Journal of Modern Optics Vol. 52, No. 8, 20 May 2005, 1123–1134
The numerical integration of fundamental diffraction integrals for converging polarized spherical waves using a two-dimensional form of Simpson’s 1/3 Rule I. J. COOPER*y, C. J. R. SHEPPARDz and M. ROYy yPhysical Optics Group, School of Physics, University of Sydney, NSW 2006, Australia zBioengineering & Diagnostic Radiology, National University of Singapore, Singapore (Received 2 April 2004; in final form 29 July 2004) A comprehensive matrix method based upon a two-dimensional form of Simpson’s 1/3 rule (2DSC method) to integrate numerically the vector form of the fundamental diffraction integrals is described for calculating the characteristics of the focal region for a converging polarized spherical wave. The only approximation needed in using the 2DSC method is the Kirchhoff boundary conditions at the aperture. The 2DSC method can be used to study the focusing of vector beams with different polarizations and profiles and for different filters over a large range of numerical apertures or Fresnel numbers.
1. Introduction The characteristics of the focal region for a converging spherical wave are found by evaluating one form or another of a fundamental diffraction integral such as the Kirchhoff, Rayleigh–Sommerfeld I or Rayleigh–Sommerfeld II [1]. The methods usually employed to evaluate such integrals require a set of approximations based upon approximations to the distance between an aperture point and an observation point [2]. This leads to replacing a spherical wavefront with a parabolic one (paraxial approximation) or the Debye approximation used by Richards and Wolf [3] in their classic paper on the electromagnetic diffraction in an optical system for linear polarized light. Mansuripur [4, 5] has described a method for vectorial diffraction based upon Fourier transforms and stationary phase in which different approximations are used depending upon the parameters of the focusing system. Based upon their study of focusing problems in various regimes with different approximations, they concluded that ‘to our knowledge there does not exist a comprehensive method of computing diffraction patterns that does not require approximations of one sort or another to the fundamental diffraction integral’ [4]. The Mansuripur method is significantly more complicated than the straightforward approach of the 2DSC method. *Corresponding author. Email:
[email protected] Journal of Modern Optics ISSN 0950–0340 print/ISSN 1362–3044 online # 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/09500340512331323439
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In the 2DSC method, a two-dimensional diffraction integral is evaluated directly without the diffraction integral being reduced to a set of one-dimensional integrals. The only assumption is that the electric field within the aperture and blocking screen are given by the Kirchhoff boundary conditions [1] and this assumption is acceptable for aperture dimensions larger than a wavelength. The 2DSC method can be used over a wide range of numerical apertures or Fresnel numbers. Figures 1 and 2 show the variation in the effective full width at half maximum (fwhm) for the average electric energy density with numerical aperture along the optical axis (Z axis) and in the focal plane respectively for the focusing of a uniform plane wave that is linearly polarized in the X direction. Along the optical axis, the fwhm is to a good approximation, proportional to 1/NA2 and in the focal plane, the fwhm proportional to 1/NA. The approximations commonly employed before integrating the diffraction integrals are valid only over a range of parameters. Therefore, when using the 2DSC method, one does not have to be so concerned with the parameters used that may invalidate the conditions of applicability of the approximations. Using the 2DSC method, one can check the validity of those approximations, since the 2DSC gives an exact evaluation of the integral provided enough aperture points are used.
2. Rayleigh–Sommerfeld diffraction integral of the first kind The 2DSC method can be used for the evaluation of any of the standard forms of the two-dimensional diffraction integrals [6]. In this paper, only the evaluation of the vector Rayleigh–Sommerfeld diffraction integral of the first kind is considered. 100 90 80
Z axis: fwhm / λ
70 60 50 40 30 20 10 0 0.0
0.2
0.4
0.6
0.8
1.0
NA
Figure 1. The effective full width at half maximum for the average electric energy density of the focal spot along the optical axis as a function of numerical aperture. The solid line is the 1=NA2 fit. Parameters: ¼ 632:8 nm, a ¼ 2:50 mm.
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XY plane: fwhm / λ
50
40
30
20
10
0 0.0
0.2
0.4
0.6
0.8
1.0
NA
Figure 2. The effective full width at half maximum for the average electric energy density of the focal spot in the focal plane as a function of numerical aperture. The solid line is the 1/NA fit. Parameters: ¼ 632:8 nm, a ¼ 2:50 mm.
