsch, A. A. Govyadinov, and R. Hillenbrand, “Phase in nanooptics,” ACS Nano **6**, 8 – 12 (2012). [CrossRef] [PubMed]

2. G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett. **9**3608–3611 (2009). [CrossRef] [PubMed]

3. E. C. Kinzel, J. C. Ginn, R. L. Olmon, D. J. Shelton, B. A. Lail, I. Brener, M. B. Sinclair, M. B. Raschke, and G. D. Boreman, “Phase resolved near-field mode imaging for the design of frequency-selective surfaces,” Opt. Express **20**, 11986–11993 (2012). [CrossRef] [PubMed]

4. E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec. **14**, 1448–1461 (2008). [CrossRef]

5. L. Novotny and N. Van Hulst, “Antennas for light,” Nature Phot. **5**, 83–90 (2011). [CrossRef]

8. R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett. **105**, 167403 (2010). [CrossRef]

9. C. Fumeaux, G. D. Boreman, W. Herrmann, F. K. Kneubhl, and H. Rothuizen, “Spatial impulse response of lithographic infrared antennas,” Appl. Opt. **38**, 37–46 (1999). [CrossRef]

10. J. Alda, C. Fumeaux, L. Codreanu, J. A. Schaefer, and G. D. Boreman, “Deconvolution method for two dimensional spatial response mapping of lithographic infrared antennas,” Appl. Opt. **38**3993–4000 (1999). [CrossRef]

11. J. M. López-Alonso, B. Monacelli, J. Alda, and G. D. Boreman, “Uncertainty analysis in the measurement of the spatial responsivity of infrared antennas,” Appl. Opt. **44**, 4557–4568 (2005). [CrossRef] [PubMed]

13. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

15. W. Chen and X. Chen, “Quantitative phase retrieval of complex-valued specimens based on noninterferometric imaging,” Appl. Opt. **50**, 2008–2015 (2011). [CrossRef] [PubMed]

16. H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. H. Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A **23**, 1179–1200(2006). [CrossRef]

17. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys. **2**, 101–104 (2006). [CrossRef]

18. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focused light beams,” Appl. Opt. **16**, 1971–1974 (1977). [CrossRef] [PubMed]

19. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B **72**, 109–113 (2001). [CrossRef]

20. J. M. Rico-García, L. M. Sánchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt. **47**, 768–775 (2008). [CrossRef] [PubMed]

21. P. Marchenko, S. Orlov, C. Huber, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Interaction of highly focused vector beams with a metal knife-edge,” Opt. Express **19**, 7244–7249 (2011). [CrossRef] [PubMed]

## 2. Theoretical background

22. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. **73**, 1434–1441 (1983). [CrossRef]

23. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. **29**, 1402–1403 (1990). [CrossRef] [PubMed]

*z*-direction the TIE associates the irradiance images at two different planes (

*I*

_{P}_{+1}and

*I*

_{P}_{−1}), located symmetrically with respect to the plane of interest, with the phase front

*φ*at

*I*.

_{P}24. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A **13**(8), 1670–1682 (1996). [CrossRef]

*N*Zernike polynomials. For instance, the phase is retrieved as decomposition in Zernike terms.

25. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**, 207–211 (1976). [CrossRef]

26. J. Alda, J. Alonso, and E. Bernabéu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A **14**, 2737–2747 (1997). [CrossRef]

*N*algebraic equations, where

*λ*, the pupil radii

*R*and the distance between planes Δ

*z*.

**M**is a

*N*×

*N*matrix that depends on the irradiance at the plane

*P*where the phase is to be recovered (see inset of Fig. 1), which elements

_{F}*M*are defined as [24

_{ij}24. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A **13**(8), 1670–1682 (1996). [CrossRef]

*i*,

*j*≤

*N*+ 1. The derivatives of the Zernike polynomials can be easily calculated following the procedure described in [25

25. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**, 207–211 (1976). [CrossRef]

*δ*

_{z}**I**corresponds to the axial derivative of intensity. This can be obtained by applying the central finite difference approximation:

*z*was moved to the constant

*N*). This is acceptable as long as Δ

_{f}*z*is small so that the ray deflections are not too large. That is indeed the case within the focal region of a laser beam, and we will check this condition in our experimental data.

