Page 1

High numerical aperture imaging with different

polarization patterns

N. Lindlein, S. Quabis, U. Peschel, G. Leuchs

Institute of Optics, Information and Photonics (Max Planck Research Group),

Friedrich Alexander University of Erlangen-Nürnberg,

Staudtstr. 7/B2, 91058 Erlangen, Germany

norbert.lindlein@optik.uni-erlangen.de

Abstract: The modulation transfer function (MTF) is calculated for

imaging with linearly, circularly and radially polarized light as well as for

different numerical apertures and aperture shapes. Special detectors are only

sensitive to one component of the electric energy density, e.g. the

longitudinal component. For certain parameters this has advantages

concerning the resolution when comparing to polarization insensitive

detectors. It is also shown that in the latter case zeros of the MTF may

appear which are purely due to polarization effects and which depend on the

aperture angle. Finally some ideas are presented how to use these results for

improving the resolution in lithography.

©2007 Optical Society of America

OCIS codes: (110.4100) Modulation Transfer Function; (260.5430) Polarization; (220.1230)

Apodization; (220.3740) Lithography.

References and links

1.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in

an aplanatic system,” Proc. R. Soc. A 253, 358-379 (1959).

M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt.

Soc. Am. A 3, 2086-2093 (1986).

M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives: erratum,”

J. Opt. Soc. Am. A 10, 382-383 (1993).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.

179, 1-7 (2000).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and

experimental tomographic reconstruction,” Appl. Phys. B B72, 109-113 (2001).

R. Dorn, S. Quabis, and G. Leuchs, “The focus of light - linear polarization breaks the rotational symmetry

of the focal spot,” J. Mod. Opt. 50, 1917-1926 (2003).

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91,

233901 (2003).

S. F. Pereira and A. S. van de Nes, “Superresolution by means of polarization, phase and amplitude pupil

masks,” Opt. Commun. 234, 119-124 (2004).

A.S. van de Nes, L. Billy, S. F. Pereira, and J. J. M. Braat, “Calculation of the vectorial field distribution in

a stratified focal region of a high numerical aperture imaging system,” Opt. Express 12, 1281-1293 (2004).

10. R. Oldenbourg and P. Török, “Point-spread functions of a polarizing microscope equipped with high-

numerical-aperture lenses,” Appl. Opt. 39, 6325-6331 (2000).

11. P. R. T. Munro and P. Török, “Vectorial, high numerical aperture study of Nomarski’s differential

interference contrast microscope,” Opt. Express 13, 6833-6847 (2005).

12. C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4,

1354-1360 (1987).

13. M. Born and E. Wolf, Principles of Optics, 6th Edition. (Cambridge University Press, Cambridge New

York Oakleigh, 1997).

14. J. W. Goodman, Introduction to Fourier optics, 2nd. Edition (McGraw--Hill, New York, 1996).

15. J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Imaging and time-resolved Spectroscopy of

single molecules at an interface,” Science 272, 255-2586 (1996).

16. K. Kamon, “Projection exposure apparatus,” United States Patent 5365371 (filed 1993).

2.

3.

4.

5.

6.

7.

8.

9.

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17. K.-H. Schuster, “Radial polarisationsdrehende optische Anordnung und Mikrolithographie-

Projektionsbelichtungsanlage damit,” European Patent 0 764 858 A2 (filed 1996) and K.-H. Schuster,

“Radial polarization-rotating optical arrangement and microlithographic projection exposure system,”

United States Patent 6885502 (filed 2002).

18. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal

polarization converters,” Opt. Lett. 21, 1948-1950 (1996).

19. D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett.

42, 492-494 (1983).

20. E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, and N. Streibl, “Form birefringence of surface relief

gratings and its angular dependence,” Opt. Commun. 89, 173-177 (1992).

21. Z. Bomzon G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by

space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285-287 (2002).

22. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Formation of linearly polarized light with axial symmetry

by use of space-variant subwavelength gratings,” Opt. Lett. 28, 510-512 (2003).

23. E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Progress in

Optics, Vol. 47, 215-289, E. Wolf, ed., (Elsevier Amsterdam 2005).

24. U. Levy, C. Tsai, L. Pang, and Y. Fainman, “Engineering space-variant inhomogeneous media for

polarization control,” Opt. Lett. 29, 1718-1720 (2004).

25. C. Tsai, U. Levy, L. Pang, and Y. Fainman, “Form-birefringent space-variant inhomogeneous medium

element for shaping point-spread functions,” Appl. Opt. 45, 1777-1784 (2006).

26. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a

parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318-1320 (2004).

1. Introduction

In modern optical lithography increasingly small structures have to be realized. This is done

by reducing the wavelength, increasing the numerical aperture, and by some other means such

as a special type of illumination or phase shift masks. The numerical aperture itself NA=n.sinϕ

can be increased by increasing either the half aperture angle ϕ or the refractive index n of the

medium in front of the target. The latter method is used in modern immersion DUV-

lithography at a vacuum wavelength of λ=193 nm by putting water between the last surface of

the lithography objective and the wafer. The refractive index of water is about n=1.44 at this

wavelength reducing the effective value λ/n to about 134 nm. That is smaller than the 157 nm

in air, formerly thought to be the next lower wavelength on the roadmap of lithography.

