Cylindrical lenses—focusing and imaging: a review [Invited]

doi:10.1364/AO.52.000538

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Cylindrical lenses—focusing and imaging: a review [Invited]

Colin J. R. Sheppard »View Author Affiliations

Applied Optics, Vol. 52, Issue 4, pp. 538-545 (2013)
http://dx.doi.org/10.1364/AO.52.000538

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▸ Article Outline
– Abstract
Introduction
Field in the Focal Region
Transfer Functions and Phase Space
Conclusions
– References and links

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Abstract

Focusing by an aberration-free cylindrical lens is analyzed in the paraxial Fresnel and Debye approximations, and expressions are given. Plots are given for the intensity in the focal region, the defocused optical transfer function (OTF), the generalized OTF, and the ambiguity function and are compared with the case of an aberration-free spherical lens. Nonparaxial lenses are also discussed.

1. Introduction

There are numerous practical examples where waves are focused in two dimensions by a cylindrical lens. Some recent important examples include plasmonics, light-sheet microcopy [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, 229–236 (1993). [CrossRef]

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

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, 1007–1009 (2004). [CrossRef]

–4

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. 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, 331–336 (2007). [CrossRef]

], line illumination microscopy [5

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

G. J. Brakenhoff and K. Visscher, “Confocal imaging with bilateral scanning and array detectors,” J. Microsc. 165, 139–146 (1992). [CrossRef]

–7

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

], and the cubic phase mask for depth-of-focus enhancement [8

E. R. Dowsky and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995). [CrossRef]

]. These are in addition to other more established areas including particle velocimetry, integrated optics, and planar optics. Although focusing of three-dimensional (3D) waves is treated in depth in most optics texts, the corresponding results for two-dimensional (2D) waves are not so well known. The most complete paper is that of Marsh [9

J. S. Marsh, “Light distribution near the focus of a two-dimensional lens,” Am. J. Phys. 52, 152–155 (1984). [CrossRef]

], but the reproduction of the illustrations in this paper were of poor quality, and there have also been several developments reported since then. Some comparisons of the behavior of 2D and 3D systems have been presented by Kelly et al. [10

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part I. 2-D system analysis,” Opt. Commun. 263, 171–179 (2006). [CrossRef]

,11

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006). [CrossRef]

]. Here we present some of the most important results, using the same notation as in the standard text by Born and Wolf [12

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

] so that the results can be compared directly with the 3D case. As many of the recent applications involve axial or 3D imaging, we also include results for out-of-focus effects. The results presented are also applicable to systems with rectangular pupils.

2. Field in the Focal Region

The amplitude in the focal region of a cylindrical lens illuminated by a plane wave in the paraxial, scalar, Fresnel regime is given by [13

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

]

U (x, z) = \sqrt{\frac{1}{λ z}} e^{- i π / 4} e^{i k z} \exp (\frac{i k x^{2}}{2 z}) \int_{- \infty}^{\infty} P (x^{'}) \exp [- \frac{i k x^{' 2}}{2} (\frac{1}{f} - \frac{1}{z})] \exp (- \frac{i k x x^{'}}{z}) d x^{'},

(1)

where

P (x^{'})

is the pupil function of the lens, assumed to be in the plane of the lens;

f

is the focal length; and the observation distance from the lens is

z = f + δ z

, and

k = 2 π / λ

. The numerical aperture of the lens is

NA = \sin α = a / f

, where the width of the pupil is

2 a

. We define normalized optical coordinates

v = \frac{k f x}{z} \sin α = \frac{k x a}{z},

(2)

u = k a^{2} (\frac{1}{f} - \frac{1}{z}) = (\frac{f}{z}) k δ z \sin^{2} α,

(3)

(to agree with those of Born and Wolf [12

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

] and Li and Wolf [14

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]

,15

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

]) and put

ξ = x^{'} / a

. The defocus coordinate is related to the defocus coefficient by

u = 2 k W_{20}

. We also introduce the Fresnel number of the system,

N_{0} = a^{2} / λ f

, and a depth-dependent Fresnel number

N (z) = a^{2} / λ z

. We then have

N (z) / N_{0} = f / z

and

u = 2 π (N_{0} - N)

. Then in general, redefining the pupil function as

P (ξ)

, the field in the focal region is

U (v, u) = \sqrt{N} e^{- i π / 4} e^{i k z} \exp (\frac{i v^{2}}{4 π N}) \int_{- 1}^{1} P (ξ) \exp (- \frac{i u ξ^{2}}{2}) \exp (- i v ξ) d ξ .

