## Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region

Optics Express, Vol. 16, Issue 7, pp. 4901-4917 (2008)

http://dx.doi.org/10.1364/OE.16.004901

Acrobat PDF (844 KB)

### Abstract

The forward problem of focusing light using a high numerical aperture lens can be described using the Debye-Wolf integral, however a solution to the inverse problem does not currently exist. In this work an inversion formula based on an eigenfunction representation is derived and presented which allows a field distribution in a plane in the focal region to be specified and the appropriate pupil plane distribution to be calculated. Various additional considerations constrain the inversion to ensure physicality and practicality of the results and these are also discussed. A number of inversion examples are given.

© 2008 Optical Society of America

## 1. Introduction

11. M. A. A. Neil, T Wilson, and R. Jus̆kaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. **197**, 219–223 (2000). [CrossRef] [PubMed]

## 2. Theory of generalised prolate spheroidal functions

### 2.1. Space-bandwidth product

*r*

_{0}outside of which the function is negligible or of little interest. The product

*c*=

*r*

_{0}Ω is then called the space-bandwidth product and is often used as a measure of the optical performance of a system [16

16. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A **13**, 470–473 (1996). [CrossRef]

17. M. A. Neifeld, “Information, resolution, and space bandwidth product,” Opt. Lett. **18**, 1477–1479 (1998). [CrossRef]

*c*. This explicit dependence is however occasionally dropped in this work for clarity with the understanding the dependence still remains.

### 2.2. Eigenfunctions of the two dimensional finite Fourier integral

*finite*two dimensional Fourier transform over a circular domain can be written in the form

_{N,n}(

*c*,

*r*), known as the circular prolate spheroidal functions, are the eigenfunctions of the

*N*order finite Hankel transform. The defining relation for these functions can thus be expressed

^{th}*J*is the

_{N}*N*order Bessel function of the first kind,

^{th}*ω*and

*r*are conjugate coordinates and λ

_{N,n}are the circular prolate spheroidal eigenvalues. Figures. 1 and 2 show the behaviour of the eigenvalues and eigenfunctions respectively which are further discussed in Section 2.4.

_{N,n}used here are scaled versions of those developed by Slepian

*φ*

_{N,n}such that

*φ*

_{N,n}are also the solutions to the wave equation when expressed in a prolate spheroidal coordinate system.

### 2.3. Orthogonality and completeness of the generalised prolate spheroidal functions

*r*≤

*r*

_{0}i.e.

*δ*is the Kronecker delta and

_{nm}*δ*(

*r*-

*r*′) is the Dirac delta function centered on

*r*=

*r*′. Furthermore the prolate functions possess the unique property that they are also orthogonal and complete on the infinite interval 0≤

*r*≤∞.

*ψ*

_{N,n}in terms of exponentials as opposed to the trigonometric functions of Eq. (1). The coefficients

*A*

_{N,n}can be calculated using the orthogonality property

### 2.4. Energy concentration property of generalised prolate spheroidal functions

*f*(

*r*,

*ϕ*) within a circular region of radius

*r*is defined as

_{0}19. B. R. Frieden “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” Prog. Opt. **9**, 311–407 (1971). [CrossRef]

_{N,n}≤1 and monotonically decrease with

*N*and

*n*and as such the encircled energy takes its maximum value of

*I*=λ

^{max}_{enc}_{0,0}when

_{N,n}is a measure of the fraction of energy contained within the circular region defined by 0≤

*r*≤

*r*

_{0}and 0≤

*ϕ*<2

*π*[20

20. D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Review **25**, 379–393 (1983). [CrossRef]

*c*=10 case it is noted that the eigenvalues drop off rapidly at

*n*~3. As such when the

*n*=0 order is plotted in Fig. 2 it is non-zero when

*r*≤0 and essentially (although not precisely) zero outside. Higher order modes,

*n*=5 and 10, however display the converse behaviour.