Increasing the half aperture angle ϕ has the natural limit of ϕ=π/2, i.e. sinϕ=1.0.

However, it is well known that the point spread function (PSF) of a linearly polarized

plane wave focused by a high numerical aperture objective to a tight spot is no longer

rotationally symmetric [1-6] because polarization effects break the symmetry. Consequently,

the modulation transfer function (MTF) for incoherent illumination which is the modulus of

the inverse Fourier transform of the PSF, will also not be rotationally symmetric. On the other

hand, the PSFs for circularly and radial polarization [7] are both rotationally symmetric, and

hence the MTFs will be rotationally symmetric in these cases as well. Thus it is very

important for modern lithography that we investigate the influence polarization effects have

on the PSF and the MTF [8-12]. In this paper we will use the MTF for incoherent

illumination, which is of course an approximation because the illumination in lithography is

partially coherent. This approximation can still give us at least an idea of which polarization

state is the most appropriate at which aperture angle.

In the next section we describe the calculation of the modulation transfer function for

different polarization states in theory. In section 3 we deal with the numerical calculation of

the MTFs for different polarization states, different aperture angles, and different apodization.

In section 4 we discuss the results. In section 5 we give some ideas of how to apply these

results to lithography and microscopy. Section 6 is the conclusion of the paper.

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2. Calculation of the modulation transfer function

In the following, we assume that the optical imaging system is aplanatic and hence fulfilling

the sine condition [13]. Moreover, we assume an ideal lens without aberrations which is

ideally anti-reflection coated so that the transmittance is one (or at least constant) at all points

of the aperture. The PSF for object points which are not far away from the optical axis will be

just a laterally shifted copy of the on-axis PSF because of the aplanatism. The calculation of

the PSF is explained in the appendix. In the case of incoherent illumination the intensity

distribution in the image plane is then a convolution of the intensity distribution in the object

plane and of the PSF. From the imaging of extended objects [13,14] it is well-known that in

this case the modulation transfer function (MTF) for incoherent illumination is the modulus of

the optical transfer function (OTF), which is the inverse Fourier transform of the point spread

function (PSF) apart from a normalization constant:

(

ν

)

()

(

ν

)()

()

∫ ∫

∞−

∫ ∫

∞−

∞+∞+

∞−

+∞+∞

∞−

+

=

dxdyyx

dxdyyxiyx

yx

yx

,PSF

2exp, PSF

,

ν

MTF

νπ

(1)

Here, x,y are the coordinates in the image plane and νx,νy are spatial frequencies in x- and y-

direction. The denominator normalizes the MTF and ensures that the MTF has the value 1 at

the spatial frequency νx=νy=0. This has to be the case because the meaning of the MTF is that

it gives the deterioration of the contrast if a grating-like object with a sinusoidal intensity

variation and spatial frequencies νx,νy is imaged. A grating with spatial frequencies zero, i.e.

with infinite period, is of course always imaged perfectly so that the MTF has to be 1 at zero

spatial frequency.

In the scalar approximation it is also quite easy to show that the MTF is proportional to the

modulus of the autocorrelation

(

λπ

/ ', '2 exp' , '' , '

yx iWyxAyxP

=

, with the amplitude distribution A and the wave

aberrations W expressed as optical path length differences in the exit pupil with coordinates x’

and y’ [14]. It is also well-known from mathematics how to calculate the autocorrelation

function of the pupil function at a certain spatial frequency. Two copies of the pupil function

are laterally shifted relative to each other and then one calculates the integral of the product of

the pupil function with the complex conjugated shifted copy of the pupil function. Since the

lateral shift depends on the spatial frequency and the pupil function is zero outside of the

aperture there is a maximum spatial frequency where the MTF has a function value different

from zero. This maximum frequency is the so called cut-off frequency νcut with:

function of the pupil function

)()()()

λ

ϕ

λ

ν

sin

2

NA

2

cut

n

==

(2)

For higher spatial frequencies the MTF is always zero because there is no overlap of the two

laterally shifted copies of the pupil function in the calculation of the autocorrelation function.

To calculate the MTF by taking into account polarization effects the PSF has of course to

be calculated according to the method described in Ref. [1] as an interference pattern of plane

waves traveling along the direction of the geometric rays from the exit pupil to the focus.

Therefore, the PSF is proportional to the sum of the electric energy densities of all three

components of the electric field in the focus. However, also in this case a kind of pupil

function can be defined for each component of the electric field and the MTF will be

proportional to the modulus of the sum of the autocorrelation functions of these three pupil

functions. Therefore, it is clear that also in the vectorial case of taking into account

polarization the maximum spatial frequency for which the MTF can be different from zero is

the cut-off frequency defined by Eq. (2).