*δ*

_{z}**I**is then expanded in a series of

*N*Zernike polynomials, and applying Eq. (2) the

*N*Zernike coefficients of the phase can be retrieved. The reliability of this process depends strongly on the experimental estimation of

*δ*

_{z}**I**[27

27. S. Barbero and L. N. Thibos, “Error analysis and correction in wavefront reconstruction from the transport-of-intensity equation,” Opt. Eng. **45**, 094001 (2006). [CrossRef]

28. M. Soto, E. Acosta, and S. Rios “Performance analysis of curvature sensors: optimum positioning of measurement planes,” Opt. Express **11**, 2577–2588 (2003). [CrossRef] [PubMed]

## 3. Experimental set- up

_{2}laser (LASY-5 Access Laser) is used as a light source. It has an output continuous power of 5 W maximum at a 10.6

*μ*m wavelength, with a high quality gaussian beam (M

^{2}∼ 1). The beam is directed to a suitable IR lens (ASPH-ZC-50-50 from ISP Optics) with an

*f*= 50.4 mm of focal length. The lens was slightly tilted (∼ 6°) to introduce some aberrations. The knife edge is located at the vicinity of the waist of the beam, and by means of DC-motor stages (PI M-415.DG) it can be moved in any

*x*,

*y*and

*z*-direction. Intensity variations, due to the obstructing knife, are observed in a thermopile based detector (Thorlabs S310C).

*z*–axis, so that the beam can be cut from any angle

*θ*. We have performed 9 cuts along different orientations with Δ

*θ*= 20° separation. Clearly the more cuts the more reliable the tomographic reconstruction will result. We will discuss the influence of the number of cuts on the angular resolution in section 5. A set of normalized measurements at a function of angle are illustrated in Fig. 3(a). These are 9 intensity profiles, arranged in columns (representation also known as sinogram). Once differentiated, these profiles will form a set of projections of the beam intensity. Because the projections at 0° and 180° are mirror images of each other, the measurements are carried out to the last angle increment before 180°. The sinogram clearly shows a strong asymmetry in the beam shape produced by the misaligned lens.

*t*≈ 2.3 s rise time) the knife edge must move accordingly. The speed of the knife (

_{r}*v*) was thus set to 3.4

*μ*m s

^{−1}and recording each intensity profile took around 4 mins (knife-edge going back and forth). The power meter console (Thor-labs PM100D), allowed a sampling frequency of ∼ 13 Hz, and was connected via USB to a computer where data was processed using MATLAB.

## 4. Measurement constraints and limitations

*v*the bandwidth of the detector sets the limit. We have derived the point spread function (PSF) of the detector by measuring the response to a step impulse. Then, by applying a deconvolution procedure to the PSF and a measured intensity profile, we obtained a restored signal. For these calculations we used the Lucy-Richardson algorithm in an iterative process that converges to the maximum likelihood solution [10

10. J. Alda, C. Fumeaux, L. Codreanu, J. A. Schaefer, and G. D. Boreman, “Deconvolution method for two dimensional spatial response mapping of lithographic infrared antennas,” Appl. Opt. **38**3993–4000 (1999). [CrossRef]

*μ*m, well below Δ

*ρ*.

29. P. Toft, “The Radon transform: theory and implementation,” Ph. D. dissertation (Technical University of Denmark, 1996), http://petertoft.dk/PhD.

20. J. M. Rico-García, L. M. Sánchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt. **47**, 768–775 (2008). [CrossRef] [PubMed]

*μ*m

^{2}with a step grid size of ∼ 1.4

*μ*m. The elliptic shape of the beam is due to astigmatism. Some artifacts from the transform, as rays emanating from the center, can be observed.

