## Fast focus field calculations

Optics Express, Vol. 14, Issue 23, pp. 11277-11291 (2006)

http://dx.doi.org/10.1364/OE.14.011277

Acrobat PDF (531 KB)

### Abstract

We present a fast calculation of the electromagnetic field near the focus of an objective with a high numerical aperture (NA). Instead of direct integration, the vectorial Debye diffraction integral is evaluated with the fast Fourier transform for calculating the electromagnetic field in the entire focal region. We generalize this concept with the chirp z transform for obtaining a flexible sampling grid and an additional gain in computation speed. Under the conditions for the validity of the Debye integral representation, our method yields the amplitude, phase and polarization of the focus field for an arbitrary paraxial input field on the objective. We present two case studies by calculating the focus fields of a 40×1.20 NA water immersion objective for different amplitude distributions of the input field, and a 100×1.45 NA oil immersion objective containing evanescent field contributions for both linearly and radially polarized input fields.

© 2006 Optical Society of America

## 1. Introduction

1. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. **30**, 755–776 (1909). [CrossRef]

2. E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field,” Proc. R. Soc. London Ser. A **253**, 349–357 (1959). [CrossRef]

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

5. P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. **36**, 2305–2312 (1997). [CrossRef] [PubMed]

7. G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, “Diffractive elements for imaging with extended depth of focus,” Opt. Eng. **44**, 058001 (2005). [CrossRef]

8. N. Huse, A. Schönle, and S.W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. **6**, 480–484 (2001). [CrossRef]

9. J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, “Performance of Fluorescence Correlation Spectroscopy for Measuring Diffusion and Concentration,” Chem. PhysChem. **6**, 2324–2336 (2005). [CrossRef]

## 2. The Debye diffraction integral as Fourier transform

_{1}and is transferred to the principal plane ℙ

_{2}. At ℙ

_{2}, the wave field is refracted and focused towards the focal point

**F**

_{2}. The point

**P**lies on the principal plane ℙ

_{2}and illustrates the focusing of a ray at ℙ

_{2}towards the focal point

**F**

_{2}. The spherical surface ℙ

_{2}is centered at

**F**

_{2}and the deflection angle

*θ*at the position

**P**is given by

*r*is the off-axis coordinate of the incident wave,

*R*the aperture stop radius,

*NA*the numerical aperture of the objective and

*n*

_{t}the index of refraction behind the ℙ

_{2}surface. In our setup, the aperture A is placed in the back focal plane, which results in a telecentric imaging system.

*inside the objective*is not obvious if the incident wave is transferred directly from the entrance pupil to the exit pupil. Within our representation, the wave propagation from the aperture plane A to the principal plane ℙ

_{1}is easily calculated with the PWS method or in most cases based on classical Fourier optics principles.

*E⃗*

_{i}(

*r*,

*ϕ*) at ℙ

_{1}is decomposed into a radial component (p-polarized) and a tangential component (s-polarized). The unit vectors for p- and s-polarization are

*ϕ*is the azimuth angle around the

*z*-axis. Upon transmission, the unit vector

*e⃗*

_{p}is deflected by

*θ*and becomes

_{2}is

*t*

_{p}(

*θ*,

*ϕ*) and

*t*

_{s}(

*θ*,

*ϕ*) are the transmission coefficients (viz pupil function, apodization) for p- and s-polarization, respectively. Accumulated phase distortions, i.e. aberrations at ℙ

_{2}, as well as attenuations, i.e. amplitude factors, are integrated in the complex parameters

*t*

_{p}and

*t*

_{s}. As we assume the incident field to be paraxial, the axial component

*E*

_{iz}is small against the lateral components

*E*

_{ix,y}and can be neglected even if the incident phase is not constant. In the Debye approximation, the transmitted field

*E⃗*

_{t}is the

*plane wave spectrum*of the focus field

*E⃗*near

**F**

_{2}. Hence, the electric field

*E⃗*at a point (

*x*,

*y*,

*z*) is obtained by integrating the propagated plane waves, viz

*z*-axis, whereas the term

*x*,

*y*,

*z*) with respect to the on-axis point (0, 0,

*z*). The integration extends over the solid angle Ω under which ℙ

_{2}is observed at

**F**

_{2}, i.e. sinΘ=

*NA*/

*n*

_{t}. The wave vector

*k⃗*

_{t}is given in spherical coordinates

*θ*and

*ϕ*by

2. E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field,” Proc. R. Soc. London Ser. A **253**, 349–357 (1959). [CrossRef]

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

*θ*,

*ϕ*)-sampling keeping dΩ=sin

*θ*d

*θ*d

*ϕ*constant is obtained by using cos

*θ*

_{m}=1-

*m*ΔΘ with

*m*∊ℕ. For

*m*∊{1…

*M*} and

*n*∊{1…

*N*}, the sampling grid is defined by

*θ*=0, a sampling point with a weight of dΩ=

*θ*, the calculation of the integrand and its integration can be merged in a single matrix product resulting in a further reduction of the computation time [4].

