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Three-dimensional focus shaping with cylindrical vector beams
Optics Communications 265 (2006) 411–417 www.elsevier.com/locate/optcom
Three-dimensional focus shaping with cylindrical vector beams Weibin Chen *, Qiwen Zhan Electro-Optics Graduate Program, University of Dayton, 300 College Park, Dayton, OH 45469-0245, USA Received 2 December 2005; received in revised form 5 April 2006; accepted 5 April 2006
Abstract A three-dimensional focus shaping technique using the combination of cylindrical polarization with binary diffractive optical element is proposed. The energy density pattern at the vicinity of the focus can be tailored in three dimensions by appropriately adjusting the parameters of the cylindrical vector beam illumination, numerical aperture of the objective lens and the design of the binary diffractive optical element. Focus with extended depth of focus that has both transversal and longitudinal flattop profile is obtained. Optical bubble that has a total dark volume surrounded by high field distributions is also shown. Potential applications of this focus shaping technique are discussed. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Focus shaping; Flattop focusing; Depth of focus; Optical bubble; Cylindrical vector beams; Polarization
1. Introduction In general, the polarization of light can be classified into two categories. The first type has spatially homogeneous distribution in terms of the state of polarization for the light wavefront. This type of polarization, including linear, circular and elliptical polarization, is most familiar to the optical community. Contrarily, the second type has spatially inhomogeneous polarization distribution. Traditionally, the second type of polarization was considered as a nuisance or aberrations in the optical design and has not drawn much research attention. However, there is an increasing interest in the spatially inhomogeneous polarization recently, mostly driven by the advances made in micro-fabrication techniques and theoretical modeling techniques that were not available previously. One example of such spatially inhomogeneous polarization that has attracted much of the interest is the so-called cylindrical vector (CV) beams. Cylindrical vector beams are solutions of Maxwell equations that obey cylindrical symmetry both in amplitude and
*
Corresponding author. E-mail address: [email protected] (W. Chen).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.04.066
polarization. Cylindrical vector beams can be divided into radial polarization, azimuthal polarization and generalized cylindrical polarization, according to the actual polarization pattern. One can use two cascaded half-wave plates to conveniently convert a radial polarization or azimuthal polarization into a generalized cylindrical vector beam, or vice versa [1]. Recently, there is a strong interest in these beams due to their peculiar properties, especially when focused under high numerical aperture (NA) conditions. Many techniques to generate radially polarized beams or azimuthally polarized beams have been reported, including interferometric techniques for converting a linearly polarized Gaussian beam into a radially polarized doughnut beam [2], the interference of two opposite hand circularly polarized Laguerre–Gaussian (LG) beams of opposite topological charge [3], using computer-generated spatially variant dielectric or metal strip subwavelength grating [4], or inserting other special designed elements into laser resonators [5,6], or using space-variant inhomogeneous media [7]. Owing to their unique properties, cylindrical vector beams may find wide applications in physics, chemistry and biology, such as particle guiding or trapping [8–12], scanning optical microscopy [13], lithography [14], laser cutting of metals [15,16], particle acceleration [17,18], and single molecule imaging [19,20].
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The propagation and focusing properties of spatially inhomogeneous polarization remain of continued interest. When focused with a high NA objective, radially polarized light leads to a strong longitudinal electric field component in the vicinity of the focus. The relative contribution of longitudinal component can be enhanced by using annular aperture. Theoretical and experimental investigations of focusing an incoming radially polarized beam into an ultrasmall spot have been reported [21–23]. The spot size can be much smaller than the diffraction limit of focused spatially homogeneously polarized beams. Besides an ultrasmall focus, it has been shown that focal distribution with extended depth of focus (DOF) can be achieved with inserting a simple diffractive optical element (DOE) in the optical path under radial polarization illumination [21]. Furthermore, using a very simple two-half-wave-plate polarization rotator, one can conveniently convert radial polarization into a generalized cylindrical vector polarization that can be used to generate focal field distribution with flattop transversal profile. In this paper, we propose to combine the techniques from [1,21] to investigate a three-dimensional focus shaping technique that uses a DOE modulated cylindrical vector beam illumination. In Section 2, we present the mathematical model of cylindrical vector beams and three-dimensional focal field distributions when these beams are focused by a high numerical objective lens with the insertion of a binary DOE. In Section 3, we show how to obtain a maximally homogenized focal field with a significantly extended DOF. This focal field has flattop profiles in both the transversal and longitudinal direction. The relationship between flattop focusing, depth of focus, sidelobe and the design of DOE, rotation angle of cylindrical vector beams is also discussed. In Section 4, the generation of an optical bubble with completely dark center surrounded by high field strength will be shown. Potential applications of this three-dimensional focus shaping technique will be proposed and discussed. 2. Proposed setup and modeling
or an azimuthal polarization, each point of the beam has a polarization rotated by U0 from its radial direction. The electric field of this beam can be expressed in a cylindrical coordinate system as ~ Eðr; uÞ ¼ LðrÞ½cos /0~ er þ sin /0~ eu
ð1Þ
where ~ er is the unit vector in the radial direction and ~ eu is the unit vector in the azimuthal direction. L(r) is the pupil function denoting the relative amplitude of the field that only depends on radial position. Thus, a generalized cylindrical vector beam can be decomposed into a radial polarization and an azimuthal polarization. The setup for the focus shaping is shown in Fig. 2. The incident light is a generalized cylindrical vector beam. An objective lens with high numerical aperture is used to produce a spherical wave converging to the focus. A binary DOE with three concentric regions (shown in Fig. 3) is placed in front of the lens to modulate the wavefront of the generalized cylindrical vector beams. Due to the symmetry and normal illumination, the phase modulation is identical for the radial and azimuthal polarization components. The numerical apertures that correspond to the outer edge of the three regions are NA1, NA2 and NA respectively, where the local transmittances are T1 = 1, T2 = 1 and T3 = 1. Thus the field distribution at the focus is given by E ¼ E1 E 2 þ E 3
ð2Þ
where E1, E2 and E3 are the focal field contribution from each annular zone that can be calculated as the following. The electric fields in the vicinity of the focal spot for radial and azimuthal polarization can be calculated by Richards–Wolf vectorial diffraction method [1,24]. The y High NA Lens
Q(r,ϕ) ϕ
Cylindrical vector incident beam
Fig. 1 shows the polarization pattern of a generalized cylindrical vector beam. Instead of a radial polarization
x
θ z
z=0 Focal Plane Binary DOE
f
Fig. 2. Focusing of a generalized cylindrical vector beam with DOE.