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3. MTFs for different polarization states and numerical apertures

The MTF is calculated numerically with the method of the last section for different

polarization states, different aperture angles, and different amplitude functions in the entrance

pupil. To avoid aliasing effects by using a Fast Fourier transformation and to have a well-

sampled MTF afterwards, the diameter of the PSF data has to be high enough. We used a PSF

field with a diameter of D=32λ/NA and 512x512 samples. So, the lateral sampling distance

between two points was just Δx=λ/(16NA) corresponding to a maximal spatial frequency

νmax=1/(2Δx)=8NA/λ. This is 4 times larger than the cut-off frequency of the MTF of Eq. (2)

so that we did not explicitly use the fact that the MTF is zero outside of the cut-off frequency,

but we can prove it in this way also numerically. Of course, by knowing that the MTF is zero

outside of the cut-off frequency, there would also be no aliasing effects if we would take a

larger lateral sampling distance of Δx=λ/(4NA), which is the largest allowed sampling

distance (or smallest sampling density) without getting aliasing effects. At the end, we have

using our small sampling distance still 128 samples of the MTF with values different from

zero, so that the lateral resolution of the MTF is high enough. Additionally, in all our

calculations a refractive index of one was assumed so that we have NA=sinϕ. This means that

the largest influence of the polarization effects onto the PSF and therefore also onto the MTF

will be for the limiting case of NA=1.0. Nevertheless, it has to be mentioned that the

polarization effects depend on the aperture angle and therefore on sinϕ, but not directly on the

NA itself which also depends on the refractive index n of the medium. Only, for a non-

immersion optical system with a refractive index of one we can use the parameters NA and

sinϕ with the same meaning.

Normally detectors are sensitive to all components of the electric field so that the MTF has

to be calculated by using the complete PSF being proportional to the sum of the electric

energy densities of all electric field components. But, as mentioned in Ref. [4], there may also

be a special detector which is only sensitive to a certain component of the electric field. By

coating a very thin photo resist layer (much smaller than the wavelength of the used light) on a

metal layer with high conductivity one can imagine that the lateral field components will

vanish in the near field shortly above the metal layer since in the metal layer electric currents

are induced which generate reverse lateral electric field components. So, only the longitudinal

electric field component will exist in the thin photo resist layer and will be detected. Another

possibility would be to use anisotropic molecules which are only sensitive to one field

component [15]. If these molecules can additionally be aligned with their symmetry axis in

one direction they would form a detector which is only sensitive to one special electric field

component. So, it would perhaps also be possible to build a detector for one of the lateral field

components only. Of course, especially this second case is in the moment of speculative

nature since it is not easy to find such molecules with a high sensitivity to only one

component of the electric field and which additionally can be aligned with their axes in one

common direction, but parallel to the substrate. However, let us assume below that we have

besides a normal detector which is sensitive to all components of the electric field also one

detector which is only sensitive to the longitudinal component (in the following named z-

component since the optical axis is along the z-axis) and also one detector which is sensitive

to one of the lateral components (in the following always the y-component of the electric

field).

In the following, it is assumed that a plane wave with a certain polarization state (linear,

circular or radial) is focused by a high NA microscope objective which fulfills the sine

condition. In the case of linear or circular polarization we assume a constant amplitude in the

entrance pupil, whereas for radial polarization first a doughnut shaped electric field

distribution

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Page 5

()

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

w

−

⎟

⎠

⎟

⎟

⎟

⎞

⎜

⎝

⎜

⎜

⎜

⎛

=

2

0

22

0

exp

0

,

yx

y

x

EyxE

(3)

with a beam waist parameter w0=0.95 raperture (raperture= radius of entrance pupil) is taken.

For linear polarization it is well-known [4, 6] that the electric field component in the focus

parallel to the direction of linear polarization of the incident light forms a nearly circular spot

similar to that assumed in the scalar case, whereas the longitudinal component with its non-

rotationally symmetric shape broadens the spot to an elliptic shape. Of course, for very high

aperture angles with sinϕ near 1.0 also the spot for the electric field component parallel to the

direction of polarization of the incident light will be elliptic to some degree. So, we will

calculate for linear polarization, which we define to be in the y-direction, the MTFs for the

two cases of using all electric field components or for using only the y-component of the

electric field.

For circular polarization we will only calculate the MTF for using all components of the

electric field.

For radial polarization [4, 7] the z-component (longitudinal component) of the electric

field forms in the focus a tight spot with a small maximum and reasonably low side lobes,

whereas the lateral components broaden the spot (especially for small values sinϕ). Therefore,

for radial polarization we will calculate the MTFs for the two cases of using all components of

the electric field or for using only the z-component.

Since for circular and radial polarization the PSF and therefore also the MTF is

rotationally symmetric, only one section of the MTF will be displayed in this case. For linear

polarization the PSF is no longer rotationally symmetric for high values sinϕ and therefore for

linear polarization we will always display a section of the MTF with spatial frequency νx in x-

direction, i.e. the grating lines are in this case in y-direction parallel to the direction of

polarization of the incident light, and a section with spatial frequency νy in y-direction, i.e. the

grating lines are here in x-direction perpendicular to the direction of linear polarization.

