## Optical testing using the transport-of-intensity equation

Optics Express, Vol. 15, Issue 12, pp. 7165-7175 (2007)

http://dx.doi.org/10.1364/OE.15.007165

Acrobat PDF (800 KB)

### Abstract

The transport-of-intensity equation links the intensity and phase of an optical source to the longitudinal variation of its intensity in the presence of Fresnel diffraction. This equation can be used to provide a simple, accurate spatial-phase measurement for optical testing of flat surfaces. The properties of this approach are derived. The experimental demonstration is performed by quantifying the surface variations induced by the magnetorheological finishing process on laser rods.

© 2007 Optical Society of America

## 1. Introduction

*et al*. [8

8. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. **77**, 2961–2964 (1996). [CrossRef] [PubMed]

14. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. **133**, 339–346 (1997). [CrossRef]

15. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. **214**, 51–61 (2004). [CrossRef] [PubMed]

16. D. Golini, S. Jacobs, W. Kordonski, and P. Dumas, “Precision optics fabrication using magnetorheological finishing,” in *Advanced Materials for Optics and Precision Structures*,
M. A. Ealey, R. A. Paquin, and T. B. Parsonage, eds., *Critical Reviews of Optical Science and Technology* (SPIE, Bellingham, WA, 1997), Vol. CR67, pp. 251–274.

17. V. Bagnoud, M. J. Guardalben, J. Puth, J. D. Zuegel, T. Mooney, and P. Dumas, “High-energy, high-average-power laser with Nd:YLF rods corrected by magnetorheological finishing,” Appl. Opt. **44**, 282–288 (2005). [CrossRef] [PubMed]

18. J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, “OMEGA EP: High-energy petawatt capability for the OMEGA laser facility,” J. Phys. IV France **133**, 75–80 (2006). [CrossRef]

19. V. Bagnoud, I. A. Begishev, M. J. Guardalben, J. Puth, and J. D. Zuegel, “5-hz, >250-mJ optical parametric chirped-pulse amplifier at 1053 nm,” Opt. Lett. **30**, 1843–1845 (2005). [CrossRef] [PubMed]

## 2. Transport-of-intensity equation

_{0}= 2

*π*/

*k*

_{0}with complex amplitude

*E*(

*x*,

*y*,

*z*) =

*z*axis in an isotropic medium of index equal to 1. The paraxial differential equation describing the free-space propagation of this optical wave is

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

*I*/∂

*z*) to the spatial intensity

*I*(

*x*,

*y*,

*z*) and spatial phase

*φ*(

*x*,

*y*,

*z*). It can, for example, be used to predict the intensity modulation due to phase-to-intensity conversion via Fresnel diffraction. For wavefront measurement of an optical wave of arbitrary intensity, the intensity of the wave is measured in two different planes (located at

*z*

_{0}and

*z*

_{0}+

*dz*along the longitudinal propagation axis) in order to approximate the derivative ∂

*I*/∂

*z*by the finite difference [

*I*(

*x*,

*y*,

*z*

_{0}+

*dz*)-

*I*(

*x*,

*y*,

*z*

_{0})]/

*dz*, as described in Fig. 1(a). Equation (2) can be written fully with the measured intensities as

*φ*. This finite-difference expression of the TIE can be numerically solved to yield the spatial phase

*φ*(

*x*,

*y*,

*z*

_{0}). Various methods for solving this equation have been described, for example, based on the development of the spatial phase on a base of Zernike polynomials [20

20. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: Matrix solution with use of zernike polynomials,” J. Opt. Soc. Am. A **12**, 1932–1941 (1995). [CrossRef]

21. 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**, 1670–1682 (1996). [CrossRef]

*I*(

*x*,

*y*,

*z*

_{0}) =

*I*

_{0}. This practical consideration allows a significant simplification of Eq. (3), which can be written as

*φ*] and FT[

*I*] are the two-dimensional spatial Fourier transforms of

*φ*and

*I*, respectively. Equation (5) allows one to write

14. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. **133**, 339–346 (1997). [CrossRef]

8. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. **77**, 2961–2964 (1996). [CrossRef] [PubMed]

*k*=

_{x}*k*= 0 stems from the fact that no information on the piston (i.e., the non-spatially varying component of the phase) can be retrieved by the TIE. The piston is in most cases of no interest in optical testing because it is relatively defined with respect to a reference optical wave and does not induce spatial modulation of a propagating optical wave. The reconstruction algorithm is schematized in Fig. 1(b). The Fourier transform of Δ

_{y}*I*=

*I*(

*x*,

*y*,

*z*

_{0}+

*dz*)/

*I*

_{0}-1 is calculated and divided by

*k*

^{2}

_{x}+

*k*

^{2}

_{y}, where this quantity is different from zero. An inverse Fourier transform of the obtained quantity leads to the phase under test after proper scaling.