Φo
NA NA2
+
Fig. 1. Generalized cylindrical vector beam with rotation from the purely radially polarization.
_
NA1
+
Fig. 3. Binary DOE with three concentric regions.
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focal field of a generalized cylindrical beam contributed from each concentric region of the DOE can be written as ~ Eðr; u; zÞ ¼ Ez~ ez þ Er~ er þ Eu~ eu
ð3Þ
where ~ ez is the unit vector in the z-direction, and Z h2 Ez ðr; u; zÞ ¼ jA cos /0 sin2 hP ðhÞLðhÞJ 0 ðkr sin hÞejkz cos h dh h1
Er ðr; u; zÞ ¼ A cos /0
Z
ð4Þ h2
sin h cos hP ðhÞLðhÞ h1
ð5Þ J 1 ðkr sin hÞejkz cos h dh Z h2 sin hP ðhÞLðhÞJ 1 ðkr sin hÞejkz cos h dh Eu ðr; u; zÞ ¼ A sin /0 h1
ð6Þ where h1 and h2 is the angular transition points determined by the inner and outer transition points of the corresponding concentric region of the binary DOE. For example, for the concentric ring with p phase shift (see Fig. 3), the integration limits are given by h1 = sin1 (NA1) and h2 = sin1 (NA2), respectively. P(h) is the pupil apodization function, k is the wave number and Jn(x) is the Bessel function of the first kind with order n. L(h) is the relative amplitude of the field, which is assumed to be dependent on the radial position only. Due to the continuity requirement, the on-axis electric field of cylindrical polarization is zero. The actual distribution of this hollow center depends on how the cylindrical polarization is generated. For simplicity, the illumination for all the examples in this paper is chosen to be a planar wavefront over the pupil, where ( 1 if sin1 ðNA0 Þ 6 h 6 sin1 ðNAÞ LðhÞ ¼ ð7Þ 0 otherwise where NA0 = 0.2 for our simulations. This means that the center of the lens pupil aperture is blocked to simulate this dark hollow center. Since the focusing properties of cylindrical vector beams are largely due to the high spatial frequency components, the effects on the field distribution by blocking the center are negligible. 3. Three-dimensional flattop focusing with extended depth of focus In this part, we explore the possibility to generate flattop focusing with extended DOF using the setup described above. In Ref. [21], similar binary DOE was proposed to generate ultrasharp focus with extended depth of focus with radially polarized illumination. Essentially, the field contributed from the middle ring destructively interferes with the field contributed from the innermost and outermost ring created a saddle in the focus and elongate the depth of focus. However, it should be possible to adjust the transition points of the middle ring to change the relative interference contribution and ‘‘fill’’ the saddle to create
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a flattop longitudinal field profile. In addition, the focal fields created by radial polarization and azimuthal polarization are mutually orthogonal and spatially separated. This property was exploited to create flattop transverse focus as reported in Ref. [1]. If a generalized cylindrical vector beam is used as illumination along with the binary DOE, one should be able to adjust the relative weighing of the radial and azimuthal components to create flattop profiles both in longitudinal and transversal directions. The design procedure is as the following. From Eqs. (3)– (6), it is clear that the only none-zero contribution for onaxis field distribution comes from the z-component. Thus, relative axial energy density distribution will not change with the rotation angle U0 because only radial component of the generalized cylindrical vector beams contributes to it. Therefore, during the first step, we set U0 = 0° and change the transition points NA1 and NA2 of the binary DOE to find a flat longitudinal field distribution with long DOF. This is done by a linear direct search through manually adjusting the two transition points with 0.01 increments in the transition points. Once this is achieved, then we obtain flattop transversal field distribution through adjusting U0. For all of the calculations in this paper, the length unit is normalized to wavelength k, and the maximum energy density is normalized to unity. First we choose an aplanatic objective lens that satisfies the sine condition. Under sine condition, the pupil apodization function is P ðhÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi cosðhÞ [25]. With fixed parameters of NA = 0.8 and NA0 = 0.2, we found the parameters for the binary DOE to be NA1 = 0.31, NA2 = 0.5 and the rotation parameter for cylindrical vector beams to be U0 = 38°. Fig. 4(a) shows the calculated result of three-dimensional energy density distribution in the vicinity of the focus. The corresponding axial and transversal energy density distributions are shown in Fig. 4(b) and (c). It can be seen that flattop profiles along both the transversal and axial distribution have been obtained. The depth of focus defined by full width half maximum (FWHM) of the jEj2 along the axial direction is calculated to be 5k. For comparison, the axial field distribution for linearly polarized distribution is also calculated and shown in Fig. 4(b). The depth of focus is calculated to be 2.4k. Clearly, an extended depth of focus is achieved with cylindrical vector polarization. This extended DOF partly is due to the destructive interference from the middle ring of the DOE. In addition, the vector projection used in vectorial diffraction theory gives larger weighing to the higher spatial frequency components in the calculation of Ez, as seen in Eq. (4). This apodization effect also contributes to the longer DOF. Besides the pupil apodization function under sine condition, we also can explore other pupil apodization functions. With NA = 0.8 and NA0 = 0.2, the parameters for the binary DOE are found to be NA1 = 0.35, NA2 = 0.55, and the rotation angle for the cylindrical vector beam is U0 = 39.5°. Fig. 5 shows the calculated results of focal field distribution for an objective lens that obeys the Helmholtz condition,
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(a) 1
1
0.8
0.8
0.6
0.6 2
0.4
|E|
|E|
2
(a)
0.2
0.4 0.2
0 4 2 0
r (λ)
-2 -4
0
-2
-4
2
0 4
4
2
4
z (λ)
2
0
(b)
-2 -4
Axial distribution
-4
z( )
1 Axial distribution
(b) 0.9
1
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0.7
0.8
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2
|E|
2
|E|
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-2
r( )
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0.2
CV beam Linear
0.1 -3
-2
-1
0 z
(c)
0.5
1
CV beam Linear
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2
3
4
0 -4
)
-3
-2
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1
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0 z( )
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2
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2
|E|
2
(c) 0.9
0.5
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0.3
0.1
0.2
0 -4
-1
0.1
-3
-2
-1
0 r( )
1
2
3
4
Fig. 4. Energy density distributions in the vicinity of focus with pupil apodization function under sine condition: (a) 3-D distribution; (b) axial distribution; (c) transversal distribution. CV: cylindrical vector beam.