Figure 1 shows the MTF curves for a full circular aperture and values of sinϕ of 0.2, 0.7,

0.8, 0.9, and 1.0. The same is done for an annular aperture with an inner radius of 90% of the

aperture radius and Fig. 2 shows the corresponding MTF curves. In both figures, there are for

the case of linear polarization curves for ν=νx (signated with “x-section”) and for ν=νy

(signated with “y-section”). Additionally, for linear and radial polarization there are curves

where all components of the electric field were taken into account for the PSF (signated with

“all components”) or only one component (signated with “y-component only” for linear

polarization or “z-component only” for radial polarization).

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Linear, all components, x-section

Linear, all components, y-section

Linear, y-component only, x-section

Linear, y-component only, y-section

Circular

Radial, all components

Radial, z-component only

0 0.20.4 0.6 0.811.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =0.2

ϕ

00.2 0.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =0.7

ϕ

00.20.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =0.8

ϕ

00.20.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =0.9

ϕ

0 0.20.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

Fig. 1. MTF curves for a full circular aperture and sinϕ ranging from 0.2 to 1.0 for different

polarization states (linear, circular and radial).

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Linear, all components, x-section

Linear, all components, y-section

Linear, y-component only, x-section

Linear, y-component only, y-section

Circular

Radial, all components

Radial, z-component only

0 0.20.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

00.2 0.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =0.9

ϕ

00.2 0.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =0.8

ϕ

00.20.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =0.7

ϕ

0 0.20.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =0.2

ϕ

Fig. 2. MTF curves for an annular aperture (inner radius is 90% of the outer radius) and sinϕ

ranging from 0.2 to 1.0 for different polarization states (linear, circular and radial).

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00.20.4 0.6 0.811.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

r =0

in

r =0.1 r

in

r =0.2 r

in

r =0.3 r

in

r =0.4 r

in

r =0.5 r

in

r =0.6 r

in

r =0.7 r

in

r =0.8 r

in

r =0.9 r

in

all components

aperture

aperture

aperture

aperture

aperture

aperture

aperture

aperture

aperture

00.20.4 0.6 0.811.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

r =0

in

r =0.1 r

in

r =0.2 r

in

r =0.3 r

in

r =0.4 r

in

r =0.5 r

in

r =0.6 r

in

r =0.7 r

in

r =0.8 r

in

r =0.9 r

in

aperture

aperture

aperture

aperture

aperture

aperture

aperture

aperture

aperture

z-component only

Fig. 3. MTF curves for the case (i) of section 3, i.e. radial polarization and different annular apertures.

Another interesting case is to show the influence of apodization effects in the entrance

pupil onto the MTF. We calculated it for the case of radial polarization at the limiting case of

sinϕ=1.0. Three cases were considered.

(i) Annular apertures with different ratios rin/raperture and homogeneous amplitude in the

entrance pupil, i.e. the electric field in the entrance pupil is of the form:

()

otherwise0

for

0

,

22

22

0

⎪

⎩

⎪

⎪

⎨

⎪

⎧

≤+≤

⎟

⎠

⎟

⎟

⎟

⎞

⎜

⎝

⎜

⎜

⎜

⎛

+

=

aperture

r

in

yxry

x

yx

E

yxE

(4)

(ii) A smoothly varying electric field in the full circular entrance pupil which is represented by

the equation

()

()

()

( )

r

22

2

0

2

0

2

0

22

2

1

22

0

and exp'exp

0

',

yxr

w

r

rEE

w

yx

y

x

yxEyxE

n

n

+=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

⇒

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

−

⎟

⎠

⎟

⎟

⎟

⎞

⎜

⎝

⎜

⎜

⎜

⎛

+=

−

(5)

Here, two different cases for the waist parameter w0 were simulated. (iia) The waist parameter

of the Gaussian function is assumed to be constant with w0=0.95 raperture. (iib) The waist

parameter is chosen in such a way that there is the maximum of the electric field amplitude of

Eq. (5) exactly at the rim of the aperture:

( )

n

w

dr

0

⎝

⎠

⎝

aperture

r

r

⇒

r

n

n

w

w

rr

nr

rEd

aperture

2

0exp2

0

2

0

2

2

1

1

==

⎟⎟

⎠

⎞

⎜⎜

⎛−

⎟⎟

⎞

⎜⎜

⎛

−=

=

+

−

(6)

Remark: There is for n≠0 really a maximum of |E| at the position of w0 (and not a

minimum or saddle point) since |E| is a non-negative function with |E|=0 at r=0. So, |E|

increases for r>0 and the only extreme value which exists has to be a maximum. For n=0 Eq.

(6) is not valid, but there is a maximum at r=0 since the function is then just a Gaussian

function exp(-r2/w0

2).