(4)

For a simple slit aperture

P (ξ) = 1

, the integral can be evaluated in terms of Fresnel integrals

C (z)

S (z)

[16

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions , 3rd ed. (Dover, 1972).

] to give

U (v, u) = \sqrt{N} e^{- i π / 4} e^{i k z} \exp (\frac{i v^{2}}{4 π N}) \exp (\frac{i v^{2}}{2 u}) \sqrt{\frac{π}{u}} [F^{*} (\frac{u + v}{\sqrt{2 u}}) + F^{*} (\frac{u - v}{\sqrt{2 u}})],

(5)

where

F (z) = C (z) + i S (z)

(6)

is the complex Fresnel integral, which can alternatively be expressed in terms of the error function of a complex argument, [16

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions , 3rd ed. (Dover, 1972).

F (z) = \frac{(1 + i)}{2} \erf [\frac{\sqrt{π}}{2} (1 - i) z] .

(7)

At the geometrical focal point

v = u = 0

, the value of the integral in Eq. (4) for

P (ξ) = 1

is equal to 2, so

U (0, 0) = 2 \sqrt{N_{0}} e^{- i π / 4} e^{i k f} .

(8)

So normalizing to unity at the focal point, we obtain

U (v, u) = \frac{1}{2} \sqrt{\frac{N}{N_{0}}} e^{i k (z - f)} \exp [\frac{i v^{2}}{2 u} (\frac{N_{0}}{N})] \sqrt{\frac{π}{u}} [F^{*} (\frac{u + v}{\sqrt{π u}}) + F^{*} (\frac{u - v}{\sqrt{π u}})] .

(9)

Equation (9) shows that the diffracted amplitude is given by interference between two terms representing diffraction at the two aperture step edges, giving two shadow edges at

v = \pm u

. Then along the optical axis,

v = 0

, and

U (0, u) = \sqrt{\frac{N}{N_{0}}} e^{i k (z - f)} \sqrt{\frac{π}{u}} F^{*} (\sqrt{\frac{u}{π}}) .

(10)

In the geometrical focal plane,

u = 0

, Eq. (4) can be readily evaluated, or asymptotic expressions for the Fresnel integrals [16

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions , 3rd ed. (Dover, 1972).

] can be used in Eq. (9), to give

U (v, 0) = \exp (\frac{i v^{2}}{4 π N_{0}}) (\frac{\sin v}{v}) .

(11)

Note that for finite values of

N_{0}

, in Eq. (10)

N / N_{0}

decreases monotonically with

u

, and so the maximum intensity is reached at a point on the axis closer to the lens than the geometrical focal point, and the intensity is not symmetrical about the geometrical focal plane. This is the so-called focal shift effect [14

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]

,15

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

,17

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957). [CrossRef]

J. J. Stamnes and S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981). [CrossRef]

J. Erkkila and M. Rogers, “Diffracted fields in the focal region of a convergent wave,” J. Opt. Soc. Am. 71, 904–905 (1981). [CrossRef]

–20

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

| δ z | ≪ f

, then

N \approx N_{0}

v = k x \sin α

u = k δ z \sin^{2} α

, and the problem reduces to the paraxial Debye approximation [12

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

]. Now

v

and

u

are directly proportional to

x

and

δ z

, respectively. The range of values of

u

for which

| δ z | ≪ f

becomes larger as

N_{0}

increases [21

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

,22

C. J. R. Sheppard, “Validity of the Debye approximation,” Opt. Lett. 25, 1660–1662 (2000). [CrossRef]

]. Contours of constant intensity for a cylindrical lens with a slit aperture in the paraxial Debye approximation are shown in Fig. 1, which can be compared with the analogous results for a circular aperture, shown in Fig. 1 and also given by Born and Wolf [12

