High numerical aperture imaging with different ... - OSA Publishing

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 [email protected]

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. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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). R. Oldenbourg and P. Török, “Point-spread functions of a polarizing microscope equipped with highnumerical-aperture lenses,” Appl. Opt. 39, 6325-6331 (2000). 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). C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354-1360 (1987). M. Born and E. Wolf, Principles of Optics, 6th Edition. (Cambridge University Press, Cambridge New York Oakleigh, 1997). J. W. Goodman, Introduction to Fourier optics, 2nd. Edition (McGraw--Hill, New York, 1996). 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). K. Kamon, “Projection exposure apparatus,” United States Patent 5365371 (filed 1993).

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Received 2 Mar 2007; revised 25 Apr 2007; accepted 25 Apr 2007; published 27 Apr 2007

30 Apr 2007 / Vol. 15, No. 9 / OPTICS EXPRESS 5827

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18. 19. 20. 21. 22. 23. 24. 25. 26.

K.-H. Schuster, “Radial polarisationsdrehende optische Anordnung und MikrolithographieProjektionsbelichtungsanlage 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). M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948-1950 (1996). D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492-494 (1983). 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). 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). 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). 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). U. Levy, C. Tsai, L. Pang, and Y. Fainman, “Engineering space-variant inhomogeneous media for polarization control,” Opt. Lett. 29, 1718-1720 (2004). 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). 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 DUVlithography 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: +∞+∞

MTF(ν x ,ν y ) =

∫ ∫ PSF

(x, y )exp(2π i (ν x x +ν y y ))dxdy

−∞−∞

+∞ +∞

∫−∞ ∫−∞ PSF

(1)

(x, y )dxdy

Here, x,y are the coordinates in the image plane and νx,νy are spatial frequencies in x- and ydirection. 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 function of the pupil function P( x' , y') = A( x' , y ') exp(2π iW (x' , y ') / λ ) , 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: ν cut = 2

NA

λ

=2

n sin ϕ

λ

(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). #80619 - $15.00 USD

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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 wellsampled 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 nonimmersion 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 zcomponent 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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E ( x, y ) =

⎛x⎞ ⎜ ⎟ ⎛ E 0 ⎜ y ⎟ exp⎜⎜ − ⎜ ⎟ ⎝ ⎜ ⎟ ⎝0⎠

x2 + y2 w02

⎞ ⎟ ⎟ ⎠

(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 nonrotationally 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 xdirection, 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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1

1

0.2

0.3

0.4

MTF 0.5

0.6

0.7

0.8

0.9

sinϕ=0.7

0.1

0.1

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MTF 0.5

0.6

0.7

0.8

0.9

sinϕ=0.2

1.8

2

0 1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/λ

1

0

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sinϕ=0.9

0.1

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MTF 0.5

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0.7

0.8

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sinϕ=0.8

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/λ

1

0

1.8

2

0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

1.8

2

0.8

0.9

sinϕ=1.0

0.1

0.2

0.3

0.4

MTF 0.5

0.6

0.7

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

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/λ

1.8

2

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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1

1

0.2

0.3

0.4

MTF 0.5

0.6

0.7

0.8

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sinϕ=0.7

0.1

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sinϕ=0.2

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2

0 1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/λ

1

0

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2

0.2

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sinϕ=0.9

0.1

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sinϕ=0.8

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/λ

1

0

1.8

2

0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

1.8

2

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sinϕ=1.0

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MTF 0.5

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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

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/λ

1.8

2

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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1 0.9

z-component only

sinϕ=1.0 rin=0 rin=0.1 raperture rin=0.2 raperture rin=0.3 raperture rin=0.4 raperture rin=0.5 raperture rin=0.6 raperture rin=0.7 raperture rin=0.8 raperture rin=0.9 raperture

0.6

0.7

0.8

rin =0 rin =0.1 raperture rin =0.2 raperture rin =0.3 raperture rin =0.4 raperture rin =0.5 raperture rin =0.6 raperture rin =0.7 raperture rin =0.8 raperture rin =0.9 raperture

0.3 0.2 0.1

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1

sinϕ=1.0

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all components

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

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0

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0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

1.8

2

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: ⎧ ⎪ E0 ⎪ E ( x, y ) = ⎨ x 2 + y 2 ⎪ ⎪ 0 ⎩