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Page 9

00.20.4 0.6 0.811.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

22

0

222

0

322

0

422

0

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

622

0

722

0

822

0

922

0

522

0

22

0

all components

w =0.95r

0aperture

00.20.4 0.6 0.811.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

22

0

222

0

322

0

422

0

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

622

0

722

0

822

0

922

0

522

0

22

0

z-component only

w =0.95r

0aperture

Fig. 4. MTF curves for the case (iia) of section 3, i.e. radial polarization and different

apodization of the aperture with the parameter n ranging from n=0 to n=9. The respective

amplitude function |E(r)| is displayed in the figure captions.

00.20.4 0.6 0.811.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

22

0

222

0

322

0

422

0

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

622

0

722

0

822

0

922

0

522

0

w =(2/n) r

0aperture

1/2

all components

0 0.20.4 0.6 0.81 1.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

22

0

222

0

322

0

422

0

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

r exp(-r /w )

622

0

722

0

822

0

922

0

522

0

w =(2/n) r

0aperture

1/2

z-component only

Fig. 5. MTF curves for the case (iib) of section 3, i.e. radial polarization and different

apodization of the aperture with the parameter n ranging from n=1 to n=9. The respective

amplitude function |E(r)| is displayed in the figure captions.

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00.20.4 0.6 0.811.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

all components

r =0.6 r

in

r exp(-r /w ) and w =0.95 r

0

r exp(-r /w ) and w =(2/3) r

0

aperture

00.20.4 0.6 0.811.2 1.4 1.6

λ

1.82

ν

in NA/

MTF

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin =1.0

ϕ

r =0.6 r

in

r exp(-r /w ) and w =0.95 r

0

r exp(-r /w ) and w =(2/3) r

0

r exp(-r /w ) and w =(2/4) r

0

aperture

z-component only

322

0aperture

322

0aperture

r exp(-r /w ) and w =(2/4) r

0

=0.71

4221/2

0 aperture

raperture

=0.82

=0.71

3221/2

0aperture

raperture

raperture

=0.82

3221/2

0aperture

raperture

4221/2

0aperture

Fig. 6. Comparison of some MTF curves using either a “hard mask” apodization via an annular

aperture (green curve for rin/raperture=0.6) or a smooth apodization via different amplitude

functions (red and black curves with the amplitude function |E| in the figure caption).

The MTF curves of the case (i) with rin/raperture ranging from 0 (full aperture) to 0.9 are

shown in Fig. 3. For the case (iia) and the power n ranging from n=0 to n=9 Fig. 4 gives the

MTF curves. Finally, Fig. 5 shows case (iib) with n ranging from n=1 to n=9. In each of the

figures the MTF is calculated using either the total electric energy density in the focus or only

the z-component.

4. Evaluation of the calculated modulation transfer functions

4.1 Full circular aperture

First, the MTF curves for the full circular aperture of Fig. 1 will be discussed for the different

values sinϕ and different polarization states. It can easily be seen that for the small value

sinϕ=0.2 which approaches the scalar case the MTF curves for linear polarization and circular

polarization coincide in x- and y-direction independent whether all electric field components

are taken or only the y-component. On the other side, the MTF curves for radial polarization

are totally different. If the sum of the electric energy density of all components is taken the

green short dashed curve results which has a zero of the contrast for a spatial frequency of

about 0.9 NA/λ with NA=0.2. So, for higher spatial frequencies there is a contrast inversion

which is of course quite bad for optical imaging. On the other side, if only the electric energy

density coming from the z-component is taken (long dashed black line), there is a quite high

contrast for high spatial frequencies. But, it has to be taken into account that the amount of

light power which is in the z-component decreases with the square of sinϕ. Therefore, for

sinϕ=0.2 there is nearly no light power in the z-component. For increasing values sinϕ, there

is an increasing difference between the two curves of the MTF with spatial frequencies in x-

and y-direction for the case of linear polarization. This is especially valid if the total electric

energy density is taken, but also in a less pronounced form if only the y-component is taken.

For a final conclusion we have to distinguish between several cases:

(i) Normal detectors are sensitive to all components of the electric field. So, in this case we

can only compare the corresponding curves for the different polarization states (blue lines for

linear polarization, dashed-dotted black line for circular polarization, and short dashed green

line for radial polarization). So, if there are small structures with different orientations radial

polarization is superior for sinϕ ≥0.9 and high spatial frequencies near the cut-off frequency.

If all structures are oriented in the same direction, i.e. grating-like structures in only one

direction, linear polarization with the polarization direction parallel to the grating lines gives

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the best solution. But, if there are also structures in the perpendicular direction there is a zero

of the MTF for linear polarization and sinϕ>0.7!

(ii) An anisotropic detector which is only sensitive to the z-component of the electric field

should be easier to realize than a detector which is only sensitive to the y-component. So, it is

more probable that the z-component of radial polarization (long dashed black line) can be

used in practice than using alone the y-component for linear polarization (red curves). Using

the z-component of radial polarization is according to the MTF curves useful for all values

sinϕ, but only for high values sinϕ there is also a high amount of the light power in this

component since it is proportional to approximately sinϕ 2. Using the y-component of linear

polarization is especially useful if the structures, which have to be imaged, are all oriented in

the same direction (grating lines in y-direction so that the spatial frequencies are in x-

direction) and if the value sinϕ is high. If there are also structures with spatial frequencies in

y-direction (dashed red lines) the contrast will decrease quite fast, although there is no zero of

the MTF below the cut-off frequency.