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

]. Another representation of results valid in the Debye approximation is shown in Fig. 2. The general features of both the cylindrical and spherical cases are similar, exhibiting “wings” near the edges of the geometrical shadow boundary,

v = \pm u

. The side lobes are weaker for the spherical case. The intensity in the focal plane is compared with that for a spherical lens in Fig. 3(a). The point spread function is narrower, but with stronger side lobes, for the cylindrical case. The intensity along the shadow edge,

v = u

, is shown in Fig. 3(b). The intensity along the axis is shown in Fig. 4(a). Again the central lobe is narrower, and the intensity at larger distances falls off more slowly, for the cylindrical case. Unlike the case of the spherical lens, there are no zeros in intensity along the axis. The zeros for the 3D case are a consequence of the fact that the Fresnel half-period zones are of equal area so that their total can exactly cancel. The phase variation along the axis, with the contribution from

\exp (i k z)

suppressed, is shown in Fig. 4(b). The Gouy phase shift through the focus is

- π / 2

rather than

- π

as it is for a spherical lens. The Gouy phase shift is clearly obvious in Fig. 4(b), whereas for the spherical lens with a circular pupil the phase cannot be unwrapped because of the presence of phase singularities along the axis. The fraction of energy

E

within a region

- v_{d} \leq v \leq v_{d}

is given by [5

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

]

E (v_{d}) = \frac{2}{π} [Si (2 v_{d}) - \frac{\sin^{2} v_{d}}{v_{d}}],

(12)

where Si is a sine integral and is shown in Fig. 5. The fraction of energy rises more quickly for the cylindrical lens.

Fig. 1. Contours of constant intensity in the focal region of (a) an aberration-free cylindrical lens and (b) a spherical lens, in the paraxial Debye regime.

Fig. 2. Intensity in the focal region of (a) an aberration-free cylindrical lens and (b) a spherical lens, in the paraxial Debye regime shown as 3D plots.

Fig. 3. Intensity (a) in the focal plane and (b) along the shadow edge,

v = u

, of a cylindrical lens (solid curve) and a spherical lens (dotted curve).

Fig. 4. (a) Intensity along the optical axis of a cylindrical lens (solid curve) and a spherical lens (dotted curve). (b) The axial phase variation, with

k z

suppressed, for a cylindrical lens (solid curve) and a spherical lens (dotted line).

Fig. 5. Fraction of energy

E

with a region

| v | \leq v_{d}

for a cylindrical lens (solid curve) and a spherical lens (dotted curve).

The case of diffraction of a plane wave by a slit corresponds to

f \to \infty

N_{0} \to 0

. Then we have

v = 2 π N (x / a)

u = - 2 π N

[15

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

]. Results for this problem for the case of a circular aperture are well known in the acoustics literature [23

J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971). [CrossRef]

For systems of high numerical aperture, we should consider polarization effects. A method analogous to that of Richards and Wolf [24

B. Richards and E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. B 69, 854–856 (1956). [CrossRef]

] can be used. The aplanatic apodization factor for a slit system is

\cos^{1 / 2} θ

, with

θ

the angle subtended by a ray relative to the optical axis, the same as for a circular aperture [25

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993). [CrossRef]

]. The

s

-polarization case is identical to the nonparaxial scalar case. In the Debye regime, for

s

polarization the electric field in the focal region is given by (neglecting a multiplying constant)

E_{y} = \int_{- α}^{α} \cos^{1 / 2} θ \exp [i k (x \sin θ + z \cos θ)] d θ,

(13)

whereas for

p

polarization it is

E_{x} = \int_{- α}^{α} \cos^{3 / 2} θ \exp [i k (x \sin θ + z \cos θ)] d θ, E_{z} = \int_{- α}^{α} \cos^{1 / 2} θ \sin θ \exp [i k (x \sin θ + z \cos θ)] d θ .

(14)

The time-averaged electric energy density in the focal region, normalized to unity at the focal point, is plotted in Figs. 6 and 7 for a cylindrical lens with angular semiaperture of 60°. The

p

-polarization case gives a broader point spread function, as a result of the apodization factor

\cos θ

in the expression for

E_{x}

, and a stronger side lobe structure as a result of the longitudinal field component in the focal region. We can also plot the Poynting vector, showing power flow in the focal region, as has been done for the case of a spherical lens [26

A. Boivin, J. Dow, and E. Wolf, “Enery flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967). [CrossRef]

]. Flow curves are shown in Fig. 8. As a result of the interchangeability of electric and magnetic fields, this figure holds for either

s

p

polarization. In Fig. 8(b) we show a closeup of a region containing a vortex and two singularities A and B at which the magnitude of the Poynting vector vanishes. The

z

component of the Poynting vector can also take negative values at some points.

