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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 6339–6345
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Pupil filters for generation of light sheets

Colin J. R. Sheppard  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 6339-6345 (2013)
http://dx.doi.org/10.1364/OE.21.006339


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Abstract

Pupil filters for cylindrical (two-dimensional) focusing with extended depth of field are investigated. An important application is in generating light sheets with uniform intensity. Filters for spherical (three-dimensional) focusing with a flat axial intensity, coupled with weak side lobes are also discussed.

© 2013 OSA

1. Introduction

In numerous applications we require to optimize depth of field for a given aperture size, i.e. while maintaining transverse spatial resolution and light collection efficiency. We investigate the use of pupil filters to increase the focal depth for cylindrical focusing systems. We note that there are many practical examples of such two-dimensional (2D) optical systems, including integrated optics (planar waveguides) and plasmonics, and also optical systems such as light sheet microscopy [1

1. A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170(3), 229–236 (1993). [CrossRef] [PubMed]

4

4. H.-U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007). [CrossRef] [PubMed]

], line illumination microscopy [5

5. C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 35(7), 1169–1185 (1988). [CrossRef]

, 6

6. R. Wolleschensky, B. Zimmermann, and M. Kempe, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11(6), 064011 (2006). [CrossRef] [PubMed]

], and cubic phase mask systems [7

7. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

].

We start by deriving performance parameters for focusing by centro-symmetric cylindrical focusing systems, in analogy with those presented previously for rotationally symmetric spherical focusing systems [8

8. C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil plane filters,” J. Opt. Soc. Am. A 5(5), 643–647 (1988). [CrossRef]

]. We then continue with development of binary phase pupil filters (i.e. those having regions of amplitude +1and 1) that give maximally-flat (MF) axial intensity, using an analytic approach (i.e. not iterative) previously investigated for rotationally symmetric pupils [9

9. C. J. R. Sheppard, “Binary phase filters with a maximally-flat response,” Opt. Lett. 36(8), 1386–1388 (2011). [CrossRef] [PubMed]

, 10

10. C. J. R. Sheppard and S. Mehta, “Three-level filter for increased depth of focus and Bessel beam generation,” Opt. Express 20(25), 27212–27221 (2012). [PubMed]

].

2. Performance parameters for symmetric, cylindrical focusing systems

We can introduce performance parameters to describe various features of the focusing system for the 2D case, in a similar way to those used for 3D systems [8

8. C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil plane filters,” J. Opt. Soc. Am. A 5(5), 643–647 (1988). [CrossRef]

]. These parameters are useful because they allow the filter performance to be investigated without the necessity for calculating the focused field. These parameters can be used as a design tool, including use as merit functions for iterative optimization. The amplitude in the focal region for a cylindrical lens with pupil function (i.e. the amplitude transmission of the lens pupil) given byP(ξ) in the scalar, paraxial, Fresnel regime can be written [15

15. C. J. R. Sheppard, “Cylindrical lenses; focusing & imaging: a review [invited],” Appl. Opt. 52, 1–8 (2013).

]
U(v,u)=N(z)eiπ/4eikzexp[iv24πN(z)]11P(ξ)exp(iuξ22)exp(ivξ)dξ,
(2)
where, for a system of width 2a and focal length f, ξ=xpupil/a and the optical coordinates in the transverse and axial directions are
v=kfxzsinα=kxaz,
(3)
u=ka2(1f1z)=(fz)kδzsin2α,
(4)
respectively (to be consistent with Born and Wolf [16

16. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

], and Li and Wolf [17

17. Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1(8), 801–808 (1984). [CrossRef]

, 18

18. Y. Li, “Three-dimensional intensity distribution in low Fresnel number focusing systems,” J. Opt. Soc. Am. A 4(8), 1349–1353 (1987). [CrossRef]

]), and N(z)=a2/λz is a depth dependent Fresnel number [19

19. C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17(4), 772–779 (2000). [CrossRef] [PubMed]

]. Here the pupil is assumed to be in the plane of the lens. The Debye approximation follows by putting N and z=f, and is equivalent to the pupil being situated in the front focal plane of the system, at z=f. For a high numerical aperture system in the scalar regime, sin2αis then more accurately replaced by 4sin2(α/2) [20