⎛ x⎞ ⎜ ⎟ ⎜ y ⎟ for r ≤ in ⎜ ⎟ ⎜0⎟ ⎝ ⎠

x 2 + y 2 ≤ raperture

(4)

otherwise

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

(

E ( x, y ) = E ' 0 x + y 2

2

⎛ x⎞ ⎟ ⎛ y ⎟ exp⎜⎜ − ⎜ ⎟ ⎝ ⎜ ⎟ ⎝0⎠

n −1 ⎜ 2 ⎜

)

(x

2

+ y2 w02

) ⎞⎟ ⇒ E (r ) = E' ⎟ ⎠

⎛

0

r n exp⎜⎜ − ⎝

r2 ⎞ 2 2 (5) ⎟ and r = x + y w02 ⎟⎠

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:

⎛⎜ ⎜⎝

⎞⎟ ⎟⎠

⎛⎜ ⎜⎝

⎞⎟ ⎟⎠

d E (r ) r n+1 r2 = nr n−1 − 2 2 exp − 2 = 0 w0 w0 dr

⇒

r =raperture

w0 =

2 raperture n

(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/w02). #80619 - $15.00 USD

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MTF 0.5

0.6

1

sinϕ=1.0

0.8

0.9

exp(-r2/w02 ) 2 2 r exp(-r /w0 ) 2 2 2 r exp(-r /w0 ) 3 2 2 r exp(-r /w0 ) 4 2 2 r exp(-r /w0 ) 5 2 2 r exp(-r /w0 ) 6 2 2 r exp(-r /w0 ) r7 exp(-r2 /w02 ) 8 2 r exp(-r /w02 ) 9 2 2 r exp(-r /w0 )

0.2

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w0=0.95 raperture

0.1

0.1

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z-component only

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exp(-r2 /w02 ) 2 2 r exp(-r /w0 ) 2 2 2 r exp(-r /w0 ) 3 2 2 r exp(-r /w0 ) 4 2 2 r exp(-r /w0 ) 5 2 2 r exp(-r /w0 ) 6 2 2 r exp(-r /w0 ) r7 exp(-r2 /w02 ) 8 2 r exp(-r /w02 ) 9 2 2 r exp(-r /w0 )

0.6

sinϕ=1.0

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1

all components

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

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r exp(-r /w 2 2 r exp(-r /w 3 2 r exp(-r /w 4 2 r exp(-r /w 5 2 r exp(-r /w 6 2 r exp(-r /w 7 r exp(-r2 /w r8 exp(-r2 /w 9 2 r exp(-r /w

1 ) ) ) ) ) ) ) ) )

sinϕ=1.0 2

2

r exp(-r /w0 ) 2 2 2 r exp(-r /w0 ) 3 2 2 r exp(-r /w0 ) 4 2 2 r exp(-r /w0 ) 5 2 2 r exp(-r /w0 ) 6 2 2 r exp(-r /w0 ) 7 2 r exp(-r /w02 ) r8 exp(-r2 /w02 ) 9 2 2 r exp(-r /w0 )

0.2

0.3

0.4

w0=(2/n)1/2 raperture

0.1

0.1

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z-component only

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2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0

0.8

2

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sinϕ=1.0

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all components

MTF 0.5

1

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.

0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

1.8

2

0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

1.8

2

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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2

2

1

1/2

r exp(-r /w0 ) and w0=(2/3) raperture=0.82 raperture

r3 exp(-r2 /w02 ) and w0=0.95 raperture 3

2

2

1/2

r exp(-r /w0 ) and w0=(2/3)

raperture=0.82 raperture

r4 exp(-r2 /w02 ) and w0=(2/4)1/2 raperture=0.71 raperture

0.6 MTF 0.5 0.4 0.3 0.2 0.1

0.1

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0.3

0.4

MTF 0.5

0.6

0.7

r4 exp(-r2 /w02 ) and w0=(2/4)1/2 raperture=0.71 raperture

sinϕ=1.0

rin=0.6 raperture

0.8

0.8

r3 exp(-r2 /w02 ) and w0=0.95 raperture 3

z-component only

0.9

sinϕ=1.0

rin=0.6 raperture

0.7

1 0.9

all components

0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

1.8

2

0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ν in NA/ λ

1.8

2

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 xand 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 xdirection) 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 ydirection (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>1, the numerical aperture NAimage will be demagnified by the same factor compared to the numerical aperture in the object space NAObj NA image =

1

β

NA Obj