*c*=20 case the eigenvalues remain close to unity up to higher orders and instead decrease at

*n*~6. When plotted the

*n*=0 mode displays the same properties as before, but now the

*n*=5 mode shows contributions for all values of

*r*considered. With an eigenvalue of 8.46×10

^{-9}the

*n*=10 order again contains negligible energy within the central region.

*c*=40 case the eigenvalues do not fall off until

*n*~13 meaning the plotted orders have only a small contribution for

*r*≥

*r*

_{0}.

## 3. The Debye-Wolf diffraction integral

*p*with Cartesian coordinates

**r**

*=(*

_{p}*x*,

_{p}*y*,

_{p}*z*) in the vicinity of the focal region of a telecentric, high NA lens as shown in Fig. 3 and can be written [21

_{p}21. E. Wolf “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy. Soc. A-Math Phy **253**, 349–357 (1959). [CrossRef]

22. B. Richards and E. Wolf “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A-Math Phy **253**, 358–379 (1959). [CrossRef]

**a**(

*s*,

_{x}*s*) is the strength vector of a geometric ray at the Gaussian reference sphere centered on the focal point,

_{y}**s**=(

*s*,

_{x}*s*,

_{y}*s*) is a unit ray vector and Θ is the domain of the exit pupil. The strength vector is easily modified by introduction of suitable optics in the pupil plane which can be written in terms of the pupil coordinates in the general form

_{z}*u*=sin

*θ*,

*g*(

*u*,

*ϕ*) and Ψ(

*u*,

*ϕ*) describe the amplitude and phase variation introduced to the incident field distribution

*ℒ*is the sub generalised Jones matrix given by

23. P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. **148**, 300–315 (1998). [CrossRef]

### 3.1. Eigenfunction expansion of the Debye-Wolf integral

*A*is a constant as per [22

22. B. Richards and E. Wolf “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A-Math Phy **253**, 358–379 (1959). [CrossRef]

*k*=2

*π*/λ is the wavenumber of the illuminating light,

*α*′=sin

*α*is the NA of the lens assumed to be in air,

*ρ*is the field of view in the focal space,

^{max}_{p}*A*

^{j}

_{m,N,n}are expansion coefficients and a switch to cylindrical polar coordinates has been made such that

*c*=

*kα*′

*ρ*since the spatial cutoff frequency of a lens is

^{max}_{p}*ω*

_{0}=

*kα*′ [24].

*E*component when compared to the original formulation [10

_{x}10. S. S. Sherif, M. R. Foreman, and P. Török, “Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system,” Opt. Express (to be published). [PubMed]

*A*

^{j}_{m,N,n}. This is done for convenience reasons only.

### 3.2. Inversion of the Debye-Wolf integral

*Q*,

*q*and integrating over a plane in the focal region. This yields the result

*N*and

*n*, namely the term for which

*Q*=

*N*and

*q*=

*n*. Thus

*A*

^{j}_{m,N,n’}but can however form the basis for numerical optimisation techniques, an example of which is given in Section 5. Unique solution can however be achieved on the focal plane i.e. when

*z*=0 whereby

_{p}*m*has now been dropped. This equation shows that the coefficients of the expansion of the weighting function are merely a scaled version of the coefficients of the expansion of the field in the focal plane as would be expected for an eigenfunction expansion. This is the basic inversion formula for the Debye-Wolf integral.

## 4. Some notes on inversion and electric field specification

### 4.1. Degrees of freedom

**a**(

*s*,

_{x}*s*) to produce a

_{y}*single*desired field component in the focal plane. Conceivably it would be possible to use Eq. (22) to calculate all three components of the strength vector, however such a naïve approach would not guarantee physicality or realisability. Maxwell’s equations mean that at best only two field components can be specified and used for inversion, however there is no restriction on which components are chosen.