The Rayleigh–Sommerfeld diffraction integral of the first kind is a very convenient formulation for the diffraction integral, since it is consistent and it is only necessary to know the electric field within the aperture and not the gradient of the field [7, 8]. Consider a plane monochromatic wave (wavelength and propagation constant k ¼ 2=) travelling in the Z direction, incident upon an aberration-free spherical lens which gives a spherical wave converging towards the geometrical focal point SðxS , yS , zS Þ. The origin Oð0, 0, 0Þ of the coordinate system is taken as the centre of the circular aperture of radius a and the Z axis corresponds to the optical axis, as shown in figure 3. The focal length is taken as f ¼ zS and the semi-aperture angle is , where tan ¼ a=f . A point within the aperture is QðxQ , yQ , 0Þ or in polar coordinates QðQ , Q , 0Þ with xQ ¼ Q cos Q and yQ ¼ Q sin Q , where 0 Q a and QS is the semi-aperture angle to the aperture point Q and rQS is the distance from the aperture point to the focal point. The distance from an observation point PðxP , yP , zP Þ to the aperture point Q is rQP. The Fresnel number is given by Nf ¼ a2 =f and the numerical aperture by NA ¼ sin . The vectorial Rayleigh–Sommerfeld diffraction integral of the first kind is expressed by equation (1) in a form suitable for the computation of the electric field EP at an observation point PðxP , yP , zP Þ, ð 2 ð a EP ¼ C
gTEQ 0
0
exp ðikrQP Þ zP ðikrQP 1ÞQ dQ dQ : r3QP
ð1Þ
C is a normalizing constant so that the average electric energy density We ¼ EP EP can be set to one for its maximum value. The apodization factor for an aplanatic optical system is g ¼ cos1=2 QS . The pupil transmission function T can describe the beam profile, for example, uniform, Gaussian, Bessel–Gauss, doughnut beams
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Y
aperture plane
Q
X
ρQ Ei EQ O
P
Q optical axis O
α
S
θQ
f = zS
a
θQS
f = zS
X Z
rQS
S
focal plane
Z
Figure 3. Geometry for the diffraction by the circular aperture for a spherical wave converging to the geometrical focus at SðxS , yS , zS Þ. QðxQ , yQ , 0Þ or QðQ , Q Þ is a point within the aperture and PðxP , yP , zP Þ is an observation point.
and beams which have a phase singularity. The pupil transmission function can represent filters placed in the front focal plane that are used for apodization or super-resolution. An annular or half aperture filter can be modelled by simply changing the limits of integration, for example, a half aperture Q1 ¼ 0 and Q2 ¼ and for an annular aperture Q1 ¼ ba and Q2 ¼ a, where 0 b < 1. EQ is the electric field in the plane of the aperture at a point Q and is given by equations (2) and (3): EQ ¼ RpEQ ¼ e EQ ,
ð2Þ
where EQ is the magnitude of the electric field for the spherical wave converging to the focus, EQ ðxQ , yQ , 0Þ ¼
exp ðikrQS Þ , rQS
ð3Þ
and p is the unit vector describing the polarization of the incident beam. The polarization p is given by [9] p ¼ px i þ py j þ 0k,
p2x þ p2y ¼ 1:
The incident beam polarization vector can also be expressed as p ¼ cos ðmf Þi sin ðmf Þj, or p ¼ sin ðmf Þi cos ðmf Þj, and if m ¼ 1, 2, 3, . . . a phase singularity may occur along the þX axis.