pffiffiffiffiffiffiffiffiffiffiffiffiffi 3 where P ðhÞ ¼ ½ cosðhÞ [25]. In this case, the depth of focus is calculated to be 5.8k. This even longer DOF is due to the further increase of the higher spatial frequency weighing due to the apodization function. Again, a threedimensional flattop focusing with extended depth of focus is obtained.
0 -4
-3
-2
-1
0 r( )
1
Fig. 5. Energy density distributions in the vicinity of focus with pupil apodization function under the Helmhotlz condition: (a) 3-D distribution; (b) axial distribution; (c) transversal distribution. CV: cylindrical vector beam.
By comparing the above two results, we find that objective lens under the Helmhotlz condition is better for flattop focusing with extended depth of focus. This is because that the input energy at the outer portion of the DOE plays
W. Chen, Q. Zhan / Optics Communications 265 (2006) 411–417
(a) 1
|E|
2
0.8 0.6 0.4 0.2 0 4 2
4 2
0 r (λ)
0
-2 -4
-2
z (λ)
-4
(b) Axial distribution 1 0.9 0.8 0.7
2
0.6 |E|
much more important role in the axial in energy density distribution. Under Helmholtz condition, this portion of the cylindrical vector beams is strongly enhanced. In Fig. 4, the energy in the main lobe volume is about 59.14% of the total energy. As a comparison, the main lobe energy in Fig. 5 is about 62.08% of the total energy, which means that more energy is focused into the main lobe if a Helmhotlz condition objective lens is used. However, the sidelobe is slightly increased. The high sidelobe is mainly due to the annular illumination and the diffractions from the abrupt changes of the binary DOE which can be somewhat eased by properly choosing design parameters and using grayscale DOE design. One such example is shown in Fig. 6, where we increased NA to 0.815, instead of the NA of 0.8 used in Fig. 5, and the angle of cylindrical polarization U0 is found to be 40°. A flattop focus with slightly lower sidelobe is obtained. It is also possible to reduce the sidelobe by using grayscale (amplitude modulation) DOE to change the local amplitude transmittance. However, both methods also decrease the depth of focus. A compromise between the depth of focus and sidelobe needs to be made. The design parameters should be appropriately chosen according to specific applications.
415
0.5 0.4
4. Generation of optical bubble
0.2 0.1 0 -4
(c)
-3
-2
-1
0 z (λ)
1
2
3
4
2
3
4
Transversal distribution
1 0.9 0.8 0.7
2
0.6 |E|
In Section 3, we explored the feasibility of adjusting the destructive interference from the middle ring to ‘‘fill’’ the saddle to create a longitudinal flattop profile. It is also possible to further increase the destructive interference to ‘‘carve’’ into the focus deeper and generate an ‘‘optical bubble’’ which has a total dark volume surrounded by high field distributions. In this case, we limit our search to radial polarization with U0 = 0°, since this may allow one to create an ultrasmall dark volume. The objective lens is assumed to obey the sine condition. Using the same manual search methods, the parameter for the binary DOE are found to be NA1 = 0.2, NA2 = 0.65 with NA = 0.8 and NA0 = 0.2. Essentially, the DOE has two concentric rings, instead of three. Numerical simulation results for this design are shown in Fig. 7. From this figure, it can be seen that a bubble-like focal field with a totally dark center is obtained.
0.3
0.5 0.4 0.3 0.2
5. Applications and discussions Flattop focusing with extended depth of focus may have important applications in laser cutting of metals, particle acceleration, materials processing and microlithography. Laser cutting is one of the fastest growing processes in industrial manufacturing. Laser machining tools offer significant advantages in productivity, precision, part quality, material utilization and flexibility. During laser machining process, laser energy from a resonator is focused on a material in order to remove the materials. Laser beams that can be focused into a flattop spot will allow faster, high quality laser cutting with lower operating costs.
0.1 0 -4
-3
-2
-1
0 r (λ)
1
Fig. 6. Energy density distributions in the vicinity of focus with a larger NA of lens: (a) 3-D distribution; (b) axial distribution; (c) transversal distribution.