*k*=

_{x}*k*= 0, the power spectral density (PSD) of the phase [22

^{y}22. A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. **41**, 154–171 (2002). [CrossRef] [PubMed]

*k*

^{2}

_{x}+

*k*

^{2}

_{y})

^{2}and with the propagation distance via

*dz*

^{2}.

## 3. Properties

*dz*and noise on the measured intensity is quantified. In particular, it is shown that the influence of the propagation distance

*dz*on the reconstruction accuracy strongly depends on the frequency content of the phase under test. While smaller distances improve the reconstruction accuracy, they also make the diagnostic more sensitive to noise because the intensity variation induced by the phase under test is smaller. Therefore, the propagation distance must be chosen as a balance between the frequency content of the phase under test and the amplitude of the noise on the measured data. Such balance has previously been studied in general terms and demonstrated with test images [15

15. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. **214**, 51–61 (2004). [CrossRef] [PubMed]

*φ*

_{test}(

*x*) =

*a*cos(2

*π*

*x*/

*δ*), where

*a*and

*δ*are the amplitude and period of the phase, respectively, is propagated along a distance using the kernel for Fresnel diffraction

*dz*, i.e., exp[

*ikx*

^{2}

*dz*/(2

*k*

_{0})]. The intensity of the obtained wave

*I*(

*x*,

*dz*) is used for phase reconstruction using Eq. (6). The wavelength was chosen as λ

_{0}= 632.8 nm.

*a*= 1 rad,

*dz*= 1 cm, where some discrepancy is visible. To understand this discrepancy, the quantity [

*I*(

*x*,

*dz*)-

*I*

_{0}]/(

*I*

_{0}

*dz*) is plotted for

*dz*= 1 mm and

*dz*= 1 cm. While the measured intensity is sinusoidal in the first case, Fresnel diffraction induces higher harmonics in the second case. As Eq. (6) implies that there is a one-to-one correspondence between harmonics in the measured intensity and phase under test, a decrease in accuracy is induced.

*φ*

_{test}and the reconstructed phase

*φ*

_{reconstructed}has been defined to quantify the phase reconstruction accuracy following

*x*

_{1},

*x*

_{2}] equal to one period of the phase. The rms error takes for value 0.2% and 3.2% for a sinusoidal phase with a 0.1-rad amplitude and a 0.2-mm period reconstructed respectively after propagation

*dz*= 1 mm and

*dz*= 1 cm, and increases to 1.8% and 14.2% for a sinusoidal phase with a 1-rad amplitude and a 0.2-mm period reconstructed respectively after propagation

*dz*= 1 mm and

*dz*= 1 cm. Figure 3 displays

*ε*

_{rms}as a function of the amplitude and period of the sinusoidal phase under test for three different propagation distances. Short propagation distance [e.g., 1 mm as plotted in Fig. 3(a)] leads to accurate phase retrieval over a large range of periods and amplitudes. As the propagation distance is increased [e.g., 1 cm in Fig. 3(b) and 10 cm in Fig. 3(c)], phases with high spatial frequencies are not reconstructed as accurately. This is due to the fact that the finite difference of the spatial intensity is not an accurate description of the longitudinal derivative of the intensity for these phases.

## 4. Experimental demonstration

16. D. Golini, S. Jacobs, W. Kordonski, and P. Dumas, “Precision optics fabrication using magnetorheological finishing,” in *Advanced Materials for Optics and Precision Structures*,
M. A. Ealey, R. A. Paquin, and T. B. Parsonage, eds., *Critical Reviews of Optical Science and Technology* (SPIE, Bellingham, WA, 1997), Vol. CR67, pp. 251–274.