4.2 Annular aperture with inner radius equal to 90% of the aperture radius

The MTF curves for the annular aperture in Fig. 2 show the remarkable effect that for all

values of sinϕ the curves of linear polarization with spatial frequencies in x-direction and

using all electric field components (blue solid lines) nearly coincide with the curves of radial

polarization using only the z-component (long dashed black lines). Only for sinϕ=1.0 there is

a small difference in both curves. An explanation for this similarity of the MTF curves is that

the PSF along a central section in x-direction approaches the theoretical scalar PSF if the light

is linearly polarized in y-direction and if an annular aperture is used. The same is valid for the

axial component of the electric energy density in the case of radially polarized light and an

annular aperture.

It can also be seen that for sinϕ=0.2, which is for linear and circular polarization nearly

equivalent to the scalar case, all curves with the exception of the curve for radial polarization

using all electric field components (green short dashed line) coincide.

Similar to the case of a full circular aperture, there are for the annular aperture zeros of the

MTF for radial polarization using all components of the electric field (green short dashed

lines) up to values sinϕ=0.7 and for linear polarization using all components of the electric

field in the case of spatial frequencies in y-direction (blue dashed lines) if sinϕ≥0.8.

If we have a detector which is only sensitive to the y-component of the electric field, the

curves for linear polarization show for increasing values sinϕ an also increasing difference

between the contrast for structures with spatial frequencies in x- (red solid lines) and y-

direction (red dashed lines). For sinϕ=1.0 the contrast for spatial frequencies in x-direction

can approach the very high value of about 0.175 for νx=1.88 NA/λ! On the other side, for

spatial frequencies in y-direction with the same modulus, i.e. νy=1.88 NA/λ, the contrast is

only 0.02, i.e. nearly ten times smaller as in the x-direction. If we consider for comparison the

same curves for the full circular aperture at the same values of the spatial frequencies we see

that there the contrast is only about 0.06 in x-direction and 0.01 in y-direction. So, the annular

aperture is useful for the imaging of structures with spatial frequencies near the cut-off

frequency, whereas it is not so useful for the imaging of small or medium spatial frequencies.

4.3 Apodization effects for radial polarization and sinϕ=1.0

Finally, the apodization effects shall be discussed for the case of radial polarization and the

limiting case sinϕ=1.0 (Figs. 3-6).

For the annular apertures with different inner radii (case (i)) Fig. 3 shows that there is

nearly no difference between the curve of the full aperture (rin=0) and the curves with

rin/raperture<0.4, especially if all components of the electric field are detected. For increasing

values rin/raperture the contrast for small and medium spatial frequencies decreases whereas the

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contrast for very high spatial frequencies near the cut-off frequency increases by shaping a

maximum of the MTF which approaches more and more the cut-off frequency. Of course, the

existence of a maximum of the MTF for high ratios rin/raperture at very high spatial frequencies

means that there exists also a minimum of the contrast at medium spatial frequencies, whereas

the MTF curve is strictly monotonic decreasing for rin/raperture≤0.3.

Similar behaviors can be seen for the smoothly apodized pupils of the cases (iia) and (iib),

but there the maxima for high spatial frequencies are less pronounced and the contrast is fairly

high over a large range of spatial frequencies. Whereas Fig. 4, i.e. case (iia), shows some

differences between the curves for very high spatial frequencies, there are nearly no

differences for very high spatial frequencies in Fig. 5, i.e. case (iib). Totally, the curves in Fig.

5, i.e. if the maximum of the amplitude is at the rim of the aperture for all different amplitude

functions, differ less than in Fig. 4, where a constant waist parameter w0=0.95 raperture is taken.

Finally, in Fig. 6 some curves of Figs. 3, 4, and 5 are combined in the same figure to show

that nearly the same MTF curves can be obtained by using either a “hard mask” apodization

via an annular aperture and homogeneous intensity or by using a smooth apodization via a

certain smooth amplitude function. There, the amplitude functions

w0=0.95 raperture or w0=0.82 raperture (i.e. maximum at the rim of the aperture) or

(

0

/ exp

wrrE

−=

compared to the annular aperture with rin/raperture=0.6.

()

2

0

23

/ exp

wrrE

−=

and

)

224

and w0=0.71 raperture (i.e. also maximum at the rim of the aperture) are

5. Application in lithography and microscopy

Polarization

converter

High NA

microscope

objective

Low NA

tube lens

Linear

polarization

Special object points

emitting like small dipoles

with axis parallel to optical axis

Radial polarisation

Fig. 7. Optical system for the imaging of an extended object which consists of small particles

which emit like a dipole with its axis parallel to the optical axis.