20. C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4(8), 1354–1360 (1987). [CrossRef]

]. Expanding as power series to quadratic terms in v,u, for a symmetrical pupil for large Fresnel number, we obtain
U(v,u)=1211P(ξ)[1(v2+iu)ξ22u2ξ48+...]dξ            =I02[1(v2+iu)2I2I0u28I4I0+...],
(5)
where
In=11P(ξ)ξndξ
(6)
is the n-th order moment of the pupil, and we have normalized to unity at the origin for a uniform, unobstructed pupil. For a real-valued pupil function (which can nevertheless be negative as in the important case of a binary phase mask), the parabolic approximation to the intensity in the focal region can therefore be written
I(v,u)=I024{1v2(I2I0)2u24[I4I0(I2I0)2]}.
(7)
For a complex pupil, which can produce an asymmetric axial intensity and a shift of the primary focus, more complicated expressions need to be used [21

21. D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28(8), 607–609 (2003). [CrossRef] [PubMed]

23

23. C. J. R. Sheppard, S. Ledesma, J. Campos, and J. C. Escalera, “Improved expressions for performance parameters for complex filters,” Opt. Lett. 32, 1713–1715 (2007). [CrossRef] [PubMed]

]. For a uniform, unobstructed aperture, I0=2,I2=2/3,I4=2/5, so we can introduce the Strehl ratio and the transverse and axial gain parameters:
S=I02,GT=3I2I0,GA=454[I4I0(I2I0)2],
(8)
to give for the focal intensity distribution
I(v,u)=S(1GTv23GAu245).
(9)
The significance of GT,GA is that the parabolic width of the transverse or axial intensity point spread function is then 1/(GT)1/2,1/(GA)1/2, respectively, compared with the values for the unobstructed pupil. The maximum value of GTfor a positive pupil is 3, corresponding to cosine fringes, as formed by two interfering plane waves. This can regarded as the 2D analog of a Bessel beam, as generated by a thin annulus, in the 3D case. As for the 3D case, we also have for the fractional power transmission of the pupil
E=1211|P(ξ)|2dξ/(Pmax)2,
(10)
so that the intensity at the focus normalized by the focused power is F=S/E. S,GT,GA,E and F are all unity for an unobstructed pupil. These parameters can be used to investigate the focal performance of various designs of mask, without the necessity to evaluate directly the focal distribution.

3. Pupil filters for maximally flat axial intensity

From Eq. (2) with P(ξ)=1, the amplitude of the focused field for a slit pupil element extending from ξ=a to ξ=b in the scalar, Fresnel regime, can be written in terms of error functions as [15

15. C. J. R. Sheppard, “Cylindrical lenses; focusing & imaging: a review [invited],” Appl. Opt. 52, 1–8 (2013).

]
U(v,u)=iπN2ueikzexp[iv2(u+2πN)4πNu]{Erf[(1+i)(buv)2u]Erf[(1+i)(auv)2u]}.
(11)
We now use the same approach that we used to calculate maximally flat filters for the rotationally symmetric case [9

9. C. J. R. Sheppard, “Binary phase filters with a maximally-flat response,” Opt. Lett. 36(8), 1386–1388 (2011). [CrossRef] [PubMed]

]. Along the axis v=0, and restricting our attention to the case of large N, we expand the error functions as power series in u, put the lowest order terms to zero to give a flat axial intensity, and solve the resulting simultaneous nonlinear equations. For three symmetric elements of alternating sign (+1,1), we can make the u2 term zero, so that GA=0. For five symmetric elements, we can make the terms in u2 and u4zero, and for seven symmetric elements we can also make the term in u6 zero. The resulting solutions for the boundaries of the pupil elements ξ=±mi are shown in Table 1

Table 1. Values of Various Parameters for Filters with 1, 3 5 or 7 Elements

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, together with S and GT calculated from Eq. (8). For three or five elements, there are two different solutions in each case. For seven elements there are four solutions. In Table 1 we list these in decreasing GT (increasing S) for each order. We also introduce the parameter W, the normalized axial length to the 90% point in intensity, which is calculated from the axial intensity.