17. V. Bagnoud, M. J. Guardalben, J. Puth, J. D. Zuegel, T. Mooney, and P. Dumas, “High-energy, high-average-power laser with Nd:YLF rods corrected by magnetorheological finishing,” Appl. Opt. **44**, 282–288 (2005). [CrossRef] [PubMed]

17. V. Bagnoud, M. J. Guardalben, J. Puth, J. D. Zuegel, T. Mooney, and P. Dumas, “High-energy, high-average-power laser with Nd:YLF rods corrected by magnetorheological finishing,” Appl. Opt. **44**, 282–288 (2005). [CrossRef] [PubMed]

*f*= 12 cm separated by the distance 2

*f*, and the distance from the surface to the first camera is

*f*. The camera is set on a translation stage in order to modify the virtual distance

*dz*between the image of the surface and the camera. The image measured from the camera is acquired by a 14-bit frame grabber for numerical processing on a computer. The relative rms noise on the acquired images was measured to be σ

_{noise}= 5 × 10

^{−3}, and the actual dynamic range on the measured intensity profiles is therefore only approximately eight bits. Images were typically recorded for a set of distances ranging from 12 mm to 36 mm. Phase reconstruction was in each case performed with only one measured image using Eq. (6). Figure 5(b) displays the evolution of the peak-to-valley of the measured phase with respect to the distance

*dz*. While detection noise influences the measurement of the peak-to-valley phase, its effect is not significant over distances larger than 15 mm in our implementation. As shown below, consistent phase measurements have been obtained over a large range of distances. One should note that the phase introduced by the surface of the rod is half the measured phase since the measurement is double pass.

*I*=

*I*/

*I*

_{0}− 1 measured as a function of the distance between the image of the surface and the detection plane. The intensity modulation increases when the propagation distance is increased. Three measured intensities and reconstructed phases for

*dz*= 12 mm,

*dz*= 24 mm, and

*dz*= 36 mm are plotted in Fig. 7. Good agreement between the retrieved phases is obtained, particularly as the propagation distance is increased. The rms error between the phases reconstructed at

*dz*= 24 mm and

*dz*= 36 mm is 0.04 rad. Finally, as a test of consistency, an optical wave with constant intensity and the reconstructed spatial phase in these three cases was generated and propagated by the corresponding distance. The normalized intensity of the calculated field after Fresnel diffraction is also plotted in Fig. 7 and shows excellent agreement with the measured intensity profiles. Increased accuracy is expected with the use of a scientific low-noise camera, which would have a dynamic range significantly higher than eight bits.

*dz*= 24 mm. The latter shows the presence of significant directionality in the surface variations, i.e., the spectral density takes significant values for

*k*= 0. The PSD of the measured intensity summed along

_{x}*x*is plotted in Fig. 8(c) for the propagation distances

*dz*= 12 mm, 24 mm, and 36 mm. As noted in Transport-of-intensity equation, the PSD is increasing because of the

*dz*dependence of the relation between the PSD of phase under test (which does not depend on the propagation distance) and the PSD of the measured intensity.

## 5. Conclusions

18. J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, “OMEGA EP: High-energy petawatt capability for the OMEGA laser facility,” J. Phys. IV France **133**, 75–80 (2006). [CrossRef]

## Acknowledgment

## References and links

1. | D. Malacara, “Optical Shop Testing,” 2nd ed., Wiley series in |

2. | M. R. Teague, “Deterministic phase retrieval: A green’s function solution,” J. Opt. Soc. Am. |

3. | M. R. Teague, “Irradiance moments: Their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. |

4. | M. R. Teague, “Image formation in terms of the transport equation,” J. Opt. Soc. Am. A |

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

6. | T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A |

7. | S. C. Woods and A. H. Greenaway, “Wave-front sensing by use of a green’s function solution to the intensity transport equation,” J. Opt. Soc. Am. A |

8. | K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. |

9. | S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy |

10. | A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. |

11. | J. R. Fienup, “Phase retrieval algorithms: A comparison,” Appl. Opt. |

12. | C. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. |

13. | C. Dorrer, “Characterization of nonlinear phase shifts by use of the temporal transport-of-intensity equation,” Opt. Lett. |

14. | T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. |

15. | D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. |

16. | D. Golini, S. Jacobs, W. Kordonski, and P. Dumas, “Precision optics fabrication using magnetorheological finishing,” in |