In this section we will discuss ideas on how to apply the results. In Ref. [4] it was

proposed that particles oscillating like a dipole with its axis parallel to the optical axis will

emit radially polarized light which can be captured by a high NA microscope objective. In the

image plane of this microscope objective a magnified image will be formed and the PSFs of

the single particles which are assumed to be incoherent to each other will show the typical

form for radially polarized light. But, due to the magnification factor β>>1, the numerical

aperture NAimage will be demagnified by the same factor compared to the numerical aperture in

the object space NAObj

1 NA

1

β

NA

<<=

Objimage

, (7)

provided the microscope objective fulfills the sine condition. In the image plane we find that

different field components are distributed differently. On one hand there is a well focused but

quite weak longitudinal component of the electric energy density. On the other hand we also

find a strong and broad lateral component. The latter decreases the resolution if the detector is

sensitive to the total electric energy density and does not distinguish between the lateral and

longitudinal component (see for comparison the green dashed MTF curve in Fig. 1 for the

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small value sinϕ=0.2). One solution would be to use a detector sensitive only to the

longitudinal component, sacrificing most of the intensity. Another and better solution is using

a polarization converting element introduced into the Fourier plane of the microscope

objective, that converts radial polarization into linear polarization (see Fig. 7). This

polarization converter may be combined with an apodizing element. This way a normal PSF

of linearly polarized light will be formed in the image plane. As was shown in section 4 the

MTF for linearly polarized light will have higher modulation than for radially polarized light

if the numerical aperture is much smaller than 1 (see again Fig. 1 for the small value

sinϕ=0.2). Of course this application in microscopy only works if each particle in the object

emits like a dipole with radial polarization. If the object emits linearly or circularly polarized

light the polarization state should not be changed in microscopy.

On the other hand, the same optical system can be applied to optical lithography if it is

used in the opposite direction (see Fig. 8). A mask which should be demagnified by a factor of

normally β=0.2 or 0.25 is illuminated with linearly polarized light. Since the numerical

aperture in the object space NAObj is in this case according to Eq. (7) smaller than or equal to

β (assuming a non-immersion system with NAimage≤1), the plane wave behind the collimating

lens will also be linearly polarized if polarization effects at the mask are neglected. Then, the

polarization converting element forms a radially polarized mode which is focused by the high

NA objective to a tight spot if the aperture angle sinϕ is larger than 0.9. So, the modulation of

the image will be increased for high spatial frequencies compared to using linearly polarized

light which would have a zero of the MTF for grating-like structures with the grating lines

perpendicular to the direction of polarization. Normally, polarization effects at the mask with

periods down to p=1/(βνcut)=2λ/NA (assuming β=0.25) cannot be neglected and locally

elliptical polarized light will result behind the mask if the grating lines are oriented with an

arbitrary angle relative to the direction of polarization of the incident light. But then, a

polarizer in front of the polarization converting element can produce a well-defined linear

polarization state without blocking too much light.

Polarization

converter

Low NA

collimating

lens

High NA

focussing

lens

Linear

polarization

Radial

polarization

Fig. 8. Optical system for optical lithography at high aperture angles. The mask emits nearly

linearly polarized light which is collimated by the first lens. The polarization converting

element forms radially polarized light from the linearly polarized plane wave coming from one

of the object points. The high NA objective behind the polarization converter focuses the light

to a tight spot.

Such a polarization converter was proposed in the context of a lithographic system in

former patents [16, 17], but for quite a different purpose. In these patents the goal was to

minimize the large angle Fresnel reflection losses at the photo resist and also in the lenses of

the projection lens. Illumination with radial polarization leads to TM polarized light at all

interfaces and for angles close to Brewster's angle nearly all light is transmitted. Of course,

this effect of radially polarized light is also an additional advantage in our proposal.

Unfortunately an efficient polarization converting element, which does not introduce any

aberrations and also works for off-axis illumination does not exist. The reason that it has also

to work for off-axis illumination is that off-axis object points cause tilted plane waves at the

polarization converting element. But there are some demonstration elements based on an array

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of properly oriented half wave plates [17], liquid crystal devices [18] or zero order gratings

[19-25]. A promising approach uses high frequency diffractive optical elements which act as

an artificial birefringent medium [19, 20]. Such elements can be used as position dependent

quarter or half wavelength plates [21-25]. One advantage such elements have over liquid

crystal devices is that they work also with UV light, at least in principle. But the very small

minimum feature sizes and quite large depths of the structures are not easy to fabricate.

6. Conclusion

We calculated the incoherent modulation transfer function for imaging with a diffraction-

limited high numerical aperture optical system for different polarization states and different

apodization at the pupil. We showed that the MTF can have zeros if all components of the

electric energy density are detected, although in the scalar approximation there would be only

zeros of the MTF for optical systems with aberrations.

We discussed the different MTF curves assuming that different types of detectors exist.

