- Email: [email protected]
Phase filter design for sharper focus of radially polarized beam
G Model IJLEO-54428; No. of Pages 3
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Phase filter design for sharper focus of radially polarized beam Jinsong Li ∗ , Ke Feng, Ling Guo College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China
a r t i c l e
i n f o
Article history: Received 19 July 2013 Accepted 11 January 2014 Available online xxx Keywords: Phase filter Super resolution Radially polarized beam Vector diffraction theory
a b s t r a c t Based on vector diffraction theory, a phase-only filter with cosine function is proposed and a sharper focal spot of radially polarized beam is obtained on the focal plane. The function parameters of the filter are optimized and a series of optimized filters for different Strehl ratio S are given. The optimization results show that there is a significant improvement on obtaining sharper focal spot with this new type of filter. Compared with other results, this type of filters has superior supperresolution effect and higher energy utilization ratio. The numerical calculation results about the phase filter may facilitate new approaches to get superresolution. © 2014 Published by Elsevier GmbH.
1. Introduction Vector polarized beams are solutions of Maxwell’s equations with spatially nonuniform polarization and they are cylindrically symmetry in both polarization and amplitude. As a kind of vector beam, radially polarized beam (RPB) has attracted much attention in recent years for its unique properties [1–12]. A small spot size can be obtained due to the strong longitudinal electric field when RPB is focusing. The capability of super resolution with RPB has been studied both theoretically and experimentally [1–8]. It may have applications in surface plasmon-polariton excitation [8], data storage [9], particle acceleration [10], laser scanning microscopy [11], etc. To sharpen the focal spot of RPB and achieve better super resolution performance, several kinds of methods are proposed including adopting the pupil filters. Different types of filters have been proposed to alter the amplitude, phase or polarization of RPB for a shaper focal spot. R. Dorn and S. Quabis adopted an annular aperture to achieve a spot size of 0.161 2 for RPB in the focal plane, which is much smaller than that for linear polarized beam 0.26 2 [12]. Kozawa and Sato [13] investigated the focusing property of RPB with a double-ring-shaped pattern and a smaller sharper focal spot can be obtained by changing the ratio of the pupil radius to the beam radius. Caballero et al. [14] utilized a shaded mask filter to improve the resolution and reduced the sidelobes’ intensity as well. Tan et al. [15] optimized a 0, two-phase distributed diffractive superresolution elements and designed a series of diffractive
superresolution elements with different super resolution performances. To our knowledge, most of the researches for a sharper focal spot of RPB are based on the amplitude pupil filter or incontinuous phase pupil filter. In this paper, a kind of continuous phase-only pupil filter with cosine function is designed for RPB and optimized with Matlab optimization toolbox. This type of filter can keep high energy utilization and realize a good super resolution performance. The theory of the focusing radially polarized beam is given in Section 2. Section 3 shows the simulation results and discussions. The conclusions are summarized in Section 4. 2. Theory High numerical aperture (NA) focusing of radially polarized beam is analyzed based on vectorial diffraction theory. The electric field in focal region of focusing RPB in cylindrical coordinates is in the form [1] ៝ ϕ, z) = Er e៝ r + Ez e៝ z E(r,
(1)
where e៝ r and e៝ z are the unit vectors in the radial and propagating directions, respectively. ER and EZ are amplitudes of the two orthogonal components and can be expressed as [1]
˛
ER (r, z) = A
U() ·
cos sin(2) · J1 (kr sin ) exp(ikz cos )d
0
(2)
∗ Corresponding author. Tel.: +86 571 86835769. E-mail address: [email protected] (J. Li).
EZ (r, z) = 2iA
˛
U() · 0
cos sin2 () · J0 (kr sin ) exp(ikz cos )d (3)
http://dx.doi.org/10.1016/j.ijleo.2014.01.132 0030-4026/© 2014 Published by Elsevier GmbH.