ð4Þ
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Circular polarization is specified by a complex term for the y component right
p ¼ 21=2 ð1i þ ijÞ,
left
p ¼ 21=2 ð1i ijÞ:
For linear polarization in the þX direction p ¼ ð1, 0, 0Þ, radial polarization p ¼ ðcos Q , sin Q , 0Þ and azimuthal polarization p ¼ ð sin Q , cos Q , 0Þ and a more complex polarization state, which possess a polarization singularity along the þX axis, is p ¼ ð cos Q , sin Q , 0Þ. A rotation matrix R applied to the incident beam polarization vector p, converts it to the unit vector e for the rotated state that describes the polarization of the electric field of the beam after the lens [10] e ¼ Rp, or 0
ex
1
0
1 þ ðcos QS 1Þ cos2 Q
ð5Þ
ðcos QS 1Þ cos Q sin Q sin QS cos Q
B C B @ ey A ¼ @ ðcos QS 1Þ cos Q sin Q 1 þ ðcos QS 1Þ sin2 Q ez sin QS cos Q sin QS sin Q 0 1 px B C @ py A: 0
1
C sin QS sin Q A
cos QS
3. 2DSC method To numerically evaluate the integral in equation (1), the diffraction integral is approximated by a sum over each aperture point that is specified by the integers m and n for an M M aperture grid (M must be an odd integer greater than three) using a two-dimensional version of Simpson’s 1/3 rule as given by equation (6) EP ðxP , yP , zP Þ ¼
M X M X m
gmn emn Tmn Smn
n
exp ðikrQSmn Þ exp ðikrQPmn Þ rQSmn r3QPmn
zP ðikrQPmn 1Þmn :
ð6Þ
Each term in equation (6) is constructed as an M M matrix. The matrix Smn is the two-dimensional Simpson’s coefficient matrix described by equation (9). The basis of the 2DSC method is Simpson’s 1/3 rule, where a one-dimensional integral is approximated by the weighted sum in equation (7) m¼M ð xM h X f ðxÞ dx ¼ f ðxm ÞSm , I¼ ð7 aÞ 3 x1 m¼1 where f(x) is evaluated at M equally spaced points along the X axis from x ¼ x1 to xM with h ¼ ðx2 x1 Þ and the Simpson’s coefficients are Sm ¼ ð1 4 2 4 . . . 4 2 4 1Þ:
ð7 bÞ
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Simpson’s 1/3 rule can be extended to evaluate double integrals [11] in Cartesian or polar form ð M ð M ð y M ð xM f ðx, yÞ dx dy ¼ f ð, Þ d d, ð8 aÞ I¼ y1
x1
1
1
m¼M X hy X n¼M hx I¼ f ðxm , yn ÞSmn , 3 3 m¼1 n¼1
ð8 bÞ
m¼M X h h X n¼M I¼ f ðm , n Þmn Smn , 3 3 m¼1 n¼1
ð8 cÞ
where f(x, y) is evaluated at M M equally spaced points in the XY plane from x ¼ x1 to xM and y ¼ y1 to yM with hx ¼ ðx2 x1 Þ and hy ¼ ðy2 y1 Þ. For the integration in polar coordinates the limits of integration are ¼ 1 to M and ¼ 1 to M with h ¼ ð2 1 Þ and h ¼ ðh2 h1 Þ. The two-dimensional Simpson’s coefficient matrix is given by Smn ¼ Sm Sn :
ð9Þ
Figure 4 shows the grid points and two-dimensional Simpson coefficients with M ¼ 9 and M ¼ 11 for integrating over a circular area. If M ¼ 4P þ 1, where P is a positive integer then some grid points will always lie along the XY axes and this may be desirable in evaluating certain integrals. The coefficients and points for 1 ¼ 0 and M ¼ 2 are identical and lie along the þX axis. If M is not of the form 4P þ 1, then the grid points may not lie along the Y axis and the negative X axis. One difficulty of performing the integration of equation (1) is the time taken to do the calculation because of the number of loops needed to scan both the aperture and observation space. Another difficulty is the nature of the complex exponential terms. However, using fast personal computers and with software such as Matlab [12], some of the loops can be eliminated by matrix operations performed on complex numbers and complex exponential functions, thereby reducing the computation time dramatically. The execution time for the simultaneous calculation of the field along the X, Y and Z axes is typically a few seconds
2
8
2 8 4 4 4 8 16 16 8 8 4 16 16 8 8 8 16 16 4 8 8 8 16 16 4 8 4 8 4 8 1 4 2 4 2 4 2 16 8 16 8 8 4 16 16 8 8 8 16 16 4 8 8 16 16 8 4 4 4 8 2
M =9
4
1
4 2 16 8 8 4 16 8 2 4 8 4 8 16 16 8 4 8 8 16 8 4 4 8 8 16 16 8 4 8 8 4 8 16 4 168 8 8 16 4 16 8 16 8 16 8 16 8 16 1 4 2 4 2 4 2 4 2 4 1 8 16 4 168 8 8 16 8 4 4 8 16 8 8 16 4 8 8 4 8 16 4 8 16 8 8 16 8 4 2 4 16 8 8 4 16 8 4 2
M = 11
Figure 4. The grid patterns for M ¼ 9 and M ¼ 11 when sampling the aperture for the numerical integration over a circular area using polar coordinates. If M ¼ 4P þ 1 (P ¼ 1, 2, 3, . . .) then some grid points will always be along the X and Y axes.
The numerical integration of fundamental diffraction integrals
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and for the field in an XY plane or the meridional plane the execution time is typically a few minutes.
