## Numerical algorithm for the retrieval of spatial coherence properties of partially coherent beams from transverse intensity measurements

Optics Express, Vol. 15, Issue 21, pp. 13613-13623 (2007)

http://dx.doi.org/10.1364/OE.15.013613

Acrobat PDF (583 KB)

### Abstract

A novel algorithm for the retrieval of the spatial mutual coherence function of the optical field of a light beam in the quasimonochromatic approximation is presented. The algorithm only requires that the intensity distribution is known in a finite number of transverse planes along the beam. The retrieval algorithm is based on the observation that a partially coherent field can be represented as an ensemble of coherent fields. Each field in the ensemble is propagated with coherent methods between neighboring planes, and the ensemble is then subjected to amplitude restrictions, much in the same way as in conventional phase recovery algorithms for coherent fields. The proposed algorithm is evaluated both for one- and two-dimensional fields using numerical simulations.

© 2007 Optical Society of America

## 1. Introduction

1. F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. **61**, 158–164 (1948). [CrossRef]

5. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Letters **72**, 1137–1140 (1993). [CrossRef]

11. B. Y. Gu, G. Z. Yang, and B. Z. Dong, “General theory for performing an optical transform,” Appl. Opt. **25**, 3197–3206 (1986). [CrossRef] [PubMed]

## 2. Theory

*u*denotes the complex amplitude of the quasimonochromatic field in the slowly-varying envelope formulation,

*u*

^{*}is its complex conjugate, and ρ = (

*x*,

*y*) denotes spatial coordinates in a plane perpendicular to the direction of the propagating field, the latter which we take to be along the

*z*-axis. For a temporally stationary field, the amount of mutual temporal coherence is solely dependent on the time difference τ =

*t*

_{2}-

*t*

_{1}. Recalling the Wiener-Khintchine theorem, the cross-spectral density function for this field is written as [12],

*S*is essentially independent of

*v*within the narrow-band frequency range of the quasimono-chromatic field. In this case the function

*S*(ρ

_{1},ρ

_{2}) is often called the (spatial) mutual coherence function. This is the function we want to calculate with our retrieval algorithm.

### 2.1. Uniqueness of the spatial coherence properties for a certain intensity distribution

15. D. Dragoman, “Unambiguous coherence retrieval from intensity measurements,” J. Opt. Soc. Am. A **20**, 290–295 (2003). [CrossRef]

## 3. The proposed retrieval algorithm

16. C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A **23**, 1616–1625 (2006). [CrossRef]

*z*, the partially coherent field can be represented by an ensemble of coherent, but mutually incoherent, fields

*N*coherent fields,

*U*(ρ,

_{i}*z*),

*i*= 1,…,

*N*. Further, ρ is the transverse coordinates in the plane. Since the fields are mutually incoherent, there is no correlation between different fields so that the intensity distribution of the representation becomes

*z*

17. H. Stark and Y.Y. Yang, *Vector space projections: A numerical approach to signal and image processing, neural nets, and optics*, Wiley, New York (1998). [PubMed]

11. B. Y. Gu, G. Z. Yang, and B. Z. Dong, “General theory for performing an optical transform,” Appl. Opt. **25**, 3197–3206 (1986). [CrossRef] [PubMed]

18. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. **32**, 1737–1746 (1993). [CrossRef] [PubMed]

*I*(ρ,

_{ref}*z*), is known in a number,

*N*, of transverse planes located at

_{t}*z*=

*z*,

_{n}*n*= 1,…,

*N*. The algorithm is based on repeated projection of the representation of the partially coherent field, in one plane after another, onto a new representation having an intensity distribution that is equal to the known intensity distribution in that plane. The amplitude of every coherent field in the representation is scaled by the same real positive multiplicative function. After the projection in one plane, each coherent field is individually propagated to the next transverse plane where the intensity distribution is known, and a new projection is performed. To be more specific, starting from the first operation after having propagated all coherent fields to a new plane, the proposed algorithm can be expressed as the following sequence of basic operations,

_{t}*P*is the propagation operator for a coherent field from the plane located in

*z*=

*z*to a plane located in

_{n}*z*=

*z*

_{n+1}. Either the two-step method, presented in [16

16. C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A **23**, 1616–1625 (2006). [CrossRef]

^{+}indicates that the coherent fields are corrected, i.e. projected, so that the intensity distribution of the field representation in the plane equals the known reference intensity distribution in each lateral position ρ. Further,