_{1}instead of ℙ

_{2}. Using Eq. (1) and (6), the integration step dΩ for a sampling over ℙ

_{2}is projected onto ℙ

_{1}, which yields

*k*

_{x},

*k*

_{y})∊ℝ

^{2}by setting |

*E⃗*

_{t}|=0 for

*r*>

*R*allows to rewrite the Debye diffraction integral as a Fourier transform of the weighted field

*E⃗*

_{t}, which finally results in

*θ*≈1 and Eq. (10) is equivalent to the Fraunhofer diffraction integral.

## 3. Numerical implementation

_{2}is used for the numerical evaluation of Eq. (10). For an equidistant sampling

*k*

_{x}=

*m*Δ

*K*and

*k*

_{y}=

*n*Δ

*K*with Δ

*K*=

*k*

_{0}

*NA*/

*M*, viz

*M*sampling points over the aperture radius, the sampling points on ℙ

_{2}are

*K*)

^{2}with the prefactor of Eq. (10) yields the numerical implementation of Eq. (10) as

*M*

^{2}≳ 100×100 sampling points over Ω, but care has to be taken in order to avoid artifacts due to sampling and aliasing. Subsequently, the necessary conditions for obtaining accurate results are investigated [10].

### 3.1. Sampling condition

11. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A **6**, 786–805 (1989). [CrossRef]

*k*

_{z}

*z*must not change by more than

*π*between neighboring sampling points in the aperture plane A. With

*K*=

*k*

_{0}

*NA*/

*M*and

*M*≳50 reveals necessary for an accurate sampling of

*ϕ*. Deviations from these sampling conditions result in granular artifacts as seen in Fig. 3(a). As a typical value for

*M*, we have chosen

*M*=125 for the focus field calculation of a 1.20 NA water immersion objective (see the example 4.1). A high accuracy is obtained for |

*z*|≲25

*λ*

_{0}, corresponding to ≈12

*µ*m at a wavelength of 488 nm.

### 3.2. Sampling step

*E⃗*is obtained for the sampling positions (

*m*Δ

*x*,

*n*Δ

*y*,

*z*). With Δ

*k*=2

*π*/

*N*Δ

*r*and Δ

*r*=

*f*Δ

*K*/

*k*

_{t}, the sampling step in the

*xy*-plane is

*N*>4

*M*is the number of FFT sampling points per transformed dimension (see also Fig. 2, where the arrows span over 2

*M*+1 samples and the padded dimension over

*N*samples). For optimal FFT performance, it is best to set

*N*=2

^{s}with

*s*∈ℕ. Respecting the condition (14),

*M*can be adjusted to fit Δ

*x*and Δ

*y*. Along the

*z*-direction, the sampling can be chosen arbitrarily by respecting the limits given above.

### 3.3. Aliasing suppression

*E⃗*

_{t}is the plane wave spectrum of the focus field

*E⃗*. Usually, the smallest area (aperture matrix) containing

*E⃗*

_{t}≠0 is transformed (see Fig. 2). The spectral product

12. M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Comm. **116**, 43–48 (1995). [CrossRef]

*E⃗*

_{t}is padded with zeros along the first dimension. In this dimension, the FFT is calculated and the result cropped. Along the second dimension, the same procedure is applied on the intermediate result. Zero padding simultaneously suppresses aliasing and refines the sampling grid for the focus field. Using two one-dimensional FFTs with intermediate cropping and zero padding minimizes the numerical processing cost.

### 3.4. Aperture rim smoothing

*U*)| for a circular aperture with radius

*R*. As already stated, the field

*U*vanishes outside the aperture for

*r*>

*R*, whereas inside the aperture for

*r*<

*R*, the field is given as

*U*=

*U*

_{0}. This discretization leads to a serrated aperture rim inducing granular artifacts at higher frequencies. Hence, the expected Airy function is only seen at low frequencies (central region in Fig. 3(a), please note the logarithmic scale). A smooth sampling of the aperture rim improves the accuracy of the spectrum [13

13. P. Luchini, “Two-dimensional numerical integration using a square mesh,” Comp. Phys. Comm. **31**, 303–310 (1984). [CrossRef]

*R*=

*R*/30 was the sampling step. The granular artifacts are efficiently reduced and the FFT approximates the Airy function with a good accuracy over a much larger area. Figure 4 shows a comparison of cross-sections of the spectra on the meridian

*k*

_{y}=0. Overall, for values |

*k*

_{x}|>60×2

*π*/256Δ

*R*, the ‘sharp’ spectrum shows granular artifacts, whereas the ‘smooth’ spectrum approximates well the Airy function.