Fig. 6. Time-averaged electric energy density in the focal region of a cylindrical lens of angular semiaperture 60° for (a)

s

polarization and (b)

p

polarization in the Debye regime shown as 3D plots.

Fig. 7. Time-averaged electric energy density (a) in the focal plane and (b) along the axis, of a cylindrical lens of angular semiaperture 60° in the Debye regime.

s

polarization is shown as solid curves and

p

polarization as dotted curves.

Fig. 8. (a) Flow curves of the Poynting vector for either an

s

- or a

p

-polarized cylindrical lens of angular semiaperture 60°. (b) Closeup of a region containing two singularities and a vortex.

For intermediate values of numerical aperture, up to a value of

α

of about 30°, Eq. (9) is a good approximation with, if defocus

δ z / f

is small so that the Debye approximation holds, a defocus coordinate given by

u = 4 k δ z \sin^{2} (\frac{α}{2}),

(15)

rather than

u = k δ z \sin^{2} α

[27

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

,28

C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003). [CrossRef]

]. As

δ z / f

increases, the variation of the defocus coordinate with defocus becomes nonlinear [28

C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003). [CrossRef]

z = \frac{u}{4 k f \sin^{2} (α / 2)} {\frac{1 - (u / 4 k f)}{1 - [u / 4 k f \sin^{2} (α / 2)]}},

(16)

exactly, which reduces to the expression of Li and Wolf for low numerical aperture [14

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]

For systems with aberration, the equivalent for the cylindrical lens case of the Zernike aberration polynomials for three dimensions is the set of Legendre polynomials, suitably normalized [29

R. Barakat, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965). [CrossRef]

3. Transfer Functions and Phase Space

In the scalar paraxial approximation (as is assumed throughout this section), the defocused pupil function for a cylindrical lens is

P (ξ) = P_{0} (ξ) \exp (\frac{1}{2} i u ξ^{2}), ξ < 1,

(17)

where

P_{0} (ξ)

is the in-focus pupil function. The coherent transfer function (CTF) can be used to calculate the spatial frequency content of the amplitude image in a coherent imaging system [13

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

] and is simply a scaled version of the pupil function, so that it is given by

c (m) = c_{0} (m) \exp (\frac{1}{2} i u m^{2}), m < 1,

(18)

where

m

is a normalized spatial frequency with a cutoff at

m = 1

, corresponding to a true spatial frequency with cutoff of

\sin α / λ

. For incoherent imaging, on the other hand, the defocused optical transfer function (OTF) can be used to give the strength of the spatial frequencies of the intensity image and is given by a scaled and normalized autocorrelation of the pupil:

C (m, u) = \frac{\int c (m^{'} + m / 2) c^{*} (m^{'} - m / 2) d m^{'}}{\int {| c (m^{'}) |}^{2} d m^{'}} .

(19)

The OTF is the Fourier transform of the intensity point spread function. For an aberration-free slit aperture,

P (ξ) = 1

, Eq. (19) becomes [30

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond Ser. A 231, 91–103 (1955). [CrossRef]

]

C (m, u) = \frac{\sin [(1 - m / 2) m u]}{m u}, | m | \leq 2 .

(20)

This is illustrated in Fig. 9 and can be compared with the analogous results for a spherical lens, also shown in Fig. 9 (and in Born and Wolf [12

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

]). The OTF for the cylindrical lens becomes negative for smaller values of defocus than it does for the spherical lens. For large defocus, the OTF tends to the result predicted by geometrical optics [30

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond Ser. A 231, 91–103 (1955). [CrossRef]

C (m, u) = \frac{\sin (m u)}{m u}, | m | \leq 2 .

(21)

Fig. 9. Defocused OTF for a cylindrical lens (solid curves) and a spherical lens (dotted curves). The curves are shown for

u = 0

, 4, 8, 12, 16.