Please cite this article in press as: J. Li, et al., Phase filter design for sharper focus of radially polarized beam, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.132
G Model
ARTICLE IN PRESS
IJLEO-54428; No. of Pages 3
J. Li et al. / Optik xxx (2014) xxx–xxx
2
7 Filter1 Filter2 Filter3 Filter4
5
Normalized intensity
Phase of filters
6
No filter Filter3 Filter4 Filter2 Filter1
0.9
4 3 2
0.8 0.7 0.6 0.5 0.4 0.3 0.2
1
0.1
0 0
0.2
0.4
0.6
-4
0.8
-2
sin
No filter Filter4 (a) Filter3 Fllter2 Filter1
0.7 0.6
(b)
ϕ() = (a cosn (b))
(7)
where n can be altered according to different cases. 3. Simulation results and discussion
0.5 0.4 0.3 0.2 0.1 -2
-1
0 Radius
1
2
Fig. 2. Normalized intensity in the focal plane for RPB with the four optimized filters.
where r and z are the radial and longitudinal coordinates of observation point in focal region, respectively. Parameter A is a constant, and k is wave number. Parameter represents the polar angle corresponding to the optical aperture, and ˛ = arcsin(NA). U() is the transmittance function of the diffractive superresolution elements. The intensity in the focal region of RPB of high numerical aperture focusing RPB can be expressed as [1]
4
by phase function. Here a continuous phase-only filter with cosine function is proposed and the phase function is written as
2
2
I(r, z) = ER (r, z) + EZ (r, z)
(5)
Here we design U() to achieve the sharper focus of RPB. Generally, U() can be written in the form: U() = T () exp[iϕ()],
The goal of superresolution technology is to control the focal volume smaller than the Airy disk. Several parameters are defined to evaluate the superresolution performance. We consider two factors, one is super resolution performance and the other is energy utilization. Here the super resolution capability is evaluated by the ratio G between the FWHM (full width at half maximum intensity) of RPB’s super resolution intensity distribution with filter and the FWHM of intensity distribution without filter. A smaller G means better super resolution capability. The ratio of the central intensities between the intensity distribution of RPB with filter and the direct focusing of RPB without filter is the Strehl ratio S. A larger S means the higher energy utilization. Therefore, the aim of optimizing filters is to reduce G to achieve better super resolution performance as well as obtain the highest Strehl ratio S (Fig. 1). Normalized intensity in the focal plane with the four optimized filters is shown in Fig. 2. It can be seen that the spot size for RPB with filter is smaller as compared to the spot size without filter. As the super resolution performance improves, the Strehl ratio S decreases. The parameters corresponding the four optimized filters are S = 0.7011, G = 0.833; S = 0.6227, G = 0.800; S = 0.5007, G = 0.677; S = 0.4000, G = 0.540, respectively.
(6)
where T() is the transmittance function of amplitude. For a phaseonly filter, T() = 1, the super resolution capability is determined
NA=0.8 NA=0.95
0.9
Normalized intensity
Normalized intensity
0.8
2
Fig. 4. The axial normalized intensity for RPB with the four filters.
Fig. 1. Phase of filter.
0.9
0 Axial distance
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -2
Fig. 3. Normalized intensity in the focal plane for RPB (a) without filter and (b) with filter 4.
-1
0 Radius
1
2
Fig. 5. Normalized intensity in the focal plane for RPB without filter in the case of NA = 0.95 and NA = 0.8.
Please cite this article in press as: J. Li, et al., Phase filter design for sharper focus of radially polarized beam, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.132
G Model
ARTICLE IN PRESS
IJLEO-54428; No. of Pages 3
J. Li et al. / Optik xxx (2014) xxx–xxx 1 No filter
Normalized intensity
0.9
Filter 1 for NA=0.95,n=2
0.8
Filter 2 for NA=0.95,n=2
0.7 0.6 0.5 0.4
G=0.724, S=0.6
G=0.713, S=0.4
0.3 0.2 0.1 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Radius Fig. 6. Normalized intensity in the focal plane for NA = 0.95, n = 2 with the two filters. Table 1 Results of optimization. Filter (n = 2)
a
b
G
S
uF
1 2 3 4
1.1797 1.3027 1.8291 1.9880
2.3766 2.1727 2.3191 2.0672
0.833 0.800 0.677 0.540
0.7011 0.6227 0.5007 0.4000
−0.5758 −1.0606 −0.6970 −1.8485
No filter Annular aperture Filter for n=2 Filter for n=3
Normalized intensity
0.9 0.8 0.7
3
other is for n = 3) are given to contrast to the annular aperture. Here three filters are all under the condition that the Strehl ratio S is 0.6. From Fig. 7, it can be seen that the supper resolution capability of the filters with cosine function is better than the annular aperture with a constant outside radius r which is the radius of optical aperture, the inner radius of annular portion is rinner which can be altered from 0 to a, the parameter of annular aperture = rinner /r. And the filter with cosine function for n = 3 performs a little better than the one for n = 2. The parameters of the filter with cosine function for n = 2 are a = 1.2768, b = 1.7251, G = 0.494, S = 0.6; respectively and the parameters of the filter with cosine function for n = 3 are a = 1.5585, b = 1.6799, G = 0.4817, S = 0.6 respectively, while the parameter of annular aperture = (rinner /r) = 0.23. 4. Conclusion Phase-only filter with cosine function is proposed. It is optimized with Matlab optimization toolbox and a sharper focused spot of radially polarized beam is obtained in the focal plane. The simulated results show that there is a significant improvement on sharper focus with this new type filter. This kind of filter can be improved further since the parameter n can be optimized according to different cases. We hope the numerical calculation results about the continuous phase filter with cosine function will help the manufacture. Acknowledgment This work was financially supported by National Natural Science Foundation of China (61108005).