The laser machining efficiency of metals strongly depends on the polarization. Spatially homogeneous polarizations have substantial disadvantages. For linear polarization, the laser–metal interaction depends upon the polarization
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(a)
Axial distribution
(b)
1 0.9 0.8 0.7
|E|
2
0.6 0.5 0.4 0.3 0.2 0.1 0 -4
-3
-2
-1
0
1
2
3
4
z(λ) (c)
Transversal distribution 0.35 0.3 0.25
mized for minimum losses nor for maximum absorption. It has been pointed out that inhomogeneous types of polarization are much better in laser cutting [15,16]. In the case of cutting metals with a large aspect ratio of sheet thickness to width, the laser cutting efficiency for a radially polarized beam is shown to be 1.5–2 times larger than for P-polarized and circularly polarized beams [15]. In order to increase the laser cutting efficiency and velocity, sharper focusing of incoming light is necessary for high-speed cutting of sheet steel. If linear or circular polarized light is used in laser cutting, the radiation intensity decreases quickly along the focal axis, giving rise to a small ratio of cutting depth to cutting width. On the contrary, using cylindrical vector beams with flattop focusing and a long depth of focus, one may significantly increase the ratio of cutting depth to cutting width. Because of the threshold character of material removing, laser cutting works only when high densities of absorbed power exceed the threshold value. It is possible to adjust the beam power to make sure such threshold condition met by the power level in the main lobe only. Thus, the relative high sidelobe may not be a problem. Optical bubble may find important applications in particle trapping and fluorescence microscopy [26,27]. For example, the resolution of fluorescence microscopy is mostly determined by the extent of the fluorescence spot. In order to improve the spatial resolution of fluorescence microscopy, stimulated emission depletion (STED) technique has been developed [26]. In the STED technique, a depletion pulse following an excitation pulse is focused into a donut shape around the focus of the excitation beam. In the region where the focal field intensity of the depletion beam is above certain threshold intensity, fluorescence is inhibited. An optical bubble that has a total dark volume surrounded by high field distributions in three dimensions may be applied in the STED microscopy to improve the spatial resolution of fluorescence microscopy.
|E|
2
0.2
6. Conclusions
0.15 0.1 0.05 0 -4
-3
-2
-1
0
1
2
3
4
r (λ) Fig. 7. Simulation results for the generation of an optical bubble: (a) 2-D distribution; (b) axial distribution; (c) transversal distribution.
orientation. The shape of laser cutting is directly related to how the linearly polarized beam is oriented with respect to the direction in which the cut is traveling. For circular polarization, these parameters are time averaged, neither opti-
We have proposed a three-dimensional focus shaping technique using the combination of generalized cylindrical vector beams and diffractive optical elements. Our simulations have shown that three-dimensional flattop focus with extended depth of focus can be obtained. By selecting an incoming light with appropriate spatial amplitude distribution, it is possible to obtain flattop focusing with a longer depth of focus and much smaller sidelobe, and to improve the quality of the flattop focus, such as the sharpness of edge and the flatness of the focal field. With an appropriate design, it is also shown that an optical bubble can be obtained. This focus shaping technique may find applications in laser micromachining, photolithography, laser printing, high resolution microscopy, and optical trapping and manipulation of particles. Experimental works for
W. Chen, Q. Zhan / Optics Communications 265 (2006) 411–417
verification of these special focal field distributions are under development and will be reported in the future. References [1] Q. Zhan, J.R. Leger, Opt. Exp. 10 (2002) 324. [2] N. Passilly, R.S. Denis, K. Ait-Ameur, F. Treussart, R. Hierle, J. Roch, J. Opt. Soc. Am. A 22 (2005) 984. [3] B. Jia, X. Gan, M. Gu, Opt. Exp. 13 (2005) 6821. [4] Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, Opt. Lett. 27 (2002) 285. [5] A.V. Nesterov, V.G. Niziev, V.P. Yakunin, J. Phys. D: Appl. Phys. 32 (1999) 2871. [6] A.A. Tovar, J. Opt. Soc. Am. A 15 (1998) 27052711. [7] S. Quabis, R. Dorn, M. Eberler, O. Glockl, G. Leuchs, Opt. Commun. 179 (2000) 1. [8] T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, Phys. Rev. Lett. 78 (1997) 4713. [9] K.T. Gahagan, G.A. Swartzlander, J. Opt. Soc. Am. B 16 (1999) 533. [10] Q. Zhan, Opt. Exp. 12 (2004) 3377.
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[11] Q. Zhan, J. Opt. A: Pure Appl. Opt. 5 (2003) 229. [12] Y.Q. Zhao, Q. Zhan, Y.L. Zhang, Y. Li, Opt. Lett. 30 (2005) 848. [13] K.S. Youngworth, T.G. Brown, Proc. SPIE 3919 (2000) 75. [14] L.E. Helseth, Opt. Commun. 191 (2001) 161. [15] V.G. Niziev, A.V. Nesterov, J. Phys. D 32 (1999) 1455. [16] A.V. Nesterov, V.G. Niziev, J. Phys. D 33 (1999) 1817. [17] C. Varin, M. Piche, Appl. Phys. B 74 (2002) S83. [18] B. Hafizi, E. Esarey, P. Sprangle, Phys. Rev. E 55 (1997) 3539. [19] B. Sick, B. Hecht, L. Novotny, Phys. Rev. Lett. 85 (2000) 4482. [20] L. Novotny, M.R. Beversluis, K.S. Youngworth, T.G. Brwon, Phys. Rev. Lett. 86 (2001) 5251. [21] C. Sun, C. Liu, Opt. Lett. 28 (2003) 99. [22] R. Dorn, S. Quabis, G. Leuchs, Phys. Rev. Lett. 91 (2003) 233901. [23] U. Levy, C. Tsai, L. Pang, Y. Fainman, Opt. Lett. 29 (2004) 1718. [24] K.S. Youngworth, T.G. Brown, Opt. Exp. 7 (2000) 77. [25] Min Gu, Advanced Optical Imaging Theory, Springer-Verlag, New York, 1999, p. 143. [26] T.A. Klar, S. Jakobs, M. Dyba, A. Egner, S.W. Hell, Proc. Natl. Acad. Sci. 97 (2000) 8206. [27] P. Torok, P.R.T. Munro, Opt. Exp. 12 (2004) 3605.