For the case of an aberration-free spherical lens, a powerful approach for understanding and computing diffraction and imaging effects is based on the principle of the generalized aperture [31

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964). [CrossRef]

]. For the spherical case, this is a 3D function that represents the pupil in

k

space, but for the case of an aberration-free cylindrical lens it is a 2D function, given by a one-dimensional (1D) Fourier transform of the defocused pupil function. The generalized aperture is given by

Π (m, s) = c_{0} (m) δ (s - \frac{m^{2}}{2}), | m | < 1,

(22)

in which

s

is an axial spatial frequency that cuts off at a value of one-half and

δ

is a Dirac delta function. The generalized aperture exists only along a parabolic line, the paraxial approximation of a circle in

k

space. Then the amplitude in the focal region can be efficiently calculated as the 2D Fourier transform of the generalized pupil [31

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964). [CrossRef]

]. We also have, by integration of Eq. (20),

c_{0} (m) = \int Π (m, s) d s

. An axial CTF, which describes the longitudinal image of an axial line object, can then be defined as a projection of the generalized aperture onto the axis (from the projection/slice theorem), giving for an aberration-free slit aperture

c_{axial} (s) = \int_{- 1}^{1} δ (s - \frac{m^{2}}{2}) d m = \frac{1}{\sqrt{2 | s |}}, 0 < s < \frac{1}{2} .

(23)

The generalized OTF is given by the inverse 2D Fourier transform of the intensity in the focal region and is thus the autocorrelation of the generalized aperture [32

L. Mertz, Transformations in Optics (Wiley, 1965).

]. It is given by the 1D Fourier transform of the defocused OTF:

G (m, s) = \int C (m, u) \exp (- i u s) d u .

(24)

For the spherical case, Frieden has shown that the generalized OTF for the paraxial case can be calculated very simply as a line integral over two displaced pupil functions [33

B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967). [CrossRef]

]. In this way it can be seen as an intermediate step in the calculation of the defocused OTF. This approach can also be applied in the cylindrical geometry to give for an aberration-free slit aperture

G (m, s) = \frac{1}{2 | m |}, | s | < | m | (2 - | m |), | m | < 2 .

(25)

This is illustrated in Fig. 10. It exhibits a singularity at the origin. The corresponding result for a spherical lens was illustrated by Sheppard and Gu [34

C. J. R. Sheppard and M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991). [CrossRef]

]. It also exhibits a singularity at the origin. Both exhibit a missing region of spatial frequencies, a double-prism shape for the cylindrical case and a double cone for the spherical case. This missing region is related to the “wings” of the corresponding point spread functions and to the inability of conventional imaging systems to produce good 3D images. The axial OTF, again appropriate for imaging of a longitudinal line object [35

B. R. Frieden, “Longitudinal image formation,” J. Opt. Soc. Am. 56, 1495–1501 (1966). [CrossRef]

], is then given by the projection of the generalized OTF to give for a aberration-free slit aperture

L (s) = \frac{1}{2} \ln (\frac{1 + \sqrt{1 - 2 | s |}}{1 - \sqrt{1 - 2 | s |}}) = \ln (\frac{1 + \sqrt{1 - 2 | s |}}{\sqrt{2 | s |}}), | s | \leq \frac{1}{2} .

(26)

This is illustrated in Fig. (11). It exhibits a singularity at zero spatial frequency. The analogous result for a spherical lens, given by Sheppard and Gu [36

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992). [CrossRef]

], is

L (s) = 1 - 2 | s |

Fig. 10. Generalized (2D) OTF

G (m, s)

for a cylindrical lens. The axes are

m

(normalized transverse spatial frequency) and

s

(normalized axial spatial frequency). Reproduced from [45

C. J. R. Sheppard and K. G. Larkin, “The three-dimensional transfer function and phase space mappings,” Optik 112, 189–192 (2001). [CrossRef]

Fig. 11. Axial OTF for a cylindrical lens.

Phase space refers to distance/spatial frequency representations such as the ambiguity function [37

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974). [CrossRef]

] and the Wigner distribution function [38

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978). [CrossRef]

]. These are useful for investigating optical propagation. The ambiguity function for an aberration-free slit aperture is [39

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983). [CrossRef]

]

A (m, v^{'}) = \frac{\sin [(2 - | m |) v^{'}]}{2 v^{'}}, | m | \leq 2 .

(27)

This is illustrated in Fig. (12). Brenner et al. [39

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983). [CrossRef]

] showed that the ambiguity function can be considered as a polar display of the defocused OTF, with defocus

u = 2 v^{'} / m

. The Wigner function is

W (m^{'}, v) = \frac{\sin [2 (1 - | m^{'} |) v]}{2 π v}, | m^{'} \leq 1 | .