### 3.5. Generalization based on the chirp z transform

*N*=2

^{s}for the FFT (

*s*∈ℕ). The corresponding number of sampling points

*M*over the aperture radius often exceeds the initial guess based Eq. (14). In such cases, the chirp z transform (CZT) is computationally faster than the FFT. In summary, the CZT (a) allows breaking the relationship between

*M*and

*N*, (b) allows an implicit frequency offset, and (c) internalizes the zero padding. Applying this generalization, we adapted the sampling step in the focus field independently of the sampling step in the input field, introduced an additional shift of the region of interest, and finally improved the computational efficiency.

*z*

_{m}∀

*m*∈[0,

*M*-1] be a discrete representation of a spatial signal

*z*(

*r*=

*m*Δ

*r*). The discrete Fourier transform (DFT) at a frequency

*k*=

*n*Δ

*k*∀

*n*∈[0,

*N*-1] is then obtained with

*k*=2

*π*/

*M*Δ

*r*and

*N*=

*M*. For Δ

*k*<2

*π*/

*M*Δ

*r*, a zero padding is implicitly contained in Eq. (17). Comparing the DFT with the CZT defined by

*a*=1 and

*w*=

*e*

^{-iΔk}for obtaining the DFT as a particular case of the general CZT. Setting

*k*

_{0}(see above). Furthermore, Eq. (18) can be rewritten as a convolution

*M*+

*N*-1) point FFTs (a third one can be precomputed) [14

14. J. L. Bakx, “Efficient computation of optical disk readout by use of the chirp z transform,” Appl. Opt. **41**, 4897–4903 (2002). [CrossRef] [PubMed]

15. Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A **1**, 801–808 (1984). [CrossRef]

16. W. Hsu and R. Barakat, “Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A **11**, 623–629 (1994). [CrossRef]

_{2}from a point (0,0,

*z*) on the axis. As a result, the sampling Δ

*k*depends upon the axial position

*z*, i.e. Δ

*k*(

*z*)=Δ

*k*(0)

*f*/(

*f*+

*z*) with Δ

*k*(0)=Δ

*k*as defined before. Using the CZT, the additional calculations remain restricted to the repeated computation of

*w*varies now with

*z*.

## 4. Selected examples

*t*

_{p}and

*t*

_{s}between the principal planes ℙ

_{1}and ℙ

_{2}need to be defined. We present the microscope objective as an optical system of only 2 optical interfaces and a convex interface into the immersion medium

*n*

_{t}. To this end, the three interfaces provide a physical model for deflection angles

*θ*∈[0,

*π*/2). The amplitude transmission efficiency, i.e. apodization, and the polarization are obtained based on the Fresnel equations.

*n*

_{g}and the immersion medium

*n*

_{t}, the Fresnel transmission coefficients are calculated for the succession of the air(

*n*

_{a})-glass(

*n*

_{g})-air(

*n*

_{a})-immersion(

*n*

_{t}) interfaces. The corresponding deflection angle

*θ*

_{ij}at each interface was chosen proportional to the difference of the index of refraction, viz

*θ*

_{ij}∝|

*n*

_{i}-

*n*

_{j}|. With

*n*

_{a}=1, the Fresnel transmission coefficients are then

### 4.1. 1.20 NA water immersion objective

*x*=Δ

*y*=20 nm, Δ

*z*=50 nm and

*M*=100, a 2.0 GHz Pentium 4 processor computed the field within a volume of 3

*µ*m×3

*µ*m× 5

*µ*m i.e. 150 ×150 ×100 sampling points in less than 40 seconds. Taking the symmetry into account, the volume was further extended to 6

*µ*m×6

*µ*m×10

*µ*m.

*y*-axis. For an underfilled aperture, the diameter of the central lobe is ≈25% larger but the side lobes vanish quickly. In both cases, the polarization leads to a larger x-extension compared to the

*y*-extension.

### 4.2. 1.45 NA oil immersion objective

*n*

_{s}=1.33, aqueous solution). This generates a partially evanescent focus field at the cover slip–sample interface. Depending upon the illumination of the aperture, the focus field can be fully propagating or fully evanescent. A fully propagating field can be calculated easily with the procedure outlined above. However, the evanescent field contribution needs an additional consideration for obtaining the total focus field.

*M*≳100 was used for |

*z*|→0.