*U*(ρ,

_{i}*z*) is the distribution in the plane at

_{n}*z*=

*z*of the

_{n}*i*:th coherent field, and

*I*(ρ,

*z*) is the intensity distribution of the entire field representation, as given by Eq. 4. The expression in Eq. 6 defines the amplitude correction factor that is required in the current transverse plane in order for the intensity of the representation to equal the known intensity in every lateral position. Equation 7 shows how this factor is multiplied to each individual coherent field in the representation to produce the corrected coherent field. By doing so the intensity of the field representation containing the modified coherent fields equals the known intensity in the plane, as desired. Finally, Eq. 8 describes how each coherent field is then individually propagated to the next transverse plane, after which the field operations continue from Eq. 6 for the fields in the plane at

_{n}*z*=

*z*

_{n+1}. The operations of the algorithm are illustrated in Fig. 1. Since the algorithm is iterative, there must be a starting distribution for all coherent fields in the plane from which the retrieval algorithm begins. In the examples in section 4, each of the coherent fields was generated from a unique random field as described in [16

16. C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A **23**, 1616–1625 (2006). [CrossRef]

*z*=

*z*by applying Eq. 5, using the ensemble of coherent fields that was obtained in this plane, i.e., {

_{n}*U*

_{1}(ρ,

*z*),

_{n}*U*

_{2}(ρ,

*z*),…,

_{n}*U*(ρ,

_{N}*z*)}.

_{n}## 4. Evaluation of the retrieval algorithm

**23**, 1616–1625 (2006). [CrossRef]

### 4.1. Retrieval for a one-dimensional field: the Gaussian-Schell beam

*x*, in the planes would be sufficient, since the field is one-dimensional) in combination with an efficient retrieval algorithm should enable us to calculate the spatial coherence properties quite accurately.

*x*

^{2}-

*x*

^{1}∣. For the Gaussian-Schell beam this dependency is also described by a Gaussian function. The intensity distribution of the Gaussian-Schell beam at any point in space can be calculated directly from an analytical expression [12]. Moreover, the mutual coherence function in any transverse plane can be described analytically as

*w*is the 1/

*e*

^{2}-radius of the intensity profile, 2σ

_{g}is the 1/

*e*

^{2}-radius of the two-point correlation function, and

*f*is the radius of the phase curvature. Of course, since the problem under study is one-dimensional, the fields and the mutual coherence function are invariant along the

*y*-axis. This means, naturally, that all fields are sampled only along the

*x*-direction, and

*x*

_{1}and

*x*

_{2}denote the

*x*-coordinates for two positions between which the mutual coherence is calculated. However, by convention we use the term radius for

*w*and σ

_{g}although the term distance would perhaps be less confusing in the one-dimensional case. In the simulations conducted the following numerical values were used for the parameters:

*w*= 2.4∙10

^{3}λ, σ

_{g}= 2.8∙10

^{3}λ, and

*f*= 1.6∙10

^{7}λ, where λ is the center wavelength of the quasimonochromatic radiation.

*E*(

*S*,

_{calc}*S*) that indicates the normalized distance between the mutual coherence function that is calculated with the retrieval algorithm,

_{ref}*S*(

_{calc}*x*

_{1},

*x*

_{2}), and the known solution,

*S*(

_{ref}*x*

_{1},

*x*

_{2}) from Eq. 9 with the mentioned numerical values of the parameters. The error measure is defined as

*E*(

*S*,

_{calc}*S*) indicates that the two functions,

_{ref}*S*and

_{calc}*S*, are similar and thus that the retrieval was successful. The error measure is normalized in the sense that

_{ref}*E*(

*S*,

_{calc}*S*) ∈ [0,1] and equals zero only if

_{ref}*S*and

_{calc}*S*are identical.