These are either sensitive to the total electric energy density which is normally the case or are

assumed to be sensitive to only one polarization component of the electric energy density. We

studied the case of linear incident polarization in which only the component of the electric

field parallel to the incident polarization is used. Grating-like structures with the grating lines

parallel to the incident polarization are imaged with the best contrast for high spatial

frequencies. But the contrast is much smaller for structures with grating lines perpendicular to

the direction of polarization. On the other hand, radially polarized light yields a rotationally

symmetric MTF. For high aperture angles, i.e. high sinϕ with a value of nearly 1.0, the

contrast is comparatively high. Thus, depending on the symmetry of the structures to be

imaged different polarization states should be used: linear polarization for grating-like

structures with only one orientation of the grating lines and radial polarization for structures

with arbitrary orientations. At this point it may be interesting to note that at high spatial

frequencies the MTF calculated in the vectorial theory (s. Figs. 1 and 2) can be higher than the

limiting MTF obtained for the unrealistic assumption of scalar fields (for comparison see the

curves in Figs. 1 and 2 for linear and circular polarization and sinϕ=0.2 which correspond

very well to the scalar MTF curve). A detector which is only sensitive to a certain electric

field component is required for exploiting this advantage.

Finally we proposed the principle design of a system which can apply these results to

lithography by converting the polarization state between the object and the image space. By

specifically controlling the polarization state an increase of the resolution should be

achievable, even in the case of the incoherent imaging of a whole object field.

7. Appendix: Calculation of the PSF of an aplanatic fast lens

The PSF is calculated using a numerical implementation of the Debye integral similar to Ref.

[1] or Refs. [2, 3]. The principal idea is that the electric field in the vicinity of the focus can be

calculated by superimposing plane wave components which propagate along the direction of

the geometrical rays to the focus. Of course, by doing this the local polarization, local phase

and local amplitude of the plane wave components have to be taken into account correctly.

In the entrance pupil of the aplanatic lens the electric field E is numerically sampled by taking

an array of equidistant rays which represent local plane wave components. Each ray number j

is associated with a polarization vector Pj which is a complex valued vector perpendicular to

the direction of propagation ej of the ray. Its modulus |Pj| is proportional to the local electric

field E if the rays are equidistant and its direction represents the direction of polarization for

locally linearly polarized light. OPDj is the optical path length of the ray in the entrance pupil.

It is zero for a real plane wave without aberrations.

For an infinite distant object point (i.e. incident plane wave) the aplanatic lens now

deflects each ray with the height

yxh

+=

in the entrance pupil in such a way that it seems

22

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Page 15

to come from a sphere (exit pupil) with radius f around the focus fulfilling the sine condition

(see Fig. 9):

fh =

ϑ

sin

(8)

ϑ

z

hj

f

ej

e’j

Pj

P’j

dF dF’

Fig. 9. Scheme of a lens fulfilling the sine condition.

Here, ϑ is the angle of the deflected ray with the optical axis (z-axis). In the focus all these

rays will have the same optical path length as in the entrance pupil (i.e. zero for an incident

plane wave without aberrations). The deflected rays now propagate along new direction

vectors e’j pointing from the exit pupil (sphere) to the focus and the new polarization vectors

P’j are calculated in such a way that the component perpendicular to the plane of deflection

remains the same and the component in the plane of deflection is rotated so that it is

perpendicular to the new direction vector e’j. The equations which are simple vector calculus

will not be shown here.

The only thing which has to be considered carefully is that the modulus |P’j| of the

polarization vector of the deflected rays is different from the modulus |Pj| of the incident rays

so that there is a scaling g(ϑ) for each polarization vector:

j

P

'

This is a consequence of energy conservation. To calculate the factor g it has to be taken into

account that the rays in the exit pupil are no longer equidistant and therefore the amplitude A’

(which is proportional to the modulus of the electric field) associated with each plane wave

component along a ray is given by

' / ''

dFA

P

=

exit pupil which each ray covers in the numerical sampling. In the plane entrance pupil the

surface area elements dF are equal for all rays since the rays were sampled equidistant.

However, in the exit pupil, which is for an aplanatic lens a sphere around the focus, the

surface area elements change geometrically by

of the plane wave component in the exit pupil is connected to the amplitude A of the plane

wave component in the entrance pupil by the well-known factor [1]

the factor g is in total:

( )

g

( )

ϑ

j

g

P

=

(9)

. Here, dF’ is the surface area element in the

ϑ

cos/'

dFdF =

(see Fig. 9). The amplitude A’

ϑ

cos'

AA =

. Therefore,

ϑ

ϑ

cos

1

=

(10)

Finally, the electric field in the focus is calculated by:

( )

r

∑

j

⎟⎠

⎞

⎜⎝

⎛

+⋅=

jjj total

ii

OPD

2

λ

'

2

λ

exp'

ππ

α

rePE

(11)

The parameter α is a constant of proportionality which can be set to 1 if we are only interested

in relative units of E. Here, the summation is done over all rays/plane wave components in the

exit pupil and as mentioned before for a plane wave without aberrations the optical path

lengths OPDj are zero. However, also aberrations of the incident wave can simply be taken

into account with our method. Another advantage is that the aperture shape can be arbitrary

(circular, annular, rectangular, etc.) by just changing the rays which are summed up. If other

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30 Apr 2007 / Vol. 15, No. 9 / OPTICS EXPRESS 5841