### 4.2. Field specification away from the focal plane

*N*×

*n*equations for

*m*×

*N*×

*n*unknowns uniquely. This restriction can however be circumvented under certain circumstances.

*e*- and

*m*- theory must instead be used [25

25. B. Karczewski and E. Wolf, “Comparison of three theoreis of electromagnetic difftraction at an aperture Part I: coherence matrices, Part II: The far field,” J. Opt. Soc. Am **56**, 1207–19 (1966). [CrossRef]

### 4.3. Extrapolation and encircled energy

*ρ*. So as to ensure the completeness of the prolate functions over the specification area it is necessary to use the appropriate space-bandwidth product

^{max}_{p}*c*=

*kα′ρ*when calculating the coefficients from Eq. (19).

^{max}_{p}*E*as shown in Fig. 4(a) over a circle of radius ~1.6 times that of the Airy disc. Expansion of the specified field in terms of generalised prolate spheroidal functions as per Eqs. (6) and (19) allows extrapolation of the field beyond this region since [19

_{x}19. B. R. Frieden “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” Prog. Opt. **9**, 311–407 (1971). [CrossRef]

*f*(

*r*) over a finite region 0≤

*r*≤

*r*

_{0}it is possible to extrapolate to all values of

*r*>0 and is a consequence of the duality of completeness and orthogonality of the circular prolate spheroidal functions.

^{-4}. This behaviour arises since the high order modes contribute significantly as shown in 4(c) and thus energy is pushed out of the specification area as discussed in Section 2.4.

### 4.4. Noise amplification

*κ*is a commonly used quantity which measures the amplification

*A*of noise and errors in the initial data to the final inversion [29] such that

*A*∝

*κ*. When using an eigenfunction inversion method the condition number can be defined as the ratio of the largest and smallest non-zero eigenvalue, that is

^{min}

_{N,n}is the value of the smallest eigenvalue used in the truncated series expansion. It is thus advisable to use orders that lie within or close to the plateau of eigenvalues of Fig. 1 to reduce noise amplification. Since small eigenvalues (high orders) correspond to high frequency components better inversion will be obtained for smoother, slower varying fields.

### 4.5. Pixelation

11. M. A. A. Neil, T Wilson, and R. Jus̆kaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. **197**, 219–223 (2000). [CrossRef] [PubMed]

*jk*

^{th}pixel of the SLM be set such that the output field is the average of the ideal profile over the domain Π

*of the pixel i.e.*

_{jk}*S*denotes the area of the

_{jk}*jk*

^{th}pixel. Minimisation can also be performed on the Gaussian reference sphere, although this would require projection back to pupil plane to determine the appropriate SLM configuration.

## 5. Examples

### 5.1. Polarisation structuring

30. T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single molecule dynamics studied by polarization modulation,” Phys. Rev. Lett. **77**, 3979–3982 (1996). [CrossRef] [PubMed]

31. B. Sick, B. Hecht, and L. Novotny “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. **85**4482–4485 (2000). [CrossRef] [PubMed]

**p**, illuminated by a field

**E**re-radiates light as if it had an effective dipole moment proportional to

**p**·

**E**. Efficient coupling thus entails minimising the longitudinal component of the focused field. Inverting a field specification of

*E*=0 gives a zero strength vector, meaning that such a specification cannot be achieved via apodisation or phase masks. However a beam with a non-uniform polarisation distribution can be used. There exist numerous methods to produce these so-called vector beams including: modification of laser cavities, via introduction of polarisation sensitive components such that only modes with the desired polarisation structure can lase [32

_{z}32. Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE. **60**, 1107–1109 (1972). [CrossRef]

33. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. **29**, 2234–2239 (1990). [CrossRef] [PubMed]

34. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. **27**, 285–287 (2002). [CrossRef]

35. K. C. Toussaint Jr., S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett **30**, 2846–2848 (2005). [CrossRef]

36. M. A. A. Neil, F. Massoumian, R. Jus̆kaitis, and T. Wilson “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. **27**, 1929–1931 (2002). [CrossRef]