Three-dimensional focus shaping with cylindrical vector beams Weibin Chen *, Qiwen Zhan Electro-Optics Graduate Program, University of Dayton, 300 College Park, Dayton, OH 45469-0245, USA Received 2 December 2005; received in revised form 5 April 2006; accepted 5 April 2006
Abstract A three-dimensional focus shaping technique using the combination of cylindrical polarization with binary diffractive optical element is proposed. The energy density pattern at the vicinity of the focus can be tailored in three dimensions by appropriately adjusting the parameters of the cylindrical vector beam illumination, numerical aperture of the objective lens and the design of the binary diffractive optical element. Focus with extended depth of focus that has both transversal and longitudinal flattop profile is obtained. Optical bubble that has a total dark volume surrounded by high field distributions is also shown. Potential applications of this focus shaping technique are discussed. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Focus shaping; Flattop focusing; Depth of focus; Optical bubble; Cylindrical vector beams; Polarization
1. Introduction In general, the polarization of light can be classified into two categories. The first type has spatially homogeneous distribution in terms of the state of polarization for the light wavefront. This type of polarization, including linear, circular and elliptical polarization, is most familiar to the optical community. Contrarily, the second type has spatially inhomogeneous polarization distribution. Traditionally, the second type of polarization was considered as a nuisance or aberrations in the optical design and has not drawn much research attention. However, there is an increasing interest in the spatially inhomogeneous polarization recently, mostly driven by the advances made in micro-fabrication techniques and theoretical modeling techniques that were not available previously. One example of such spatially inhomogeneous polarization that has attracted much of the interest is the so-called cylindrical vector (CV) beams. Cylindrical vector beams are solutions of Maxwell equations that obey cylindrical symmetry both in amplitude and
*
Corresponding author. E-mail address: [email protected] (W. Chen).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.04.066
polarization. Cylindrical vector beams can be divided into radial polarization, azimuthal polarization and generalized cylindrical polarization, according to the actual polarization pattern. One can use two cascaded half-wave plates to conveniently convert a radial polarization or azimuthal polarization into a generalized cylindrical vector beam, or vice versa [1]. Recently, there is a strong interest in these beams due to their peculiar properties, especially when focused under high numerical aperture (NA) conditions. Many techniques to generate radially polarized beams or azimuthally polarized beams have been reported, including interferometric techniques for converting a linearly polarized Gaussian beam into a radially polarized doughnut beam [2], the interference of two opposite hand circularly polarized Laguerre–Gaussian (LG) beams of opposite topological charge [3], using computer-generated spatially variant dielectric or metal strip subwavelength grating [4], or inserting other special designed elements into laser resonators [5,6], or using space-variant inhomogeneous media [7]. Owing to their unique properties, cylindrical vector beams may find wide applications in physics, chemistry and biology, such as particle guiding or trapping [8–12], scanning optical microscopy [13], lithography [14], laser cutting of metals [15,16], particle acceleration [17,18], and single molecule imaging [19,20].
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The propagation and focusing properties of spatially inhomogeneous polarization remain of continued interest. When focused with a high NA objective, radially polarized light leads to a strong longitudinal electric field component in the vicinity of the focus. The relative contribution of longitudinal component can be enhanced by using annular aperture. Theoretical and experimental investigations of focusing an incoming radially polarized beam into an ultrasmall spot have been reported [21–23]. The spot size can be much smaller than the diffraction limit of focused spatially homogeneously polarized beams. Besides an ultrasmall focus, it has been shown that focal distribution with extended depth of focus (DOF) can be achieved with inserting a simple diffractive optical element (DOE) in the optical path under radial polarization illumination [21]. Furthermore, using a very simple two-half-wave-plate polarization rotator, one can conveniently convert radial polarization into a generalized cylindrical vector polarization that can be used to generate focal field distribution with flattop transversal profile. In this paper, we propose to combine the techniques from [1,21] to investigate a three-dimensional focus shaping technique that uses a DOE modulated cylindrical vector beam illumination. In Section 2, we present the mathematical model of cylindrical vector beams and three-dimensional focal field distributions when these beams are focused by a high numerical objective lens with the insertion of a binary DOE. In Section 3, we show how to obtain a maximally homogenized focal field with a significantly extended DOF. This focal field has flattop profiles in both the transversal and longitudinal direction. The relationship between flattop focusing, depth of focus, sidelobe and the design of DOE, rotation angle of cylindrical vector beams is also discussed. In Section 4, the generation of an optical bubble with completely dark center surrounded by high field strength will be shown. Potential applications of this three-dimensional focus shaping technique will be proposed and discussed. 2. Proposed setup and modeling
or an azimuthal polarization, each point of the beam has a polarization rotated by U0 from its radial direction. The electric field of this beam can be expressed in a cylindrical coordinate system as ~ Eðr; uÞ ¼ LðrÞ½cos /0~ er þ sin /0~ eu
ð1Þ
where ~ er is the unit vector in the radial direction and ~ eu is the unit vector in the azimuthal direction. L(r) is the pupil function denoting the relative amplitude of the field that only depends on radial position. Thus, a generalized cylindrical vector beam can be decomposed into a radial polarization and an azimuthal polarization. The setup for the focus shaping is shown in Fig. 2. The incident light is a generalized cylindrical vector beam. An objective lens with high numerical aperture is used to produce a spherical wave converging to the focus. A binary DOE with three concentric regions (shown in Fig. 3) is placed in front of the lens to modulate the wavefront of the generalized cylindrical vector beams. Due to the symmetry and normal illumination, the phase modulation is identical for the radial and azimuthal polarization components. The numerical apertures that correspond to the outer edge of the three regions are NA1, NA2 and NA respectively, where the local transmittances are T1 = 1, T2 = 1 and T3 = 1. Thus the field distribution at the focus is given by E ¼ E1 E 2 þ E 3
ð2Þ
where E1, E2 and E3 are the focal field contribution from each annular zone that can be calculated as the following. The electric fields in the vicinity of the focal spot for radial and azimuthal polarization can be calculated by Richards–Wolf vectorial diffraction method [1,24]. The y High NA Lens
Q(r,ϕ) ϕ
Cylindrical vector incident beam
Fig. 1 shows the polarization pattern of a generalized cylindrical vector beam. Instead of a radial polarization
x
θ z
z=0 Focal Plane Binary DOE
f
Fig. 2. Focusing of a generalized cylindrical vector beam with DOE.