0.6 0.5 0.4
References
0.3 0.2 0.1 0
0.5
1
1.5
Radius Fig. 7. Normalized intensity in the focal plane for NA = 0.95 with two types of filters.
To show effect of the filter directly, normalized intensity in the focal plane without filter is shown in Fig. 3(a) while normalized intensity in the focal plane with filter 4 is shown in Fig. 3(b). It can be seen that focal spot size with filter is smaller than the focal spot size without filter, however the intensity and region of sidelobes are increasing. Fig. 4 shows the axial normalized intensity with the four filters and it can be seen that the focal plane moves toward the pupil. It is well known that if NA is increasing, the focal spot for RPB will become smaller even without filter because of the stronger longitudinal component as shown in Fig. 5. Under this condition, using the same method and increasing NA from 0.8 to 0.95, we simulated the results for n = 2 to see whether this type of filter can further sharpen the focal spot. Fig. 6 demonstrated that this type of filter with cosine function worked well on condition that NA = 0.95. Two optimized filters are presented here. The parameters of Filter 1 and filter 2 for NA = 0.95, n = 2 are a = 1.2768, b = 1.7251 G = 0.724, S = 0.6000; a = 1.9263, b = 1.7679, G = 0.713, S = 0.4000, respectively (Table 1). When NA = 0.95, to show the performance of the filter, two continuous phase filters with cosine function (one is for n = 2 and the
[1] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (77–87) (2000) 7. [2] S. Quabis, R. Dorn, M. Eberler, et al., Focusing light to a tighter spot, Opt. Commun. 179 (2000) 1–7. [3] S. Quabis, R. Dorn, M. Eberler, et al., The focus of light—theoretical calculation and experimental tomographic reconstruction, Appl. Phys. B 72 (109–113) (2001) 9. [4] C.C. Sun, Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation, Opt. Lett. 28 (2003) 99–101. [5] E.Y.S. Yew, C.J.R. Sheppard, Tight focusing of radially polarized Gaussian and Bessel–Gauss beams, Opt. Lett. 32 (2007) 3417–3419. [6] C.J.R. Sheppard, E.Y.S. Yew, Performance parameters for focusing of radial polarization, Opt. Lett. 33 (2008) 497–499. [7] C.J.R. Sheppard, A. Choudhury, Annular pupils, radial polarization, and super resolution, Appl. Opt. 43 (2004) 4322–4327. [8] N.M. Mojarad, M. Agio, Tailoring the excitation of localized surface plasmonpolariton resonances by focusing radially-polarized beams, Opt. Express 17 (2009) 117–122. [9] Y.J. Zhang, J.P. Bai, Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams, Opt. Express 17 (2009) 3698–3706. [10] Y.I. Salamin, Acceleration in vacuum of bare nuclei by tightly focused radially polarized laser light, Opt. Lett. 32 (2007) 3462–3464. [11] Y. Kozawa, T. Hibi, A. Sato, et al., Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam, Opt. Express 19 (2011) 15947–15954. [12] R. Dorn, S. Quabis, G. Leuchs, Sharper focus for a radially polarized light beam, Phys. Rev. Lett. 91 (1–4) (2003) 233901. [13] Y. Kozawa, S. Sato, Focusing property of a double-ringshaped radially polarized beam, Opt. Lett. 31 (2006) 820–822. [14] M.T. Caballero, C. Ibanez-Lopez, M. Martinez-Corral, Shaded-mask filtering: novel strategy for improvement of resolution in radial-polarization scanning microscopy, Opt. Eng. 45 (2006) 098003. [15] Q. Tan, K. Cheng, Z. Zhou, et al., Diffractive super resolution elements for radially polarized light, J. Opt. Soc. Am. A 27 (2010) 1355–1360.