4. Results The summation used in equation (6) with the two-dimensional Simpson’s coefficients converges rapidly for small values of M for points near the geometrical focus, usually, M < 100 gives accurate results. Figure 5 shows the average electric energy density along the Z axis for a uniform beam linearly polarized in the X direction for aperture points 7 7 and 49 49. Even for an aperture grid 7 7, the 2DSC method predicts the variation of the central maximum quite well. Richards and Wolf [3] describe a method for calculating the electromagnetic diffraction in an optical system for linear polarized light based upon the Debye approximation. Their method results in three one-dimensional integrals containing Bessel functions. The average electric energy density, We along the X, Y and Z axes were calculated by the 2DSC methods in less than three seconds, about the same time that is required to evaluate the set of one-dimensional Richards and Wolf’s integrals. The values for the average energy density along the X and Y axes calculated by the 2DSC are identical with the values quoted by Richards and Wolf. The Richards and Wolf method is invalid for optical systems with low Fresnel numbers as it does not predict the asymmetry in the axial average energy density along the optical axis or the focal shift towards the aperture. The approximation method proposed by Hsu and Barakat [13] is often used for low Fresnel number
1
0.8
We
0.6
0.4 M=7
M = 49
0.2
0
-5
0
5
(zP-zS) / λ Figure 5. The average electric energy density along the optical axis for M ¼ 7 and M ¼ 49. Parameters: ¼ 632:8 nm, a ¼ 2:50 mm, NA ¼ 0:8.
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systems, but can give inaccurate results. Hsu and Barakat applied the Stratton–Chu theory of electromagnetic scattering to develop a Kirchhoff formalism of the diffracted waves by a circular aperture when the Fresnel number of the optical system is small (Nf < 10). The approximations used by Hsu and Barakat lead to a set of one-dimensional integrals. Hsu and Barakat calculated the diffracted vector fields in the focal region for a small Fresnel number, Nf ¼ 5, system with parameters: ¼ 32:2 mm, a ¼ 310 mm, f ¼ 510:2 mm and ¼ 31:28 . Figure 6 gives the average electric energy density We along the optical axis about the geometrical focus for the Hsu and Barakat and the 2DSC methods. The execution time for both programs were similar and so there is no time advantage of performing a set of one-dimensional integrations compared with doing only one double integral. Figure 6 shows that there is a small focal shift towards the aperture and the side lobes are more pronounced from the central maximum on the aperture side. The approximation method of Hsu and Barakat accurately describes the average electric energy density very near the geometrical focus and along the optical axis for zP > zS and correctly predicts the positions of the focal shift in the maximum energy density. However, in the region between the aperture and the position of the maximum energy density, the results of Hsu and Barakat give poor agreement with the 2DSC results for the position and size of the side lobes. Sheppard and Torok [14] used an improved set of approximations for the Hsu and Barakat parameters and the results of the 2DSC method are in agreement with their results. One advantage of the 2DSC method is that it gives an accurate estimate of the integral over a wider range of Fresnel numbers that many of the approximation methods commonly used. Interest is being shown in the use of radial and azimuthal polarized light beams [15], because the average electric energy density distributions are cylindrical symmetric in the focal plane. The average electric energy distribution is not symmetrical in the focal plane for linearly polarized light since the total electric
1
0.8
We
0.6
0.4
0.2
0 -0.4
H&B
2dSc
-0.2
0
0.2
0.4
axial position (zP - f) (m)
Figure 6. The axial electric energy density We about the geometrical focal point. The thicker line is for the Hsu and Barakat calculation and the thinner line is 2DSC calculation. Parameters: ¼ 32:2 mm, a ¼ 310:0 mm, f ¼ 510:2 mm, Nf ¼ 5:0.
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WeY 15 10
v
PY
5 0 -5 -10 -15 -15
-10
-5
0
5
10
15
5
10
15
5
10
15
v
PX
WeZ 15 10
vPY
5 0 -5 -10 -15 -15
-10
-5
0
v
PX
We 15 10
vPY
5 0 -5 -10 -15 -15
-10
-5
0 v
PX
Figure 7. The total average electric energy density We and components WeX , WeY , WeZ in the focal plane for a spherical lens illuminated with a uniform plane linearly polarized wave in the X direction. The axes are in the optical units of Richards and Wolf. Parameters: ¼ 400 nm, a ¼ 1:71 mm, NA ¼ 0:97.