Φo
NA NA2
+
Fig. 1. Generalized cylindrical vector beam with rotation from the purely radially polarization.
_
NA1
+
Fig. 3. Binary DOE with three concentric regions.
W. Chen, Q. Zhan / Optics Communications 265 (2006) 411–417
focal field of a generalized cylindrical beam contributed from each concentric region of the DOE can be written as ~ Eðr; u; zÞ ¼ Ez~ ez þ Er~ er þ Eu~ eu
ð3Þ
where ~ ez is the unit vector in the z-direction, and Z h2 Ez ðr; u; zÞ ¼ jA cos /0 sin2 hP ðhÞLðhÞJ 0 ðkr sin hÞejkz cos h dh h1
Er ðr; u; zÞ ¼ A cos /0
Z
ð4Þ h2
sin h cos hP ðhÞLðhÞ h1
ð5Þ J 1 ðkr sin hÞejkz cos h dh Z h2 sin hP ðhÞLðhÞJ 1 ðkr sin hÞejkz cos h dh Eu ðr; u; zÞ ¼ A sin /0 h1
ð6Þ where h1 and h2 is the angular transition points determined by the inner and outer transition points of the corresponding concentric region of the binary DOE. For example, for the concentric ring with p phase shift (see Fig. 3), the integration limits are given by h1 = sin1 (NA1) and h2 = sin1 (NA2), respectively. P(h) is the pupil apodization function, k is the wave number and Jn(x) is the Bessel function of the first kind with order n. L(h) is the relative amplitude of the field, which is assumed to be dependent on the radial position only. Due to the continuity requirement, the on-axis electric field of cylindrical polarization is zero. The actual distribution of this hollow center depends on how the cylindrical polarization is generated. For simplicity, the illumination for all the examples in this paper is chosen to be a planar wavefront over the pupil, where ( 1 if sin1 ðNA0 Þ 6 h 6 sin1 ðNAÞ LðhÞ ¼ ð7Þ 0 otherwise where NA0 = 0.2 for our simulations. This means that the center of the lens pupil aperture is blocked to simulate this dark hollow center. Since the focusing properties of cylindrical vector beams are largely due to the high spatial frequency components, the effects on the field distribution by blocking the center are negligible. 3. Three-dimensional flattop focusing with extended depth of focus In this part, we explore the possibility to generate flattop focusing with extended DOF using the setup described above. In Ref. [21], similar binary DOE was proposed to generate ultrasharp focus with extended depth of focus with radially polarized illumination. Essentially, the field contributed from the middle ring destructively interferes with the field contributed from the innermost and outermost ring created a saddle in the focus and elongate the depth of focus. However, it should be possible to adjust the transition points of the middle ring to change the relative interference contribution and ‘‘fill’’ the saddle to create
413
a flattop longitudinal field profile. In addition, the focal fields created by radial polarization and azimuthal polarization are mutually orthogonal and spatially separated. This property was exploited to create flattop transverse focus as reported in Ref. [1]. If a generalized cylindrical vector beam is used as illumination along with the binary DOE, one should be able to adjust the relative weighing of the radial and azimuthal components to create flattop profiles both in longitudinal and transversal directions. The design procedure is as the following. From Eqs. (3)– (6), it is clear that the only none-zero contribution for onaxis field distribution comes from the z-component. Thus, relative axial energy density distribution will not change with the rotation angle U0 because only radial component of the generalized cylindrical vector beams contributes to it. Therefore, during the first step, we set U0 = 0° and change the transition points NA1 and NA2 of the binary DOE to find a flat longitudinal field distribution with long DOF. This is done by a linear direct search through manually adjusting the two transition points with 0.01 increments in the transition points. Once this is achieved, then we obtain flattop transversal field distribution through adjusting U0. For all of the calculations in this paper, the length unit is normalized to wavelength k, and the maximum energy density is normalized to unity. First we choose an aplanatic objective lens that satisfies the sine condition. Under sine condition, the pupil apodization function is P ðhÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi cosðhÞ [25]. With fixed parameters of NA = 0.8 and NA0 = 0.2, we found the parameters for the binary DOE to be NA1 = 0.31, NA2 = 0.5 and the rotation parameter for cylindrical vector beams to be U0 = 38°. Fig. 4(a) shows the calculated result of three-dimensional energy density distribution in the vicinity of the focus. The corresponding axial and transversal energy density distributions are shown in Fig. 4(b) and (c). It can be seen that flattop profiles along both the transversal and axial distribution have been obtained. The depth of focus defined by full width half maximum (FWHM) of the jEj2 along the axial direction is calculated to be 5k. For comparison, the axial field distribution for linearly polarized distribution is also calculated and shown in Fig. 4(b). The depth of focus is calculated to be 2.4k. Clearly, an extended depth of focus is achieved with cylindrical vector polarization. This extended DOF partly is due to the destructive interference from the middle ring of the DOE. In addition, the vector projection used in vectorial diffraction theory gives larger weighing to the higher spatial frequency components in the calculation of Ez, as seen in Eq. (4). This apodization effect also contributes to the longer DOF. Besides the pupil apodization function under sine condition, we also can explore other pupil apodization functions. With NA = 0.8 and NA0 = 0.2, the parameters for the binary DOE are found to be NA1 = 0.35, NA2 = 0.55, and the rotation angle for the cylindrical vector beam is U0 = 39.5°. Fig. 5 shows the calculated results of focal field distribution for an objective lens that obeys the Helmholtz condition,
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(a) 1
1
0.8
0.8
0.6
0.6 2
0.4
|E|
|E|
2
(a)
0.2
0.4 0.2
0 4 2 0
r (λ)
-2 -4
0
-2
-4
2
0 4
4
2
4
z (λ)
2
0
(b)
-2 -4
Axial distribution
-4
z( )
1 Axial distribution
(b) 0.9
1
0.8
0.9
0.7
0.8
0.6
0.7
0.5
0.6
2
|E|
2
|E|
0
-2
r( )
0.4 0.3
0.4
0.2
0.3
0 -4
0.2
CV beam Linear
0.1 -3
-2
-1
0 z
(c)
0.5
1
CV beam Linear
0.1
2
3
4
0 -4
)
-3
-2
Transversal distribution
1
0.8
0.9
0.7
0.8
0.6
0.7
0.5
0.6 |E|
0.4
0 z( )
1
2
3
4
2
3
4
Transversal distribution 1
2
|E|
2
(c) 0.9
0.5
0.3
0.4
0.2
0.3
0.1
0.2
0 -4
-1
0.1
-3
-2
-1
0 r( )
1
2
3
4
Fig. 4. Energy density distributions in the vicinity of focus with pupil apodization function under sine condition: (a) 3-D distribution; (b) axial distribution; (c) transversal distribution. CV: cylindrical vector beam.