Please cite this article in press as: J. Li, et al., Phase filter design for sharper focus of radially polarized beam, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.132
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Phase filter design for sharper focus of radially polarized beam Jinsong Li ∗ , Ke Feng, Ling Guo College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China
a r t i c l e
i n f o
Article history: Received 19 July 2013 Accepted 11 January 2014 Available online xxx Keywords: Phase filter Super resolution Radially polarized beam Vector diffraction theory
a b s t r a c t Based on vector diffraction theory, a phase-only filter with cosine function is proposed and a sharper focal spot of radially polarized beam is obtained on the focal plane. The function parameters of the filter are optimized and a series of optimized filters for different Strehl ratio S are given. The optimization results show that there is a significant improvement on obtaining sharper focal spot with this new type of filter. Compared with other results, this type of filters has superior supperresolution effect and higher energy utilization ratio. The numerical calculation results about the phase filter may facilitate new approaches to get superresolution. © 2014 Published by Elsevier GmbH.
1. Introduction Vector polarized beams are solutions of Maxwell’s equations with spatially nonuniform polarization and they are cylindrically symmetry in both polarization and amplitude. As a kind of vector beam, radially polarized beam (RPB) has attracted much attention in recent years for its unique properties [1–12]. A small spot size can be obtained due to the strong longitudinal electric field when RPB is focusing. The capability of super resolution with RPB has been studied both theoretically and experimentally [1–8]. It may have applications in surface plasmon-polariton excitation [8], data storage [9], particle acceleration [10], laser scanning microscopy [11], etc. To sharpen the focal spot of RPB and achieve better super resolution performance, several kinds of methods are proposed including adopting the pupil filters. Different types of filters have been proposed to alter the amplitude, phase or polarization of RPB for a shaper focal spot. R. Dorn and S. Quabis adopted an annular aperture to achieve a spot size of 0.161 2 for RPB in the focal plane, which is much smaller than that for linear polarized beam 0.26 2 [12]. Kozawa and Sato [13] investigated the focusing property of RPB with a double-ring-shaped pattern and a smaller sharper focal spot can be obtained by changing the ratio of the pupil radius to the beam radius. Caballero et al. [14] utilized a shaded mask filter to improve the resolution and reduced the sidelobes’ intensity as well. Tan et al. [15] optimized a 0, two-phase distributed diffractive superresolution elements and designed a series of diffractive
superresolution elements with different super resolution performances. To our knowledge, most of the researches for a sharper focal spot of RPB are based on the amplitude pupil filter or incontinuous phase pupil filter. In this paper, a kind of continuous phase-only pupil filter with cosine function is designed for RPB and optimized with Matlab optimization toolbox. This type of filter can keep high energy utilization and realize a good super resolution performance. The theory of the focusing radially polarized beam is given in Section 2. Section 3 shows the simulation results and discussions. The conclusions are summarized in Section 4. 2. Theory High numerical aperture (NA) focusing of radially polarized beam is analyzed based on vectorial diffraction theory. The electric field in focal region of focusing RPB in cylindrical coordinates is in the form [1] ៝ ϕ, z) = Er e៝ r + Ez e៝ z E(r,
(1)
where e៝ r and e៝ z are the unit vectors in the radial and propagating directions, respectively. ER and EZ are amplitudes of the two orthogonal components and can be expressed as [1]
˛
ER (r, z) = A
U() ·
cos sin(2) · J1 (kr sin ) exp(ikz cos )d
0
(2)
∗ Corresponding author. Tel.: +86 571 86835769. E-mail address: [email protected] (J. Li).
EZ (r, z) = 2iA
˛
U() · 0
cos sin2 () · J0 (kr sin ) exp(ikz cos )d (3)
http://dx.doi.org/10.1016/j.ijleo.2014.01.132 0030-4026/© 2014 Published by Elsevier GmbH.
Please cite this article in press as: J. Li, et al., Phase filter design for sharper focus of radially polarized beam, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.132
G Model
ARTICLE IN PRESS
IJLEO-54428; No. of Pages 3
J. Li et al. / Optik xxx (2014) xxx–xxx
2
7 Filter1 Filter2 Filter3 Filter4
5
Normalized intensity
Phase of filters
6
No filter Filter3 Filter4 Filter2 Filter1
0.9
4 3 2
0.8 0.7 0.6 0.5 0.4 0.3 0.2
1
0.1
0 0
0.2
0.4
0.6
-4
0.8
-2
sin
No filter Filter4 (a) Filter3 Fllter2 Filter1
0.7 0.6
(b)
ϕ() = (a cosn (b))
(7)
where n can be altered according to different cases. 3. Simulation results and discussion
0.5 0.4 0.3 0.2 0.1 -2
-1
0 Radius
1
2
Fig. 2. Normalized intensity in the focal plane for RPB with the four optimized filters.