(28)

This can be compared with that for a circular aperture [40

M. J. Bastiaans and P. G. J. van der Mortel, “Wigner distrubution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996). [CrossRef]

]. Interestingly, the Wigner function is a rescaled, renormalized, and rotated version of the ambiguity function. This is true for any even-valued pupil function. The projections of the Wigner function, or sections of the ambiguity function, at different angles are related to the fractional Fourier transform of the aperture [41

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). [CrossRef]

], although a scaling parameter need be defined to determine the fractional order [42

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997). [CrossRef]

,43

C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998). [CrossRef]

]. Various other relationships between phase-space functions and the generalized OTF were given by Larkin and Sheppard [44

K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999). [CrossRef]

,45

C. J. R. Sheppard and K. G. Larkin, “The three-dimensional transfer function and phase space mappings,” Optik 112, 189–192 (2001). [CrossRef]

]. The generalized CTF for a nonparaxial confocal reflection system with cylindrical lenses was presented by Sheppard et al. [25

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993). [CrossRef]

Fig. 12. Ambiguity function for a cylindrical lens. The horizontal axis is

m

and the vertical axis is

v^{'} = u m / 2

. Gray levels range from

- 0.25

(black) to 1 (white).

4. Conclusions

We have presented results for focusing by a cylindrical lens in the Fresnel approximation, using the terminology of Born and Wolf, which apply for systems satisfying the paraxial Debye approximation and also for systems of noninfinite values of Fresnel number. Expressions and plots for systems satisfying the paraxial Debye approximation are given for the intensity in the focal region, the defocused OTF, the generalized OTF, the ambiguity function, and the Wigner function. These results are applicable in the design of optical systems such as microscopes with light-sheet illumination. Many generalizations for focusing in cases when the Fresnel approximation breaks down have been presented by Stamnes [46

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

], but this case is beyond the scope of the present paper.

Acknowledgments

The author acknowledges support from SMART and thanks S. B. Mehta for useful discussions.

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, 229–236 (1993). [CrossRef]
2.	E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azzam, “Thin laser sheet microscope for microbial oceanography,” Opt. Express 10, 145–154 (2002). [CrossRef]
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, 1007–1009 (2004). [CrossRef]
4.	H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. 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, 331–336 (2007). [CrossRef]
5.	C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 35, 1169–1185 (1988). [CrossRef]
6.	G. J. Brakenhoff and K. Visscher, “Confocal imaging with bilateral scanning and array detectors,” J. Microsc. 165, 139–146 (1992). [CrossRef]
7.	R. Wolleschensky and B. Zimmermann, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006). [CrossRef]
8.	E. R. Dowsky and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995). [CrossRef]
9.	J. S. Marsh, “Light distribution near the focus of a two-dimensional lens,” Am. J. Phys. 52, 152–155 (1984). [CrossRef]
10.	D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part I. 2-D system analysis,” Opt. Commun. 263, 171–179 (2006). [CrossRef]
11.	D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006). [CrossRef]
12.	M. Born and E. Wolf, Principles of Optics , 6th ed. (Pergamon, 1993).
13.	J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
14.	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]
15.	Y. Li, “Three-dimensional intensity distribution in low Fresnel number focusing systems,” J. Opt. Soc. Am. A 4, 1349–1353 (1987). [CrossRef]
16.	M. Abramowitz and I. Stegun, Handbook of Mathematical Functions , 3rd ed. (Dover, 1972).
17.	G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957). [CrossRef]
18.	J. J. Stamnes and S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981). [CrossRef]
19.	J. Erkkila and M. Rogers, “Diffracted fields in the focal region of a convergent wave,” J. Opt. Soc. Am. 71, 904–905 (1981). [CrossRef]
20.	C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000). [CrossRef]
21.	E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981). [CrossRef]
22.	C. J. R. Sheppard, “Validity of the Debye approximation,” Opt. Lett. 25, 1660–1662 (2000). [CrossRef]
23.	J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971). [CrossRef]
24.	B. Richards and E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. B 69, 854–856 (1956). [CrossRef]
25.	C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993). [CrossRef]
26.	A. Boivin, J. Dow, and E. Wolf, “Enery flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967). [CrossRef]
27.	C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987). [CrossRef]
28.	C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003). [CrossRef]
29.	R. Barakat, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965). [CrossRef]
30.	H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond Ser. A 231, 91–103 (1955). [CrossRef]
31.	C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964). [CrossRef]
32.	L. Mertz, Transformations in Optics (Wiley, 1965).
33.	B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967). [CrossRef]
34.	C. J. R. Sheppard and M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991). [CrossRef]
35.	B. R. Frieden, “Longitudinal image formation,” J. Opt. Soc. Am. 56, 1495–1501 (1966). [CrossRef]
36.	C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992). [CrossRef]
37.	A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974). [CrossRef]
38.	M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978). [CrossRef]
39.	K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983). [CrossRef]
40.	M. J. Bastiaans and P. G. J. van der Mortel, “Wigner distrubution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996). [CrossRef]
41.	A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). [CrossRef]
42.	J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997). [CrossRef]
43.	C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998). [CrossRef]
44.	K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999). [CrossRef]
45.	C. J. R. Sheppard and K. G. Larkin, “The three-dimensional transfer function and phase space mappings,” Optik 112, 189–192 (2001). [CrossRef]
46.	J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(110.4850) Imaging systems : Optical transfer functions

