## High numerical aperture imaging with different polarization patterns

Optics Express, Vol. 15, Issue 9, pp. 5827-5842 (2007)

http://dx.doi.org/10.1364/OE.15.005827

Acrobat PDF (563 KB)

### 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

## 1. Introduction

*n*˙ sin

*φ*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. sin

*φ*=1.0.

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). [CrossRef]

7. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially
polarized light beam,” Phys. Rev. Lett. **91**, 233901 (2003). [CrossRef] [PubMed]

8. 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). [CrossRef]

## 2. Calculation of the modulation transfer function

*x*,

*y*are the coordinates in the image plane and ν

_{x},ν

_{y}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

*ν*

_{x}=

*ν*

_{y}=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

*ν*

_{x},

*ν*

_{y}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.

*P*(

*x*′,

*y*′) =

*A*(

*x*′,

*y*′)exp(2

*πiW*(

*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 ν

_{cut}with:

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). [CrossRef]

## 3. MTFs for different polarization states and numerical apertures

*D*=32

*λ*/NA and 512x512 samples. So, the lateral sampling distance between two points was just ∆

*x*=

*λ*/(16NA) corresponding to a maximal spatial frequency

*ν*

_{max}=1/(2∆

*x*)=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 ∆

*x*=

*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=sin

*φ*. 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 sin

*φ*, 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 sin

*φ*with the same meaning.

4. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter
spot,” Opt. Commun. **179**, 1–7
(2000). [CrossRef]

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). [CrossRef]

*w*

_{0}=0.95

*r*

_{aperture}(

*r*

_{aperture}= radius of entrance pupil) is taken.

4. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter
spot,” Opt. Commun. **179**, 1–7
(2000). [CrossRef]

*φ*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.

4. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter
spot,” Opt. Commun. **179**, 1–7
(2000). [CrossRef]

7. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially
polarized light beam,” Phys. Rev. Lett. **91**, 233901 (2003). [CrossRef] [PubMed]

*φ*). 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.

*φ*and therefore for linear polarization we will always display a section of the MTF with spatial frequency

*ν*

_{x}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

*ν*

_{y}in y-direction, i.e. the grating lines are here in x-direction perpendicular to the direction of linear polarization.

*φ*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*=

*ν*

_{x}(signated with “x-section”) and for

*ν*=

*ν*

_{y}(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).

*φ*=1.0. Three cases were considered.

- Annular apertures with different ratios
*r*_{in}/*r*_{aperture}and homogeneous amplitude in the entrance pupil, i.e. the electric field in the entrance pupil is of the form: - A smoothly varying electric field in the full circular entrance pupil which is represented by the equation

*w*

_{0}were simulated. (iia) The waist parameter of the Gaussian function is assumed to be constant with

*w*

_{0}=0.95

*r*

_{aperture}. (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*≠0 really a maximum of |

*E*̱| at the position of

*w*

_{0}(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(-

*r*

^{2}/

*w*

_{0}

^{2}).

*r*

_{in}/

*r*

_{aperture}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

*φ*and different polarization states. It can easily be seen that for the small value sin

*φ*=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 sin

*φ*. Therefore, for sin

*φ*=0.2 there is nearly no light power in the z-component. For increasing values sin

*φ*, 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.

- 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 sin
*φ*≥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 the best solution. But, if there are also structures in the perpendicular direction there is a zero of the MTF for linear polarization and sin*φ*>0.7! - 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 sin
*φ*, but only for high values sin*φ*there is also a high amount of the light power in this component since it is proportional to approximately sin*φ*^{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 sin*φ*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

*φ*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 sin

*φ*=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.

*φ*=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.

*φ*=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 sin

*φ*≥0.8.

*φ*an also increasing difference between the contrast for structures with spatial frequencies in x- (red solid lines) and y-direction (red dashed lines). For sin

*φ*=1.0 the contrast for spatial frequencies in x-direction can approach the very high value of about 0.175 for

*ν*

_{x}=1.88 NA/

*λ*! On the other side, for spatial frequencies in y-direction with the same modulus, i.e.

*ν*

_{y}=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

*φ*=1.0 (Figs. 3–6).

*r*

_{in}=0) and the curves with

*r*

_{in}/

*r*

_{aperture}<0.4, especially if all components of the electric field are detected. For increasing values

*r*

_{in}/

*r*

_{aperture}the contrast for small and medium spatial frequencies decreases whereas the 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

*r*

_{in}/

*r*

_{aperture}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

*r*

_{in}/

*r*

_{aperture}≤0.3.

*w*

_{0}=0.95

*r*

_{aperture}is taken.

*E*̱| =

*r*

^{3}exp(-

*r*

^{2}/

*w*

^{2}

_{0}) and

*w*

_{0}=0.95

*r*

_{aperture}or

*w*

_{0}=0.82

*r*

_{aperture}(i.e. maximum at the rim of the aperture) or |

*E*̱| =

*r*

^{4}exp(-

*r*

^{2}/

*w*

^{2}

_{0}) and

*w*

_{0}=0.71

*r*

_{aperture}(i.e. also maximum at the rim of the aperture) are compared to the annular aperture with

*r*

_{in}/

*r*

_{aperture}=0.6.