where r and z are the radial and longitudinal coordinates of observation point in focal region, respectively. Parameter A is a constant, and k is wave number. Parameter represents the polar angle corresponding to the optical aperture, and ˛ = arcsin(NA). U() is the transmittance function of the diffractive superresolution elements. The intensity in the focal region of RPB of high numerical aperture focusing RPB can be expressed as [1]
4
by phase function. Here a continuous phase-only filter with cosine function is proposed and the phase function is written as
2
2
I(r, z) = ER (r, z) + EZ (r, z)
(5)
Here we design U() to achieve the sharper focus of RPB. Generally, U() can be written in the form: U() = T () exp[iϕ()],
The goal of superresolution technology is to control the focal volume smaller than the Airy disk. Several parameters are defined to evaluate the superresolution performance. We consider two factors, one is super resolution performance and the other is energy utilization. Here the super resolution capability is evaluated by the ratio G between the FWHM (full width at half maximum intensity) of RPB’s super resolution intensity distribution with filter and the FWHM of intensity distribution without filter. A smaller G means better super resolution capability. The ratio of the central intensities between the intensity distribution of RPB with filter and the direct focusing of RPB without filter is the Strehl ratio S. A larger S means the higher energy utilization. Therefore, the aim of optimizing filters is to reduce G to achieve better super resolution performance as well as obtain the highest Strehl ratio S (Fig. 1). Normalized intensity in the focal plane with the four optimized filters is shown in Fig. 2. It can be seen that the spot size for RPB with filter is smaller as compared to the spot size without filter. As the super resolution performance improves, the Strehl ratio S decreases. The parameters corresponding the four optimized filters are S = 0.7011, G = 0.833; S = 0.6227, G = 0.800; S = 0.5007, G = 0.677; S = 0.4000, G = 0.540, respectively.
(6)
where T() is the transmittance function of amplitude. For a phaseonly filter, T() = 1, the super resolution capability is determined
NA=0.8 NA=0.95
0.9
Normalized intensity
Normalized intensity
0.8
2
Fig. 4. The axial normalized intensity for RPB with the four filters.
Fig. 1. Phase of filter.
0.9
0 Axial distance
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -2
Fig. 3. Normalized intensity in the focal plane for RPB (a) without filter and (b) with filter 4.
-1
0 Radius
1
2
Fig. 5. Normalized intensity in the focal plane for RPB without filter in the case of NA = 0.95 and NA = 0.8.
Please cite this article in press as: J. Li, et al., Phase filter design for sharper focus of radially polarized beam, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.132
G Model
ARTICLE IN PRESS
IJLEO-54428; No. of Pages 3
J. Li et al. / Optik xxx (2014) xxx–xxx 1 No filter
Normalized intensity
0.9
Filter 1 for NA=0.95,n=2
0.8
Filter 2 for NA=0.95,n=2
0.7 0.6 0.5 0.4
G=0.724, S=0.6
G=0.713, S=0.4
0.3 0.2 0.1 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Radius Fig. 6. Normalized intensity in the focal plane for NA = 0.95, n = 2 with the two filters. Table 1 Results of optimization. Filter (n = 2)
a
b
G
S
uF
1 2 3 4
1.1797 1.3027 1.8291 1.9880
2.3766 2.1727 2.3191 2.0672
0.833 0.800 0.677 0.540
0.7011 0.6227 0.5007 0.4000
−0.5758 −1.0606 −0.6970 −1.8485
No filter Annular aperture Filter for n=2 Filter for n=3
Normalized intensity
0.9 0.8 0.7
3
other is for n = 3) are given to contrast to the annular aperture. Here three filters are all under the condition that the Strehl ratio S is 0.6. From Fig. 7, it can be seen that the supper resolution capability of the filters with cosine function is better than the annular aperture with a constant outside radius r which is the radius of optical aperture, the inner radius of annular portion is rinner which can be altered from 0 to a, the parameter of annular aperture = rinner /r. And the filter with cosine function for n = 3 performs a little better than the one for n = 2. The parameters of the filter with cosine function for n = 2 are a = 1.2768, b = 1.7251, G = 0.494, S = 0.6; respectively and the parameters of the filter with cosine function for n = 3 are a = 1.5585, b = 1.6799, G = 0.4817, S = 0.6 respectively, while the parameter of annular aperture = (rinner /r) = 0.23. 4. Conclusion Phase-only filter with cosine function is proposed. It is optimized with Matlab optimization toolbox and a sharper focused spot of radially polarized beam is obtained in the focal plane. The simulated results show that there is a significant improvement on sharper focus with this new type filter. This kind of filter can be improved further since the parameter n can be optimized according to different cases. We hope the numerical calculation results about the continuous phase filter with cosine function will help the manufacture. Acknowledgment This work was financially supported by National Natural Science Foundation of China (61108005).