## 5. Zernike expansion considerations

*n*and

*m*are mode ordering numbers which denote the radial and the azimuthal orders respectively.

*n*and

*m*.

### 5.1. Radial limit

*n*and

*m*,

*R*. Then, since the normalized minimum resolvable spatial feature is Δ

*ρ*/

*R*the associated angular spatial frequency is

*k*]. The radial part of the Fourier transform of the Zernike polynomials (

_{max}*(*

_{n}*k*)) is a function that depends on the

*n*th-order Bessel function of the first kind [32

32. G. M. Dai, “Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,” Opt. Lett. **31**, 501–503 (2006). [CrossRef] [PubMed]

*n*. Therefore high order Zernike polynomials will contribute to carry high frequency noise. Our criterion is based on including only the terms necessary to describe the data.

*(*

_{n}*k*)| in the mentioned range as a function of the radial degree:

*n*,

*β*is a measure of the amount of power spectral density that lies within the range, see Fig. 4. Consider a particular

*k*,

_{max}*β*decreases with increasing

*n*, because the central frequency of the Fourier transform lies further. Alternatively, as

*k*increases, the spectral density of more polynomials lay within the range and therefore the power ratio

_{max}*β*increases. Four cases are plotted to illustrate this behavior.

*β*falls 1% (a −0.45 dB change). To some extent, this threshold is taken arbitrarily: due to the high spatial resolution,

*k*is large, and the curves fall slowly. Since our measurement system is slow no information is carried by high spatial frequencies. Therefore including high order polynomials in the expansion becomes less and less relevant. More importantly, if we employ too few polynomials we are filtering the signal: smoothing of the maps and loss of contrast will result.

_{max}33. S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A **374**, 2798–2800 (2006). [CrossRef]

*k*= 25.6. In this case

_{max}*β*is above the predefined threshold (

*β*≥ 0.99), for

*n*= 21 (see Fig. 4). Therefore polynomials to the radial order 21 will be employed in the expansion. That accounts for the first 253 polynomials in the Noll notation [25

_{max}25. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**, 207–211 (1976). [CrossRef]

### 5.2. Azimuthal limit

*m*|

*=*

_{max}*m*′, i.e. the maximum azimuthal index that should be employed in the expansion corresponds with the number of knife-edge cuts in our measurements.

*θ*angular variations, is computed from the discrete 1D Fourier transform of each projection. In the 2D spatial frequency domain that implies that these 1D transforms are arranged in lines at Δ

*θ*incremental angles. The final image is obtained by integrating all the inverse 1D transforms. By considering polynomials with angular frequencies higher than the ones present in the 2D transform we may be introducing signals that were not there to begin with.

*n*= 4 and |

_{max}*m*|

*= 2. If we arrange the Zernike polynomials in a triangle, as illustrated in Fig. 5, the vertical position sets its*

_{max}*n*order and the horizontal its

*m*order. The radial constrain sets limits to the height of this triangle and the azimuthal condition establishes the maximum width. Only the functions within these limits, drawn in a gray box, should be called upon in the expansion. If we consider the radial constraint we would use a set of polynomials with

*n*≤ 4, a total number of

*N*= 15. However taking into account the azimuthal limit we would have to remove four of them:

*N*′ = 11 polynomials.

*n*≤ 21 (

*N*= 253) the ones with |

*m*| > 9. Leaving the series expansion with

*N*′ = 169 terms.

## 6. Phase retrieval results and discussion

*z*= 108

*μ*m as explained in section 3. Fig. 6(a) shows the contour irradiance maps with the propagation

*z*-axis illustrated vertically. An astigmatic Gaussian beam clearly evolves from plane 1 to 7. Actually, from the obtained measurement we may conclude that the effect of the inclined lens is to produce a beam propagating as an orthogonal astigmatic beam having their orthogonal waists located at different planes [34]. The axis of the orthogonal beam are rotated about 45° with respect to the

*x*and

*y*-axis. One of the orthogonal beam waist is located around plane 2 and the other is around plane 6. This can be deduced from the evolution of the width, and more importantly, from the evolution of the phase front, that shows an almost cylindrical wavefront at planes 2 and 6, being the axis of the cylinders oriented along orthogonal directions. In between these two planes the wavefront has a saddle shape, as it should be expected for this type of beams, showing an almost circular beam in between the orthogonal waists. Clearly the astigmatism introduced by the misaligned lens dominates over other aberrations that may be present in the beam under study.