*k⃗*

_{t}is replaced by

*n*

_{s}sin

*θ*′=

*n*

_{t}sin

*θ*. The unit vector

*e⃗*

_{r}for p-polarization becomes

*y*- than the

*x*-direction for linear polarization (Fig. 9(a)). The focal volume is reduced to about 1/7 compared to the former water immersion objective. Selecting a radially polarized input field results in a rotationally symmetric focus field as shown in Fig. 9(b). On the optical axis, the electric field becomes purely

*z*-polarized. For a distance

*z*≲0.3

*µ*m, this

*z*-component is dominant. Further away from the cover slip–sample interface, the

*xy*-components prevail, which results in an annular field distribution.

*E*

_{s}/cos

*θ*for the linear polarization. At the critical angle (

*NA*=1.33), the field amplitude triples in the

*x*-direction and doubles in the y-direction, respectively, hence marking the abrupt transition from propagating to evanescent fields.

## 5. Conclusions

17. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Comm. **39**, 205–210 (1981). [CrossRef]

18. P. Török, “Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A **15**, 3009–3015 (1998). [CrossRef]

15. Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A **1**, 801–808 (1984). [CrossRef]

16. W. Hsu and R. Barakat, “Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A **11**, 623–629 (1994). [CrossRef]

## Acknowledgements

## References and links

1. | P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. |

2. | E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field,” Proc. R. Soc. London Ser. A |

3. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A |

4. |
Typically, a good accuracy is achieved for |

5. | P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. |

6. | J.J. Stamnes, |

7. | G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, “Diffractive elements for imaging with extended depth of focus,” Opt. Eng. |

8. | N. Huse, A. Schönle, and S.W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. |

9. | J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, “Performance of Fluorescence Correlation Spectroscopy for Measuring Diffusion and Concentration,” Chem. PhysChem. |

10. |
For simplification, the sample indices |

11. | M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A |

12. | M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Comm. |

13. | P. Luchini, “Two-dimensional numerical integration using a square mesh,” Comp. Phys. Comm. |

14. | J. L. Bakx, “Efficient computation of optical disk readout by use of the chirp z transform,” Appl. Opt. |

15. | Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A |

16. | W. Hsu and R. Barakat, “Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A |

17. | E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Comm. |

18. | P. Török, “Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(180.0180) Microscopy : Microscopy

(220.2560) Optical design and fabrication : Propagating methods

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: October 3, 2006

Revised Manuscript: October 27, 2006

Manuscript Accepted: October 28, 2006

Published: November 13, 2006

**Virtual Issues**

Vol. 1, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Marcel Leutenegger, Ramachandra Rao, Rainer A. Leitgeb, and Theo Lasser, "Fast focus field calculations," Opt. Express **14**, 11277-11291 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11277

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### References

- P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30,755-776 (1909). [CrossRef]
- E. Wolf, "Electromagnetic diffraction in optical systems, I. An integral representation of the image field," Proc. R. Soc. London Ser. A 253,349-357 (1959). [CrossRef]
- B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253,358-379 (1959). [CrossRef]
- Typically, a good accuracy is achieved for M & 50 and N & 200 sampling points.
- P. Török and P. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Appl. Opt. 36,2305-2312 (1997). [CrossRef] [PubMed]
- J. J. Stamnes, Waves in Focal Regions: propagation, diffraction and focusing of light, sound and water waves, (Hilger, Bristol UK 1986).
- G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44,058001 (2005). [CrossRef]
- N. Huse, A. Schönle, and S. W. Hell, "Z-polarized confocal microscopy," J. Biomed. Opt. 6,480-484 (2001). [CrossRef]
- J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, "Performance of Fluorescence correlation Spectroscopy for measuring diffusion and concentration," Chem. Phys. Chem. 6,2324-2336 (2005). [CrossRef]
- For simplification, the sample indices kl and mn will be omitted further on.
- M. Mansuripur, "Certain computational aspects of vector diffraction problems," J. Opt. Soc. Am. A 6,786-805 (1989). [CrossRef]
- M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116,43-48 (1995). [CrossRef]
- P. Luchini, "Two-dimensional numerical integration using a square mesh," Comput. Phys. Commun. 31,303-310 (1984). [CrossRef]
- J. L. Bakx, "Efficient computation of optical disk readout by use of the chirp z transform," Appl. Opt. 41,4897-4903 (2002). [CrossRef] [PubMed]
- Y. Li and E. Wolf, "Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers," J. Opt. Soc. Am. A 1,801-808 (1984). [CrossRef]
- W. Hsu and R. Barakat, "Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems," J. Opt. Soc. Am. A 11,623-629 (1994). [CrossRef]
- E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39,205-210 (1981). [CrossRef]
- P. Török, "Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number," J. Opt. Soc. Am. A 15,3009-3015 (1998). [CrossRef]

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