The overall S versus GTperformance is illustrated in Fig. 1
Fig. 1 The variation of the Strehl ratio S with transverse gain GTfor different filters. The maximally flat filters 3a, 5a, 7a lie on the dashed line indicated. The maximally flat filters 3b, 5b, 7d lie on another dashed line. The behavior for general symmetrical 3 element binary phase filters is also shown as solid lines: there are two branches, shown in red for GT1 and in greenGT1.
. The behavior for general symmetrical 3 element binary phase filters is also shown: there are two branches, for GT1 and GT1.

The first solutions listed in Table 1 for each number of elements (Nos. 3a, 5a, 7a) approximate to a cosine beam, with strong side lobes, becoming closer to a cosine beam as the number of elements increases (GT3). The axial and transverse intensity variations calculated from Eq. (2) are shown in Fig. 2(a)
Fig. 2 The axial (a, c, e) and transverse (b, d, f) intensity variations of the different maximally flat filters. (a, b) show filters 3a, 5a, 7a. (c, d) show filters 7b, 7c. (e, f) show filters 3b, 5b, 7d. The curves for an unobstructed slit pupil are shown as dotted lines.
and 2(b), where they are plotted against the optical coordinates given by Eqs. (3), 4. The last solutions listed in Table 1 for each number of elements (Nos. 3b, 5b, 7d) exhibit GT1 and weak side lobes, with the axial and transverse intensities shown in Fig. 2(e) and 2(f). As the number of elements increases, this solution tends to a plane wave (GT0). The other solutions for seven elements (Nos. 7b, 7c) are shown in Fig. 2(c) and 2(d): they also have strong side lobes. The side lobe behavior is indicated by the parameter S/(GT)1/2in Table 1, which is a measure of the fraction of the power in the central lobe. It is seen that the last solutions for each number of elements exhibit much larger values for this parameter, indicating that the side lobes are weak.

Because the length of the focus is proportional toWsin2α and the width of the beam is inversely proportional toGTsinα, an overall performance parameter WGTis independent of sinα. This increases with the number of elements, as shown in Table 1. For solution 7d, the transverse width is about twice that for an unobstructed slit aperture, but the focal depth is increased nearly six times, and the Strehl ratio is also quite high at 0.35. Hence to maintain a particular depth of focus, a higher numerical aperture can be used for a greater number of elements, thus giving a smaller transverse width.

4. Spherical focusing with extended focal depth

5. Discussion

We have shown numerically how binary phase filters to increase the focal depth of a cylindrically focusing system can be generated using an analytic method. The resulting designs giving a narrow transverse peak exhibit very strong side lobes. While these may be useful for generating arrays of flat-topped fringes, they are unsuitable for applications where a single light sheet, or a line in 2D, is desired. On the other hand, designs with a broad transverse peak have weak side lobes and a uniform axial intensity. The combination of transverse width and depth of field can give a superior performance to that of an unobstructed slit aperture. These designs are suitable for generating light sheets, or for producing a line focus in two-dimensional systems such as planar waveguides and plasmonics.

The fact that it is possible to produce a flat axial behavior, over an extended range, with transverse behavior very different from a cosine beam is in itself intriguing.

References and links

1.

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170(3), 229–236 (1993). [CrossRef] [PubMed]

2.

E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azam, “Thin laser light sheet microscope for microbial oceanography,” Opt. Express 10(2), 145–154 (2002). [CrossRef] [PubMed]

3.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004). [CrossRef] [PubMed]

4.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007). [CrossRef] [PubMed]

5.

C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 35(7), 1169–1185 (1988). [CrossRef]

6.

R. Wolleschensky, B. Zimmermann, and M. Kempe, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11(6), 064011 (2006). [CrossRef] [PubMed]

7.

E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

8.

C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil plane filters,” J. Opt. Soc. Am. A 5(5), 643–647 (1988). [CrossRef]

9.

C. J. R. Sheppard, “Binary phase filters with a maximally-flat response,” Opt. Lett. 36(8), 1386–1388 (2011). [CrossRef] [PubMed]

10.

C. J. R. Sheppard and S. Mehta, “Three-level filter for increased depth of focus and Bessel beam generation,” Opt. Express 20(25), 27212–27221 (2012). [PubMed]

11.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2(4), 105–112 (1978). [CrossRef]

12.