*δ*

_{z}**I**within the aperture (of radius

*R*= 200

*μ*m). Loss of intensity in a region arises through energy flow across the boundary of the region. For a phase solution to exist this value must be zero.

*δ*

_{z}**I**is computed from non-consecutive planes (

*I*

_{P}_{+1}and

*I*

_{P}_{−1}), where

*P*goes from 2 to 6. Integration of

*δ*

_{z}**I**yields a value below 10

^{−6}for any

*P*and it has a minimum, below 10

^{−8}, at

*P*= 4. We will consider that negligible.

*N*′ Zernike polynomials calculated previously and the parameter

*N*= 110. From each group of three consecutive irradiance planes (

_{f}*I*

_{P}_{+1},

*I*,

_{P}*I*

_{P}_{−1}), a phase map could be generated. Hence a total number of five phase maps were recovered and they are plotted in Fig. 6(b). For instance, to retrieve the phase map at plane 2, the irradiance maps from planes 1, 2 and 3 were employed. Astigmatism clearly governs its propagation and change of curvature from positive to negative can also be seen as the wavefront crosses plane 4. Fig. 7(a) shows the phase map recovered at this plane.

*N*′ expansion set of polynomials, plotted in Fig. 7(a), and the one produced with the

*N*set (at the same plane). Fig. 7(b) shows the difference between both wavefronts. Clearly the extra number of polynomials has an impact in the phase retrieval. The highest differences are in the order of 10%, and they are particularly located at the edges of the aperture. This supports the idea that, as we explained in section 5, the extra Zernike polynomials with |

*m*| > |

*m*|

*, (polynomials with strong angular changes), introduce more uncertainty in the final recovered wavefront.*

_{max}35. J. M. Braat, S. Van Haver, A. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point spread functions,” in *Progress in Optics*, E. Wolf, ed. (Elsevier, 2008), 349–468. [CrossRef]

## 7. Conclusions

## Acknowledgments

## References and links

1. | P. S. Carney, B. Deutsch, A. A. Govyadinov, and R. Hillenbrand, “Phase in nanooptics,” ACS Nano |

2. | G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett. |

3. | E. C. Kinzel, J. C. Ginn, R. L. Olmon, D. J. Shelton, B. A. Lail, I. Brener, M. B. Sinclair, M. B. Raschke, and G. D. Boreman, “Phase resolved near-field mode imaging for the design of frequency-selective surfaces,” Opt. Express |

4. | E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec. |

5. | L. Novotny and N. Van Hulst, “Antennas for light,” Nature Phot. |

6. | J. Alda, J. M. Rico-García, J. M. López-Alonso, and G. D. Boreman, “Optical antennas for nanophotonics applications,” Nanotechnology |

7. | M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antenna,” Science |

8. | R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett. |

9. | C. Fumeaux, G. D. Boreman, W. Herrmann, F. K. Kneubhl, and H. Rothuizen, “Spatial impulse response of lithographic infrared antennas,” Appl. Opt. |

10. | J. Alda, C. Fumeaux, L. Codreanu, J. A. Schaefer, and G. D. Boreman, “Deconvolution method for two dimensional spatial response mapping of lithographic infrared antennas,” Appl. Opt. |

11. | J. M. López-Alonso, B. Monacelli, J. Alda, and G. D. Boreman, “Uncertainty analysis in the measurement of the spatial responsivity of infrared antennas,” Appl. Opt. |

12. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik |

13. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

14. | J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A |

15. | W. Chen and X. Chen, “Quantitative phase retrieval of complex-valued specimens based on noninterferometric imaging,” Appl. Opt. |