_{ref}*z*= {

*z*

_{1}, …,

*Z*

_{4}} = {0, ⅓, ⅔, 1} ∙ 8 ∙ 10

^{6}λ. Thereafter the analytically calculated intensity distributions are used to retrieve the mutual coherence function

*S*(

*x*

_{1},

*x*

_{2}) in one of the planes (the one with

*z*=

*z*

_{1}= 0 in our example) with the described retrieval algorithm. In this investigation the retrieval algorithm was executed for a sufficient number of iterations for it to converge. It was found that the accuracy of the retrieved spatial coherence function depends on several factors. First an analysis of the accuracy as a function of the number of fields,

*N*, included in the ensemble was performed. Figure 2 shows the error measure,

*E*(

*S*,

_{calc}*S*), for the retrieved coherence function,

_{ref}*S*(

_{calc}*x*

_{1},

*x*

_{2}), in the plane

*z*=

*z*

_{1}as a function of TV. It is evident that the accuracy increases with the number of fields in the ensemble, although a fair agreement can be obtained with quite a low number of fields (< 100). Second, an analysis of the convergence properties of the retrieval algorithm was conducted. Figure 3 shows the error measure,

*E*(

*S*,

_{calc}*S*), of the retrieved coherence function as a function of the number of iterations, where one iteration constitutes a full cycle of going through all transverse planes once and a backpropagation of each coherent field from the final plane at

_{ref}*z*=

*z*to the starting plane at

_{Nt}*z*=

*z*

_{1}. In this numerical simulation the number of fields,

*N*, included in the ensemble was large enough not to affect the results of the simulations; see Fig. 2 for information on the minimum number of fields needed. Three different cases were analyzed for the same beam, but where the transverse planes were located differently before and after the waist of the focused beam. The separation of the planes was uniform for each of the three cases. In Fig. 3, the case shown in the top inset is for the planes at the previously mentioned locations

*z*= {

*z*

_{1},…,

*z*

_{4}}. The simulations indicate that the convergence rate may be considerably slower for an unfortunate choice of the positions of the transverse planes. Intuitively, these results should not be too surprising, as the retrieval is based on how the coherence properties influence the propagation behavior. For this reason the planes should be separated far enough for the propagation to have considerably changed the intensity distribution. The focal region of a beam displays a particularly profound change of the intensity distribution over a rather limited propagation distance. Inserting transverse planes before and after the beam waist should thus facilitate the extraction of coherence information and reduce the required number of iterations in the retrieval.

### 4.2. Retrieval for a two-dimensional field: the twisted Gaussian-Schell beam

*n*to

*n*+1, where these two planes are the ones between which the element is inserted. For this propagation, the propagation operator in Eq. 8 is modified to a sequence of three operations so that

*z*is the position of the inserted element and

_{element}*P*→

_{zn}*z*and

_{element}*P*→

_{zelement}*z*

_{n+1}are the same free space coherent propagation operators as the one in Eq. 8, only here the propagation is from the plane at

*z*=

*z*to the plane at

_{n}*z*=

*z*, and from that plane to the plane at

_{element}*z*=

*z*

_{n+1}, respectively. The

*P*operator is the spatial complex transmission function of the inserted element—for a lens it is thus simply a parabolically varying phase function.

_{element}14. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A **11**, 1818–1825 (1994). [CrossRef]

_{1}= (

*x*

_{1},

*y*

_{1}) and ρ

_{2}= (

*x*

_{2},

*y*

_{2}) are used for the coordinates of two points in the transverse plane. The parameter

*u*is the twist parameter, for which the value

*u*= 2.0∙10

^{-8}λ

^{-1}is assigned in our simulations. The other beam parameters have the same value as for the one-dimensional beam.

*E*(

*S*,

_{calc}*S*), is again calculated to compare the retrieved spatial coherence function,

_{ref}*S*, with

_{calc}*S*. The error measure as a function of the number of iterations is shown in Fig. 4. As expected, when we use an ordinary rotationally symmetric lens the retrieved spatial coherence function does not converge toward our reference

_{ref}*S*. Although the algorithm itself converges it is unable to find the desired solution because the problem is not unique. Clearly, as explained, inserting the non-rotationally symmetric cylinder lens can restore the uniqueness which enables the retrieved spatial coherence function to converge toward the actual spatial coherence function.