pffiffiffiffiffiffiffiffiffiffiffiffiffi 3 where P ðhÞ ¼ ½ cosðhÞ [25]. In this case, the depth of focus is calculated to be 5.8k. This even longer DOF is due to the further increase of the higher spatial frequency weighing due to the apodization function. Again, a threedimensional flattop focusing with extended depth of focus is obtained.
0 -4
-3
-2
-1
0 r( )
1
Fig. 5. Energy density distributions in the vicinity of focus with pupil apodization function under the Helmhotlz condition: (a) 3-D distribution; (b) axial distribution; (c) transversal distribution. CV: cylindrical vector beam.
By comparing the above two results, we find that objective lens under the Helmhotlz condition is better for flattop focusing with extended depth of focus. This is because that the input energy at the outer portion of the DOE plays
W. Chen, Q. Zhan / Optics Communications 265 (2006) 411–417
(a) 1
|E|
2
0.8 0.6 0.4 0.2 0 4 2
4 2
0 r (λ)
0
-2 -4
-2
z (λ)
-4
(b) Axial distribution 1 0.9 0.8 0.7
2
0.6 |E|
much more important role in the axial in energy density distribution. Under Helmholtz condition, this portion of the cylindrical vector beams is strongly enhanced. In Fig. 4, the energy in the main lobe volume is about 59.14% of the total energy. As a comparison, the main lobe energy in Fig. 5 is about 62.08% of the total energy, which means that more energy is focused into the main lobe if a Helmhotlz condition objective lens is used. However, the sidelobe is slightly increased. The high sidelobe is mainly due to the annular illumination and the diffractions from the abrupt changes of the binary DOE which can be somewhat eased by properly choosing design parameters and using grayscale DOE design. One such example is shown in Fig. 6, where we increased NA to 0.815, instead of the NA of 0.8 used in Fig. 5, and the angle of cylindrical polarization U0 is found to be 40°. A flattop focus with slightly lower sidelobe is obtained. It is also possible to reduce the sidelobe by using grayscale (amplitude modulation) DOE to change the local amplitude transmittance. However, both methods also decrease the depth of focus. A compromise between the depth of focus and sidelobe needs to be made. The design parameters should be appropriately chosen according to specific applications.
415
0.5 0.4
4. Generation of optical bubble
0.2 0.1 0 -4
(c)
-3
-2
-1
0 z (λ)
1
2
3
4
2
3
4
Transversal distribution
1 0.9 0.8 0.7
2
0.6 |E|
In Section 3, we explored the feasibility of adjusting the destructive interference from the middle ring to ‘‘fill’’ the saddle to create a longitudinal flattop profile. It is also possible to further increase the destructive interference to ‘‘carve’’ into the focus deeper and generate an ‘‘optical bubble’’ which has a total dark volume surrounded by high field distributions. In this case, we limit our search to radial polarization with U0 = 0°, since this may allow one to create an ultrasmall dark volume. The objective lens is assumed to obey the sine condition. Using the same manual search methods, the parameter for the binary DOE are found to be NA1 = 0.2, NA2 = 0.65 with NA = 0.8 and NA0 = 0.2. Essentially, the DOE has two concentric rings, instead of three. Numerical simulation results for this design are shown in Fig. 7. From this figure, it can be seen that a bubble-like focal field with a totally dark center is obtained.
0.3
0.5 0.4 0.3 0.2
5. Applications and discussions Flattop focusing with extended depth of focus may have important applications in laser cutting of metals, particle acceleration, materials processing and microlithography. Laser cutting is one of the fastest growing processes in industrial manufacturing. Laser machining tools offer significant advantages in productivity, precision, part quality, material utilization and flexibility. During laser machining process, laser energy from a resonator is focused on a material in order to remove the materials. Laser beams that can be focused into a flattop spot will allow faster, high quality laser cutting with lower operating costs.
0.1 0 -4
-3
-2
-1
0 r (λ)
1
Fig. 6. Energy density distributions in the vicinity of focus with a larger NA of lens: (a) 3-D distribution; (b) axial distribution; (c) transversal distribution.