16. | H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. H. Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A |

17. | H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys. |

18. | A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focused light beams,” Appl. Opt. |

19. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B |

20. | J. M. Rico-García, L. M. Sánchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt. |

21. | P. Marchenko, S. Orlov, C. Huber, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Interaction of highly focused vector beams with a metal knife-edge,” Opt. Express |

22. | M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. |

23. | F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. |

24. | T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A |

25. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

26. | J. Alda, J. Alonso, and E. Bernabéu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A |

27. | S. Barbero and L. N. Thibos, “Error analysis and correction in wavefront reconstruction from the transport-of-intensity equation,” Opt. Eng. |

28. | M. Soto, E. Acosta, and S. Rios “Performance analysis of curvature sensors: optimum positioning of measurement planes,” Opt. Express |

29. | P. Toft, “The Radon transform: theory and implementation,” Ph. D. dissertation (Technical University of Denmark, 1996), http://petertoft.dk/PhD. |

30. | R. C. Gonzalez, R. E. Woods, and S. L. Eddins, |

31. | M. Born and E. Wolf, |

32. | G. M. Dai, “Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,” Opt. Lett. |

33. | S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A |

34. | J. Alda, “Laser and Gaussian beam propagation and transformation,” in |

35. | J. M. Braat, S. Van Haver, A. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point spread functions,” in |

**OCIS Codes**

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(100.5070) Image processing : Phase retrieval

(100.6950) Image processing : Tomographic image processing

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Image Processing

**History**

Original Manuscript: June 20, 2012

Revised Manuscript: July 23, 2012

Manuscript Accepted: July 23, 2012

Published: October 3, 2012

**Citation**

Manuel Silva-López, José María Rico-García, and Javier Alda, "Measurement limitations in knife-edge tomographic phase retrieval of focused IR laser beams," Opt. Express **20**, 23875-23886 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23875

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### References

- P. S. Carney, B. Deutsch, A. A. Govyadinov, and R. Hillenbrand, “Phase in nanooptics,” ACS Nano6, 8 – 12 (2012). [CrossRef] [PubMed]
- G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett.93608–3611 (2009). [CrossRef] [PubMed]
- E. C. Kinzel, J. C. Ginn, R. L. Olmon, D. J. Shelton, B. A. Lail, I. Brener, M. B. Sinclair, M. B. Raschke, and G. D. Boreman, “Phase resolved near-field mode imaging for the design of frequency-selective surfaces,” Opt. Express20, 11986–11993 (2012). [CrossRef] [PubMed]
- E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec.14, 1448–1461 (2008). [CrossRef]
- L. Novotny and N. Van Hulst, “Antennas for light,” Nature Phot.5, 83–90 (2011). [CrossRef]
- J. Alda, J. M. Rico-García, J. M. López-Alonso, and G. D. Boreman, “Optical antennas for nanophotonics applications,” Nanotechnology16, S230, (2005). [CrossRef]
- M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antenna,” Science332, 702–704, (2011). [CrossRef] [PubMed]
- R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett.105, 167403 (2010). [CrossRef]
- C. Fumeaux, G. D. Boreman, W. Herrmann, F. K. Kneubhl, and H. Rothuizen, “Spatial impulse response of lithographic infrared antennas,” Appl. Opt.38, 37–46 (1999). [CrossRef]
- J. Alda, C. Fumeaux, L. Codreanu, J. A. Schaefer, and G. D. Boreman, “Deconvolution method for two dimensional spatial response mapping of lithographic infrared antennas,” Appl. Opt.383993–4000 (1999). [CrossRef]
- J. M. López-Alonso, B. Monacelli, J. Alda, and G. D. Boreman, “Uncertainty analysis in the measurement of the spatial responsivity of infrared antennas,” Appl. Opt.44, 4557–4568 (2005). [CrossRef] [PubMed]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 237–246 (1972).
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21, 2758–2769 (1982). [CrossRef] [PubMed]
- J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15, 1662–1669 (1998). [CrossRef]
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