ToC Category:
Imaging Systems

History
Original Manuscript: October 1, 2012
Manuscript Accepted: November 4, 2012
Published: January 24, 2013

Virtual Issues
(2013) Advances in Optics and Photonics

Citation
Colin J. R. Sheppard, "Cylindrical lenses—focusing and imaging: a review [Invited]," Appl. Opt. 52, 538-545 (2013)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-52-4-538

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References

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, 229–236 (1993). [CrossRef]
E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azzam, “Thin laser sheet microscope for microbial oceanography,” Opt. Express 10, 145–154 (2002). [CrossRef]
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, 1007–1009 (2004). [CrossRef]
H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. 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, 331–336 (2007). [CrossRef]
C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 35, 1169–1185 (1988). [CrossRef]
G. J. Brakenhoff and K. Visscher, “Confocal imaging with bilateral scanning and array detectors,” J. Microsc. 165, 139–146 (1992). [CrossRef]
R. Wolleschensky and B. Zimmermann, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006). [CrossRef]
E. R. Dowsky and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995). [CrossRef]
J. S. Marsh, “Light distribution near the focus of a two-dimensional lens,” Am. J. Phys. 52, 152–155 (1984). [CrossRef]
D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part I. 2-D system analysis,” Opt. Commun. 263, 171–179 (2006). [CrossRef]
D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006). [CrossRef]
M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).
J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
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]
Y. Li, “Three-dimensional intensity distribution in low Fresnel number focusing systems,” J. Opt. Soc. Am. A 4, 1349–1353 (1987). [CrossRef]
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 3rd ed. (Dover, 1972).
G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957). [CrossRef]
J. J. Stamnes and S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981). [CrossRef]
J. Erkkila and M. Rogers, “Diffracted fields in the focal region of a convergent wave,” J. Opt. Soc. Am. 71, 904–905 (1981). [CrossRef]
C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000). [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]
C. J. R. Sheppard, “Validity of the Debye approximation,” Opt. Lett. 25, 1660–1662 (2000). [CrossRef]
J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971). [CrossRef]
B. Richards and E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. B 69, 854–856 (1956). [CrossRef]
C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993). [CrossRef]
A. Boivin, J. Dow, and E. Wolf, “Enery flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967). [CrossRef]
C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987). [CrossRef]
C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003). [CrossRef]
R. Barakat, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965). [CrossRef]
H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond Ser. A 231, 91–103 (1955). [CrossRef]
C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964). [CrossRef]
L. Mertz, Transformations in Optics (Wiley, 1965).
B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967). [CrossRef]
C. J. R. Sheppard and M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991). [CrossRef]
B. R. Frieden, “Longitudinal image formation,” J. Opt. Soc. Am. 56, 1495–1501 (1966). [CrossRef]
C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992). [CrossRef]
A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974). [CrossRef]
M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978). [CrossRef]
K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983). [CrossRef]
M. J. Bastiaans and P. G. J. van der Mortel, “Wigner distrubution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996). [CrossRef]
A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). [CrossRef]
J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997). [CrossRef]
C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998). [CrossRef]
K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999). [CrossRef]
C. J. R. Sheppard and K. G. Larkin, “The three-dimensional transfer function and phase space mappings,” Optik 112, 189–192 (2001). [CrossRef]
J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

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