36. M. A. A. Neil, F. Massoumian, R. Jus̆kaitis, and T. Wilson “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. **27**, 1929–1931 (2002). [CrossRef]

*et al.*[35

35. K. C. Toussaint Jr., S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett **30**, 2846–2848 (2005). [CrossRef]

*g*(

*u*,

*ϕ*)=1 and Ψ(

*u*,

*ϕ*)=0. It has been observed that azimuthally polarised light when focused has a very weak longitudinal component [37

37. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express **7**77–87 (2000). [CrossRef] [PubMed]

^{-17}; a number most likely attributable to numerical noise and inversion hence gives a suitable solution.

### 5.2. Extended depth of field

38. W. T Welford “Use of Annular Apertures to Increase Focal Depth,” J. Opt. Soc. Am. **50**, 749–753 (1960). [CrossRef]

41. T. C. Poon and M. Motamedi, “Optical/digital incoherent image processing for extended depth of field,” Appl. Opt. **26**, 4612–4615 (1987). [CrossRef] [PubMed]

*x*polarised. Consequently only the

*E*field component contributes to the axial behaviour which is thus specified as

_{x}*E*

_{0}is a constant and

*w*denotes the half width of the rect function. On axis Eq. (20) reduces to

_{|N|,n}(0)=0 for

*N*≠0 i.e. only

*N*=0 orders contribute on axis. Using Eqs. (19), (26) and (27) it is possible to numerically optimise the coefficients to find a good solution to the problem. A popular method of doing this is that of simulated annealing [43

43. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science **220**, 671–680 (1983). [CrossRef] [PubMed]

*T*which is slowly reduced. In simulated annealing a loss function is defined which is analogous to the energy in an annealing process. For the current example this was taken as the Hilbert angle

*ψ*as defined by

_{H}*E*(0,0,

_{x}*z*) and

_{p}*E*(0,0,

^{opt}_{x}*z*) respectively [42] ranging from 0 if they are identical, to

_{p}*π*/2 if they are orthogonal. Suitable truncation points for termination of the series in Eq. (27) can be determined as discussed in [10

10. S. S. Sherif, M. R. Foreman, and P. Török, “Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system,” Opt. Express (to be published). [PubMed]

*m*

_{0}=42. Rejecting eigenvalues smaller than 10

^{-5}, so as to limit noise amplification gave

*n*

_{0}=9. The resulting intensity profile as found from the 378 optimised coefficients is shown in Fig. 6(a) as compared to the desired rect function. The minimum Hilbert angle found was approximately 7π/200.

44. C. W. McCutcheon, “Generalised Aperture and the Three-Dimensional Diffraction Image,” J. Opt. Soc. Am. **54**, 240–244 (1964). [CrossRef]

45. J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett **10**, 520–522 (1985). [CrossRef] [PubMed]

### 5.3. Superresolution

46. T. di Francia, “Super-gain antennas and, optical resolving power,” Nuovo Cimento **9**, 426–438 (1952). [CrossRef]

47. Z. S. Hegedus and V. Sarafis, “Superresoliving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A **3**, 1892–1896 (1986). [CrossRef]

*E*field component as a Dirac delta function centered on the origin. Only one component of the focused field is specified since this introduces 2 degrees of freedom into the inversion problem as required for polarisation structuring.

_{x}*E*is hence written in the form

_{x}_{|N|,n}(0)=0 for

*N*≠0 immediately gives

*e*

^{2}

*=1-*

_{x}*e*

^{2}

*a quadratic equation in terms of*

_{y}*e*can be found, from which the required incident field distributions can be found. In practice however this method does not achieve superresolution for the simple reason that insufficient control is exerted on the

_{x}*y*and

*z*components of the focused field. As such when a delta function is specified for the

*x*component energy is pushed into the

*y*component. The resultant focused distribution is then essentially identical to that of a uniformly

*y*polarised beam for which there is no resolution improvement.