17. | V. Bagnoud, M. J. Guardalben, J. Puth, J. D. Zuegel, T. Mooney, and P. Dumas, “High-energy, high-average-power laser with Nd:YLF rods corrected by magnetorheological finishing,” Appl. Opt. |

18. | J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, “OMEGA EP: High-energy petawatt capability for the OMEGA laser facility,” J. Phys. IV France |

19. | V. Bagnoud, I. A. Begishev, M. J. Guardalben, J. Puth, and J. D. Zuegel, “5-hz, >250-mJ optical parametric chirped-pulse amplifier at 1053 nm,” Opt. Lett. |

20. | T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: Matrix solution with use of zernike polynomials,” J. Opt. Soc. Am. A |

21. | 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 |

22. | A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. |

**OCIS Codes**

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

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 13, 2007

Revised Manuscript: April 12, 2007

Manuscript Accepted: May 1, 2007

Published: May 29, 2007

**Citation**

C. Dorrer and J. D. Zuegel, "Optical testing using the transport-of-intensity equation," Opt. Express **15**, 7165-7175 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7165

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### References

- D. Malacara, "Optical Shop Testing," 2nd ed., Wiley series in Pure and Applied Optics (Wiley, New York, 1992).
- M. R. Teague, "Deterministic phase retrieval: A green’s function solution," J. Opt. Soc. Am. 73, 1434−1441 (1983). [CrossRef]
- M. R. Teague, "Irradiance moments: Their propagation and use for unique retrieval of phase," J. Opt. Soc. Am. 72, 1199−1209 (1982). [CrossRef]
- M. R. Teague, "Image formation in terms of the transport equation," J. Opt. Soc. Am. A 2, 2019−2026 (1985). [CrossRef]
- F. Roddier, "Wavefront sensing and the irradiance transport equation," Appl. Opt. 29, 1402−1403 (1990). [CrossRef] [PubMed]
- T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942−1946 (1995). [CrossRef]
- S. C. Woods and A. H. Greenaway, "Wave-front sensing by use of a green’s function solution to the intensity transport equation," J. Opt. Soc. Am. A 20, 508−512 (2003). [CrossRef]
- K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996). [CrossRef] [PubMed]
- S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000). [CrossRef] [PubMed]
- A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998). [CrossRef]
- J. R. Fienup, "Phase retrieval algorithms: A comparison," Appl. Opt. 21, 2758−2769 (1982). [CrossRef] [PubMed]
- C. Dorrer and I. Kang, "Complete temporal characterization of short optical pulses by simplified chronocyclic tomography," Opt. Lett. 28, 1481−1483 (2003). [CrossRef] [PubMed]
- C. Dorrer, "Characterization of nonlinear phase shifts by use of the temporal transport-of-intensity equation," Opt. Lett. 30, 3237−3239 (2005). [CrossRef] [PubMed]
- T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339-346 (1997). [CrossRef]
- D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004). [CrossRef] [PubMed]
- D. Golini, S. Jacobs, W. Kordonski, and P. Dumas, "Precision optics fabrication using magnetorheological finishing," in Advanced Materials for Optics and Precision Structures, M. A. Ealey, R. A. Paquin, and T. B. Parsonage, eds., Critical Reviews of Optical Science and Technology (SPIE, Bellingham, WA, 1997), Vol. CR67, pp. 251−274.
- V. Bagnoud, M. J. Guardalben, J. Puth, J. D. Zuegel, T. Mooney, and P. Dumas, "High-energy, high-average-power laser with Nd:YLF rods corrected by magnetorheological finishing," Appl. Opt. 44, 282−288 (2005). [CrossRef] [PubMed]
- J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006). [CrossRef]
- V. Bagnoud, I. A. Begishev, M. J. Guardalben, J. Puth, and J. D. Zuegel, "5-hz, >250-mJ optical parametric chirped-pulse amplifier at 1053 nm," Opt. Lett. 30, 1843−1845 (2005). [CrossRef] [PubMed]
- T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: Matrix solution with use of zernike polynomials," J. Opt. Soc. Am. A 12, 1932−1941 (1995). [CrossRef]
- 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, 1670−1682 (1996). [CrossRef]
- A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, "Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components," Appl. Opt. 41, 154-171 (2002). [CrossRef] [PubMed]

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