The laser machining efficiency of metals strongly depends on the polarization. Spatially homogeneous polarizations have substantial disadvantages. For linear polarization, the laser–metal interaction depends upon the polarization
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W. Chen, Q. Zhan / Optics Communications 265 (2006) 411–417
(a)
Axial distribution
(b)
1 0.9 0.8 0.7
|E|
2
0.6 0.5 0.4 0.3 0.2 0.1 0 -4
-3
-2
-1
0
1
2
3
4
z(λ) (c)
Transversal distribution 0.35 0.3 0.25
mized for minimum losses nor for maximum absorption. It has been pointed out that inhomogeneous types of polarization are much better in laser cutting [15,16]. In the case of cutting metals with a large aspect ratio of sheet thickness to width, the laser cutting efficiency for a radially polarized beam is shown to be 1.5–2 times larger than for P-polarized and circularly polarized beams [15]. In order to increase the laser cutting efficiency and velocity, sharper focusing of incoming light is necessary for high-speed cutting of sheet steel. If linear or circular polarized light is used in laser cutting, the radiation intensity decreases quickly along the focal axis, giving rise to a small ratio of cutting depth to cutting width. On the contrary, using cylindrical vector beams with flattop focusing and a long depth of focus, one may significantly increase the ratio of cutting depth to cutting width. Because of the threshold character of material removing, laser cutting works only when high densities of absorbed power exceed the threshold value. It is possible to adjust the beam power to make sure such threshold condition met by the power level in the main lobe only. Thus, the relative high sidelobe may not be a problem. Optical bubble may find important applications in particle trapping and fluorescence microscopy [26,27]. For example, the resolution of fluorescence microscopy is mostly determined by the extent of the fluorescence spot. In order to improve the spatial resolution of fluorescence microscopy, stimulated emission depletion (STED) technique has been developed [26]. In the STED technique, a depletion pulse following an excitation pulse is focused into a donut shape around the focus of the excitation beam. In the region where the focal field intensity of the depletion beam is above certain threshold intensity, fluorescence is inhibited. An optical bubble that has a total dark volume surrounded by high field distributions in three dimensions may be applied in the STED microscopy to improve the spatial resolution of fluorescence microscopy.
|E|
2
0.2
6. Conclusions
0.15 0.1 0.05 0 -4
-3
-2
-1
0
1
2
3
4
r (λ) Fig. 7. Simulation results for the generation of an optical bubble: (a) 2-D distribution; (b) axial distribution; (c) transversal distribution.
orientation. The shape of laser cutting is directly related to how the linearly polarized beam is oriented with respect to the direction in which the cut is traveling. For circular polarization, these parameters are time averaged, neither opti-
We have proposed a three-dimensional focus shaping technique using the combination of generalized cylindrical vector beams and diffractive optical elements. Our simulations have shown that three-dimensional flattop focus with extended depth of focus can be obtained. By selecting an incoming light with appropriate spatial amplitude distribution, it is possible to obtain flattop focusing with a longer depth of focus and much smaller sidelobe, and to improve the quality of the flattop focus, such as the sharpness of edge and the flatness of the focal field. With an appropriate design, it is also shown that an optical bubble can be obtained. This focus shaping technique may find applications in laser micromachining, photolithography, laser printing, high resolution microscopy, and optical trapping and manipulation of particles. Experimental works for
W. Chen, Q. Zhan / Optics Communications 265 (2006) 411–417
verification of these special focal field distributions are under development and will be reported in the future. References [1] Q. Zhan, J.R. Leger, Opt. Exp. 10 (2002) 324. [2] N. Passilly, R.S. Denis, K. Ait-Ameur, F. Treussart, R. Hierle, J. Roch, J. Opt. Soc. Am. A 22 (2005) 984. [3] B. Jia, X. Gan, M. Gu, Opt. Exp. 13 (2005) 6821. [4] Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, Opt. Lett. 27 (2002) 285. [5] A.V. Nesterov, V.G. Niziev, V.P. Yakunin, J. Phys. D: Appl. Phys. 32 (1999) 2871. [6] A.A. Tovar, J. Opt. Soc. Am. A 15 (1998) 27052711. [7] S. Quabis, R. Dorn, M. Eberler, O. Glockl, G. Leuchs, Opt. Commun. 179 (2000) 1. [8] T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, Phys. Rev. Lett. 78 (1997) 4713. [9] K.T. Gahagan, G.A. Swartzlander, J. Opt. Soc. Am. B 16 (1999) 533. [10] Q. Zhan, Opt. Exp. 12 (2004) 3377.
417
[11] Q. Zhan, J. Opt. A: Pure Appl. Opt. 5 (2003) 229. [12] Y.Q. Zhao, Q. Zhan, Y.L. Zhang, Y. Li, Opt. Lett. 30 (2005) 848. [13] K.S. Youngworth, T.G. Brown, Proc. SPIE 3919 (2000) 75. [14] L.E. Helseth, Opt. Commun. 191 (2001) 161. [15] V.G. Niziev, A.V. Nesterov, J. Phys. D 32 (1999) 1455. [16] A.V. Nesterov, V.G. Niziev, J. Phys. D 33 (1999) 1817. [17] C. Varin, M. Piche, Appl. Phys. B 74 (2002) S83. [18] B. Hafizi, E. Esarey, P. Sprangle, Phys. Rev. E 55 (1997) 3539. [19] B. Sick, B. Hecht, L. Novotny, Phys. Rev. Lett. 85 (2000) 4482. [20] L. Novotny, M.R. Beversluis, K.S. Youngworth, T.G. Brwon, Phys. Rev. Lett. 86 (2001) 5251. [21] C. Sun, C. Liu, Opt. Lett. 28 (2003) 99. [22] R. Dorn, S. Quabis, G. Leuchs, Phys. Rev. Lett. 91 (2003) 233901. [23] U. Levy, C. Tsai, L. Pang, Y. Fainman, Opt. Lett. 29 (2004) 1718. [24] K.S. Youngworth, T.G. Brown, Opt. Exp. 7 (2000) 77. [25] Min Gu, Advanced Optical Imaging Theory, Springer-Verlag, New York, 1999, p. 143. [26] T.A. Klar, S. Jakobs, M. Dyba, A. Egner, S.W. Hell, Proc. Natl. Acad. Sci. 97 (2000) 8206. [27] P. Torok, P.R.T. Munro, Opt. Exp. 12 (2004) 3605.