0.6 0.5 0.4
References
0.3 0.2 0.1 0
0.5
1
1.5
Radius Fig. 7. Normalized intensity in the focal plane for NA = 0.95 with two types of filters.
To show effect of the filter directly, normalized intensity in the focal plane without filter is shown in Fig. 3(a) while normalized intensity in the focal plane with filter 4 is shown in Fig. 3(b). It can be seen that focal spot size with filter is smaller than the focal spot size without filter, however the intensity and region of sidelobes are increasing. Fig. 4 shows the axial normalized intensity with the four filters and it can be seen that the focal plane moves toward the pupil. It is well known that if NA is increasing, the focal spot for RPB will become smaller even without filter because of the stronger longitudinal component as shown in Fig. 5. Under this condition, using the same method and increasing NA from 0.8 to 0.95, we simulated the results for n = 2 to see whether this type of filter can further sharpen the focal spot. Fig. 6 demonstrated that this type of filter with cosine function worked well on condition that NA = 0.95. Two optimized filters are presented here. The parameters of Filter 1 and filter 2 for NA = 0.95, n = 2 are a = 1.2768, b = 1.7251 G = 0.724, S = 0.6000; a = 1.9263, b = 1.7679, G = 0.713, S = 0.4000, respectively (Table 1). When NA = 0.95, to show the performance of the filter, two continuous phase filters with cosine function (one is for n = 2 and the
[1] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (77–87) (2000) 7. [2] S. Quabis, R. Dorn, M. Eberler, et al., Focusing light to a tighter spot, Opt. Commun. 179 (2000) 1–7. [3] S. Quabis, R. Dorn, M. Eberler, et al., The focus of light—theoretical calculation and experimental tomographic reconstruction, Appl. Phys. B 72 (109–113) (2001) 9. [4] C.C. Sun, Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation, Opt. Lett. 28 (2003) 99–101. [5] E.Y.S. Yew, C.J.R. Sheppard, Tight focusing of radially polarized Gaussian and Bessel–Gauss beams, Opt. Lett. 32 (2007) 3417–3419. [6] C.J.R. Sheppard, E.Y.S. Yew, Performance parameters for focusing of radial polarization, Opt. Lett. 33 (2008) 497–499. [7] C.J.R. Sheppard, A. Choudhury, Annular pupils, radial polarization, and super resolution, Appl. Opt. 43 (2004) 4322–4327. [8] N.M. Mojarad, M. Agio, Tailoring the excitation of localized surface plasmonpolariton resonances by focusing radially-polarized beams, Opt. Express 17 (2009) 117–122. [9] Y.J. Zhang, J.P. Bai, Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams, Opt. Express 17 (2009) 3698–3706. [10] Y.I. Salamin, Acceleration in vacuum of bare nuclei by tightly focused radially polarized laser light, Opt. Lett. 32 (2007) 3462–3464. [11] Y. Kozawa, T. Hibi, A. Sato, et al., Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam, Opt. Express 19 (2011) 15947–15954. [12] R. Dorn, S. Quabis, G. Leuchs, Sharper focus for a radially polarized light beam, Phys. Rev. Lett. 91 (1–4) (2003) 233901. [13] Y. Kozawa, S. Sato, Focusing property of a double-ringshaped radially polarized beam, Opt. Lett. 31 (2006) 820–822. [14] M.T. Caballero, C. Ibanez-Lopez, M. Martinez-Corral, Shaded-mask filtering: novel strategy for improvement of resolution in radial-polarization scanning microscopy, Opt. Eng. 45 (2006) 098003. [15] Q. Tan, K. Cheng, Z. Zhou, et al., Diffractive super resolution elements for radially polarized light, J. Opt. Soc. Am. A 27 (2010) 1355–1360.
Please cite this article in press as: J. Li, et al., Phase filter design for sharper focus of radially polarized beam, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.132