## 5. Application in lithography and microscopy

**179**, 1–7
(2000). [CrossRef]

_{image}will be demagnified by the same factor compared to the numerical aperture in the object space NA

_{Obj}

*φ*=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 sin

*φ*=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.

*β*=0.2 or 0.25 is illuminated with linearly polarized light. Since the numerical aperture in the object space NA

_{Obj}is in this case according to Eq. (7) smaller than or equal to β (assuming a non-immersion system with NA

_{image}≤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 sin

*φ*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/(

*βν*

_{cut})=2

*λ*/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.

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).

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). [CrossRef] [PubMed]

19. D. C. Flanders, “Submicrometer periodicity gratings
as artificial anisotropic dielectrics,”
Appl. Phys. Lett. **42**, 492–494
(1983). [CrossRef]

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). [CrossRef]

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). [CrossRef]

## 6. Conclusion

*φ*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 sin

*φ*=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.

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

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). [CrossRef]

2. M. Mansuripur, “Distribution of light at and near
the focus of high-numerical-aperture objectives,”
J. Opt. Soc. Am. A **3**, 2086–2093
(1986). [CrossRef]

3. 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). [CrossRef]

**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

**P**

_{j}which is a complex valued vector perpendicular to the direction of propagation

**e**

_{j}of the ray. Its modulus |

**P**

_{j}| 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. OPD

_{j}is the optical path length of the ray in the entrance pupil. It is zero for a real plane wave without aberrations.

*f*around the focus fulfilling the sine condition (see Fig. 9):

**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.

**P**’

_{j}| of the polarization vector of the deflected rays is different from the modulus |

**P**

_{j}| of the incident rays so that there is a scaling

*g*(ϑ) for each polarization vector:

*A*’ (which is proportional to the modulus of the electric field) associated with each plane wave component along a ray is given by

*A*′= |

**P**′|/

*dF*′ . Here,

*dF*’ is the surface area element in the 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

*dF*′=

*dF*/cosϑ (see Fig. 9). The amplitude

*A*’ 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

**253**, 358–379
(1959). [CrossRef]

*A*′=

*A*√cosϑ. Therefore, the factor

*g*is in total:

*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 OPD

_{j}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 focusing elements are taken just the parameter

*g*has to be changed and the new direction vectors

**e**’ and polarization vectors

**P**’ have to be calculated accordingly. For reflection at a parabolic mirror the factor

*g*is for example just constant and equal to 1 because by reflection the cross-section and amplitude of a plane wave component is not changed.

## Acknowledgments

## 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 |

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

3. | M. Mansuripur, “Distribution of light at and near
the focus of high-numerical-aperture objectives:
erratum,” J. Opt. Soc. Am. A |

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

5. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical
calculation and experimental tomographic
reconstruction,” Appl. Phys. B |

6. | R. Dorn, S. Quabis, and G. Leuchs, “The focus of light - linear
polarization breaks the rotational symmetry of the focal
spot,” J. Mod. Opt. |

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

8. | S. F. Pereira and A. S. van de Nes, “Superresolution by means of
polarization, phase and amplitude pupil masks,”
Opt. Commun. |

9. | 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 |

10. | R. Oldenbourg and P. Török, “Point-spread functions of a
polarizing microscope equipped with high-numerical-aperture
lenses,” Appl. Opt. |

11. | P. R. T. Munro and P. Török, “Vectorial, high numerical aperture
study of Nomarski’s differential interference contrast
microscope,” Opt. Express |

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

13. | M. Born and E. Wolf, |

14. | J. W. Goodman, |

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 |

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

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. |

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

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. |

21. | Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized
beams generated by space-variant dielectric subwavelength
gratings,” Opt. Lett. |

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. |

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

24. | U. Levy, C. Tsai, L. Pang, and Y. Fainman, “Engineering space-variant
inhomogeneous media for polarization control,”
Opt. Lett. |

25. | C. Tsai, U. Levy, L. Pang, and Y. Fainman, “Form-birefringent space-variant
inhomogeneous medium element for shaping point-spread
functions,” Appl. Opt. |

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. |

**OCIS Codes**

(110.4100) Imaging systems : Modulation transfer function

(220.1230) Optical design and fabrication : Apodization

(220.3740) Optical design and fabrication : Lithography

(260.5430) Physical optics : Polarization

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 2, 2007

Revised Manuscript: April 25, 2007

Manuscript Accepted: April 25, 2007

Published: April 27, 2007

**Citation**

N. Lindlein, S. Quabis, U. Peschel, and G. Leuchs, "High numerical aperture imaging with different polarization patterns," Opt. Express **15**, 5827-5842 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5827

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### References

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