_{ref}### 4.3. Evaluation of the sensitivity to noise in intensity data and measurement setup inaccuracy

## 5. Conclusions

## References and links

1. | F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. |

2. | B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. |

3. | J. Schwider, “Continuous Lateral Shearing Interferometer,” Appl. Opt. |

4. | C. Chang, p. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. |

5. | M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Letters |

6. | M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite-Gaussian modes,” Appl. Opt. |

7. | H. Laabs, B. Eppich, and H. Weber, “Modal decomposition of partially coherent beams using the ambiguity function,” J. Opt. Soc. Am. A |

8. | R. Borghi, G. Guattari, L. de la Torre, F. Gori, and M. Santarsiero, “Evaluation of the spatial coherence of a light beam through transverse intensity measurements,” J. Opt. Soc. Am. A |

9. | M. Ježek and Z. Hradil, “Reconstruction of spatial, phase, and coherence properties of light,” J. Opt. Soc. Am. A |

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

11. | B. Y. Gu, G. Z. Yang, and B. Z. Dong, “General theory for performing an optical transform,” Appl. Opt. |

12. | L. Mandel and E. Wolf, |

13. | F. Gori and M. Santarsiero, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A |

14. | A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A |

15. | D. Dragoman, “Unambiguous coherence retrieval from intensity measurements,” J. Opt. Soc. Am. A |

16. | C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A |

17. | H. Stark and Y.Y. Yang, |

18. | J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. |

19. | J. W. Goodman, |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(100.3190) Image processing : Inverse problems

(100.5070) Image processing : Phase retrieval

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: July 10, 2007

Revised Manuscript: September 12, 2007

Manuscript Accepted: October 1, 2007

Published: October 3, 2007

**Citation**

Christer Rydberg and Jörgen Bengtsson, "Numerical algorithm for the retrieval of spatial coherence properties of partially coherent beams from transverse intensity measurements," Opt. Express **15**, 13613-13623 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13613

Sort: Year | Journal | Reset

### References

- F. Zernike, "Diffraction and optical image formation," Proc. Phys. Soc. 61,158-164 (1948). [CrossRef]
- B. J. Thompson and E. Wolf, "Two-beam interference with partially coherent light," J. Opt. Soc. Am. 47,895-902 (1957). [CrossRef]
- J. Schwider, "Continuous Lateral Shearing Interferometer," Appl. Opt. 23, 4403-4409 (1983). [CrossRef]
- C. Chang, P. Naulleau, E. Anderson, and D. Attwood, "Spatial coherence characterization of undulator radiation," Opt. Commun. 182, 25-34 (2000). [CrossRef]
- M. G. Raymer, M. Beck and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Letters 72, 1137-1140 (1993). [CrossRef]
- M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, "Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite-Gaussian modes," Appl. Opt. 38, 5272-5281 (1999). [CrossRef]
- H. Laabs, B. Eppich, and H. Weber, "Modal decomposition of partially coherent beams using the ambiguity function," J. Opt. Soc. Am. A 19, 497-504 (2002). [CrossRef]
- R. Borghi, G. Guattari, L. de la Torre, F. Gori, and M. Santarsiero, "Evaluation of the spatial coherence of a light beam through transverse intensity measurements," J. Opt. Soc. Am. A 20, 1763-1770 (2003). [CrossRef]
- M. Je?zek and Z. Hradil, "Reconstruction of spatial, phase, and coherence properties of light," J. Opt. Soc. Am. A 21, 1407-1416 (2004). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
- B. Y. Gu, G. Z. Yang, and B. Z. Dong, "General theory for performing an optical transform," Appl. Opt. 25, 3197-3206 (1986). [CrossRef] [PubMed]
- L. Mandel and E. Wolf, Optical coherence and quantum optics, Cambridge University Press, Cambridge (1995).
- F. Gori and M. Santarsiero, "Coherence and the spatial distribution of intensity," J. Opt. Soc. Am. A 10, 673-679 (1993). [CrossRef]
- A. T. Friberg, E. Tervonen, and J. Turunen, "Interpretation and experimental demonstration of twisted Gaussian Schell-model beams," J. Opt. Soc. Am. A 11,1818-1825 (1994). [CrossRef]
- D. Dragoman, "Unambiguous coherence retrieval from intensity measurements," J. Opt. Soc. Am. A 20, 290-295 (2003). [CrossRef]
- C. Rydberg and J. Bengtsson, "Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function," J. Opt. Soc. Am. A 23, 1616-1625 (2006). [CrossRef]
- H. Stark and Y.Y. Yang, Vector space projections: A numerical approach to signal and image processing, neural nets, and optics, Wiley, New York (1998). [PubMed]
- J. R. Fienup, "Phase-retrieval algorithms for a complicated optical system," Appl. Opt. 32, 1737-1746 (1993). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier optics, McGraw-Hill, New York, 1996.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.