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0.8
We
0.6
0.4
0.2
0
0
1
2
xP/λ
Figure 8. Total average energy density in the focal plane for a beam exhibiting both a phase and azimuthal polarization singularity. Parameters: ¼ 400 nm, a ¼ 1:71 mm, NA ¼ 0:966.
average energy density is made up of contributions from its X, Y and Z components, as shown in figure 7. For the beam polarized in the þX direction, the X component of the electric energy density provides the largest contribution to the total electric energy density and the distribution is similar to the Airy disc. The Y component of the average electric energy density has four peaks, one in each quadrant, but this contribution to the total is very small. The Z component has two peaks with one on each side of the Y axis which results in the non symmetrical distribution for the total average electric energy density about the optical axis, with contours near the focal point approximately elliptical and the distribution becoming complex away from focal point with closed loops formed around certain points in the X and Y directions. With the advent of liquid crystal displays, the polarization of light beams can now be manipulated to give radial and azimuthal polarization and beams that have polarization and phase singularities. Knowledge of both the transverse and longitudinal components of the average electric field is now important [16]. The 2DSC method can easily calculate the focal spot characteristics for such beams. As an example, the focusing by an aplanatic lens of a beam exhibiting phase singularities and azimuthal polarization singularities as given by equation (10) is considered, Tmn ¼ 2mn exp ðiQmn Þð sin Qmn i þ cos Qmn jÞ:
ð10Þ
The average electric energy density is shown in figure 8. The average electric energy density in the focal plane is cylindrical symmetric. This combined phase and polarization singularity results in an interesting behaviour in that it gives a purely transverse electric field in the focal region but it does not exhibit a dark centre as a true azimuthal polarized beam.
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One proposal to create a radially polarized beam is to use a half wave plate cut into four quadrants which gives linear polarization in the four quadrants directed outward at 45 (pseudo-radial polarization). The 2DSC method can easily calculate the electric field in the focal region by summing the contributions of the linearly polarized light from each of the four quadrants. One interesting and important feature of the focusing of a wave by a lens is the Gouy phase shift where the converging light experiences a phase shift as it passes through its focus [17]. Using the 2DSC method, the variation of phase of the electric field along the optical axis can be calculated. This shows that away from the geometrical focus along the optical axis the spatial period of the phase is one wavelength, however, in a region very close to the focus, the spatial period is stretched and is larger than one wavelength and this stretching represents the Gouy phase change.
5. Conclusion The 2DSC method described in this paper provides a simple yet comprehensive method for evaluating the fundamental diffraction integrals which does not require approximations of one sort or another. The method is based upon a matrix version of a two-dimensional form of Simpson’s 1/3 rule with a set of two-dimensional Simpson coefficients to numerically integrate the vector form of the Rayleigh–Sommerfeld diffraction integral of the first kind. Normally, less than 100 100 aperture points are required with an execution time for calculating the average electric energy density along an axis taking a few seconds and for a plane, a few minutes. The 2DSC method can be applied to a wide range of input parameters, for optical systems with large and small Fresnel number or numerical aperture. This makes the 2DSC method very useful in checking the accuracy of the approximation methods commonly used. Optical systems can be modelled for a wide range of incoming beams with different polarization states and beam profiles, different filter functions in the front focal plane and for non-circular apertures.
References [1] M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999). [2] J.J. Stamnes (Editor), Selected Papers on Electromagnetic Fields in the Focal Region, SPIE Milestone Series, Vol. MS168 (SPIE Optical Engineering Press, Bellingham, WA, 2001). [3] B. Richards and E. Wolf, Proc. Roy. Soc. A 253 358 (1959). [4] M. Mansuripur, J. Opt. Soc. Am. A 3 2086 (1986). [5] M. Mansuripur, J. Opt. Soc. Am. A 6 786 (1989). [6] I.J. Cooper, C.J.R. Sheppard and M. Sharma, Optik 113 293 (2002). [7] I.J. Cooper and C.J.R. Sheppard, Optik 114 298 (2003). [8] A. Sommerfeld, Optics—Lectures on Theoretical Physics, Vol. 4 (Academic Press, New York, 1964). [9] G.R. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winstonce-Hall, New York, 1968).
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[10] P. Torok and C.J.R. Sheppard, The Role of Pinhole Size in High-Aperture Two- and Three-Photon Microscopy, Confocal and Two-Photon Microscopy (Wiley-Liss, New York, 2002). [11] P.J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic Press, New York, 1975), pp. 269–270. [12] D. Hanselman and B. Littlefield, Mastering Matlab 5 (Prentice-Hall, Upper Saddle River, NJ, 1998). [13] W. Hsu and R. Barakat, J. Opt. Soc. Am. A 11 623 (1994). [14] C.J.R. Sheppard and P. Torok, Optics and Optoelectronics, Vol. 1 (Narosa, New Delhi, 1999). [15] R. Quabis, R. Dorn, M. Eberler et al., Opt. Commun. 179 1 (2000). [16] K.S. Youngworth and T.G. Brown, Opt. Express 7 7 (2000). [17] S. Feng and H.G. Wiful, Opt. Lett. 26 485 (2001).