*E*and

_{x}*E*focused field components to be Dirac delta functions. By the same logic this means

_{y}*A*

^{x}_{N,n}=

*A*

^{y}_{N,n}as given by Eq. (33). Since

*a*(

_{j}*u*,

*ϕ*)=∑

^{∞}

_{n=0}

*A*

^{j}_{0,n}Φ

_{0,n}(

*c*,

*u*) the required incident field distributions can be found using Eqs. (12), (33) and are given by

*ℒ*denotes the

_{pq}*pq*

^{th}element of

*ℒ*as given in Eq. (14). The (1-

*u*

^{2})

^{-1/2}factor is required to conserve energy when projecting from the surface of the Gaussian reference sphere to the pupil plane [48].

*A*

^{x}_{N,n}and

*A*

^{y}_{N,n}are both real the weighting functions

*a*(

_{x}*u*,

*ϕ*) and

*a*(

_{y}*u*,

*ϕ*) are real. Projecting back to the pupil plane is not a complex operation and hence the field in the pupil plane is also real. It is thus apparent that the field in the pupil plane is linearly polarised and no phase modulation is necessary. Apodisation is however necessary as can be seen by considering

*n*=

*n*

_{0}, meaning the pupil and focal plane field distributions will differ from the ideal case in a way that is dependent on the truncation point. Figure 7(a) represents the required pupil plane distribution for

*n*

_{0}=1 whilst Fig. 8 shows the corresponding optical distribution in the focal plane. The shown distributions were calculated assuming

*NA*=0.966 and a value of

*c*=4 corresponding to a field of view in the focal plane approximately the size of the Airy disc.

*E*and

_{x}*E*distributions as compared to a clear aperture with uniform illumination, however there is little gain in the intensity focal spot. This again arises from a redistribution of energy to the unconstrained field component

_{y}*E*which is then dominant in the final intensity profile. Furthermore due to the presence of the apodising mask this arrangement also has a low optical efficiency as can be seen in the plot of the Strehl intensity ratio as shown in Fig. 7(b) as a function of the truncation order

_{z}*n*

_{0}. At high

*n*

_{0}this quantity loses its meaning however since the central peak essentially vanishes with respect to the large sidelobes as would be expected from the discussion in Section 4.3. The performance of this particular superresolution setup consequently worsens as

*n*

_{0}is increased.

## 6. Conclusions

*N*=0 modes have non-zero coefficients.

## Acknowledgments

## References and links

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18. | J. C. Heurtley, “Hyperspheroidal functions - optical resonators with circular mirrors,” Proc. Symp. on Quasi Optics, New York p.367 (1964) |

19. | B. R. Frieden “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” Prog. Opt. |

20. | D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Review |

21. | E. Wolf “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy. Soc. A-Math Phy |

22. | B. Richards and E. Wolf “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A-Math Phy |

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24. | J. W. Goodman, |

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32. | Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE. |

33. | S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. |

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35. | K. C. Toussaint Jr., S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett |

36. | M. A. A. Neil, F. Massoumian, R. Jus̆kaitis, and T. Wilson “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. |

37. | K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express |

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45. | J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett |

46. | T. di Francia, “Super-gain antennas and, optical resolving power,” Nuovo Cimento |

47. | Z. S. Hegedus and V. Sarafis, “Superresoliving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A |

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**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.1960) Diffraction and gratings : Diffraction theory

(100.3190) Image processing : Inverse problems

(180.0180) Microscopy : Microscopy

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 27, 2008

Revised Manuscript: March 25, 2008

Manuscript Accepted: March 25, 2008

Published: March 26, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Matthew R. Foreman, Sherif S. Sherif, Peter R. T. Munro, and Peter Török, "Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region," Opt. Express **16**, 4901-4917 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4901

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