J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

13.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987). [CrossRef]

14.

L. W. Casperson, D. G. Hall, and A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14(12), 3341–3348 (1997). [CrossRef]

15.

C. J. R. Sheppard, “Cylindrical lenses; focusing & imaging: a review [invited],” Appl. Opt. 52, 1–8 (2013).

16.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

17.

Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1(8), 801–808 (1984). [CrossRef]

18.

Y. Li, “Three-dimensional intensity distribution in low Fresnel number focusing systems,” J. Opt. Soc. Am. A 4(8), 1349–1353 (1987). [CrossRef]

19.

C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17(4), 772–779 (2000). [CrossRef] [PubMed]

20.

C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4(8), 1354–1360 (1987). [CrossRef]

21.

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28(8), 607–609 (2003). [CrossRef] [PubMed]

22.

M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. 241(4-6), 249–253 (2004). [CrossRef]

23.

C. J. R. Sheppard, S. Ledesma, J. Campos, and J. C. Escalera, “Improved expressions for performance parameters for complex filters,” Opt. Lett. 32, 1713–1715 (2007). [CrossRef] [PubMed]

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(050.1940) Diffraction and gratings : Diffraction
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(110.6980) Imaging systems : Transforms
(350.6980) Other areas of optics : Transforms

ToC Category:
Diffraction and Gratings

History
Original Manuscript: November 29, 2012
Revised Manuscript: January 11, 2013
Manuscript Accepted: January 11, 2013
Published: March 6, 2013

Citation
Colin J. R. Sheppard, "Pupil filters for generation of light sheets," Opt. Express 21, 6339-6345 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6339


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References

  1. A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc.170(3), 229–236 (1993). [CrossRef] [PubMed]
  2. E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azam, “Thin laser light sheet microscope for microbial oceanography,” Opt. Express10(2), 145–154 (2002). [CrossRef] [PubMed]
  3. J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science305(5686), 1007–1009 (2004). [CrossRef] [PubMed]
  4. H.-U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods4(4), 331–336 (2007). [CrossRef] [PubMed]
  5. C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt.35(7), 1169–1185 (1988). [CrossRef]
  6. R. Wolleschensky, B. Zimmermann, and M. Kempe, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt.11(6), 064011 (2006). [CrossRef] [PubMed]
  7. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt.34(11), 1859–1866 (1995). [CrossRef] [PubMed]
  8. C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil plane filters,” J. Opt. Soc. Am. A5(5), 643–647 (1988). [CrossRef]
  9. C. J. R. Sheppard, “Binary phase filters with a maximally-flat response,” Opt. Lett.36(8), 1386–1388 (2011). [CrossRef] [PubMed]
  10. C. J. R. Sheppard and S. Mehta, “Three-level filter for increased depth of focus and Bessel beam generation,” Opt. Express20(25), 27212–27221 (2012). [PubMed]
  11. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust.2(4), 105–112 (1978). [CrossRef]
  12. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987). [CrossRef] [PubMed]
  13. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987). [CrossRef]
  14. L. W. Casperson, D. G. Hall, and A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A14(12), 3341–3348 (1997). [CrossRef]
  15. C. J. R. Sheppard, “Cylindrical lenses; focusing & imaging: a review [invited],” Appl. Opt.52, 1–8 (2013).
  16. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).
  17. Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A1(8), 801–808 (1984). [CrossRef]
  18. Y. Li, “Three-dimensional intensity distribution in low Fresnel number focusing systems,” J. Opt. Soc. Am. A4(8), 1349–1353 (1987). [CrossRef]
  19. C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A17(4), 772–779 (2000). [CrossRef] [PubMed]
  20. C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A4(8), 1354–1360 (1987). [CrossRef]
  21. D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett.28(8), 607–609 (2003). [CrossRef] [PubMed]
  22. M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun.241(4-6), 249–253 (2004). [CrossRef]
  23. C. J. R. Sheppard, S. Ledesma, J. Campos, and J. C. Escalera, “Improved expressions for performance parameters for complex filters,” Opt. Lett.32, 1713–1715 (2007). [CrossRef] [PubMed]

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