Light field shaping by tailoring both phase and polarization Jingjing Hao, Zhongliang Yu, Hao Chen, Zhaozhong Chen, Hui-Tian Wang, and Jianping Ding* National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China *Corresponding author: [email protected] Received 27 September 2013; revised 5 December 2013; accepted 18 December 2013; posted 20 December 2013 (Doc. ID 197198); published 31 January 2014
We propose a method to generate a vectorial focal field with reconfigurable distributions for both the intensity and polarization state. The three-dimensional focal volume was configured by modulating the phase and polarization of the incident light. The incident light yielding the desired field was determined based on an iterative scheme involving vectorial diffraction calculations and fast Fourier transforms. Optical experiments on vectorial field shaping were performed to validate the feasibility of our method. This method may have applications in optical tweezers, such as for realizing the optical manipulation of particles via polarization modulation in addition to phase control. © 2014 Optical Society of America OCIS codes: (050.1960) Diffraction theory; (260.5430) Polarization; (120.5060) Phase modulation. http://dx.doi.org/10.1364/AO.53.000785
1. Introduction
Focus shaping is applicable in a variety of fields ranging from laser-based material processing to microscopic imaging [1–6]. The common procedure for focus shaping is to configure the phase and/or amplitude profiles of an incident beam so that it yields the desired irradiance distribution in the focal volume. Recently, vector beams with space-variant polarization have drawn much attention as they are able to significantly influence the focusing behavior of light [7–10]. Microscopic optics can greatly benefit from the polarization engineering of light beams in this way [11–13], and consequently, there is a growing interest in not only adjusting the overall intensity distribution, but also in manipulating the polarization state distribution in the focal volume [14–19]. In a previous paper, we reported on the use of polarization-only modulation of incident beams to implement focus shaping [20]. The study had demonstrated the functionality of polarization 1559-128X/14/040785-07$15.00/0 © 2014 Optical Society of America
manipulation. On the other hand, it is well known that the tailoring of the wavefront (or phase profile) of a light beam is preferred for focus shaping because, normally, the phase plays the most important role in controlling the propagation behavior of the light field. Therefore, it is natural to expect that the combined modulation of the polarization and phase would provide more elaborate and versatile structuring of the focus field. This work demonstrates the possibility of simultaneous tailoring of both polarization and phase. Moreover, the previous work could only achieve a centro-symmetric irradiance pattern with respect to the optical axis across the transverse plane in the focal volume. This was due to the fact that the polarization-only modulation employed in the previous work allowed only real-valued modulation to the incident beams. The synthesized incident beams had no phase profile and thus yielded a Hermitian distribution in the focal field. In this paper, we build on this work and incorporate phase modulation in addition to polarization modulation into the vector beams so that they can generate the desired intensity distribution and polarization state in the focal volume. We propose an iterative 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
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scheme for realizing specific polarization and intensity configurations in the focal volume by optimizing the modulation of both the polarization and phase of the incident beams. 2. Principles
Consider the geometry shown in Fig. 1. An incoming light beam with an elaborately structured phase profile and polarization state is focused by a lens to produce the desired distribution in the focal volume. The proposed scheme enables light beams to be spacevariantly modulated in both phase and polarization, which is realized in an optimization procedure for achieving the prescribed focal field. We started with an inhomogeneously polarized incident beam across whose cross section a space-variant linear polarization and an arbitrary phase profile were distributed. For simplicity, we assumed a uniform amplitude distribution across the entrance plane of the focusing lens (although our approach allows for amplitude modulations of the beam as well). We employed a circular polarization basis to express the incident vector beam, as follows:
1 x;y;z
1 E eiδ− Ei x; y; z p0 eiδ x;y;z −i i 2 1 0 δ x;y;z−δ− x;y;z cos 2 δ x;y;zδ− x;y;z @ 2 E0 ei A − x;y;z sin δ x;y;z−δ 2 cos αx; y; z ; (1) E0 eiβx;y;z sin αx; y; z
⇀
where δ x; y; z and δ− x; y; z are the phase factors imposed on the left- and right-hand circular polarization components, respectively, E0 is the amplitude of the incident light, αx; y; z δ x; y; z − δ− x; y; z∕2, and βx;y;z δ x;y;z δ− x;y;z∕2. The last equality of Eq. (1) indicates that the incident beam has a global phase factor represented by
Fig. 1. Focusing of incident light beams with space-variant phase and polarization.
vectorial diffraction integral that can be written in terms of a Fourier transform (FT) [20–22]. For the sake of simplicity, we restricted our attention to the paraxial focusing in which the longitudinal component of the focal field can be omitted. Following Eq. (1), calculations of the field distribution were performed by decomposing the circular polarization into a pair of orthogonal basis vectors, the left- and righthand circular polarization components. Letting the left- and right-hand circular polarization compo⇀
nents of the incident field be denoted by Ei x0 ; y0 ⇀
and Ei− x0 ; y0 , respectively, h i 8⇀ 1 < Ei x0 ; y0 E0 x0 ; y0 −i h1i ; : ⇀ Ei− x0 ; y0 E0− x0 ; y0 i
p and where E0 x0 ; yp 0 E0 ∕ 2 expiδ x0 ; y0 E0− x0 ; y0 E0 ∕ 2 expiδ− x0 ; y0 denote the complex amplitudes of polarization components. ⇀
Accordingly, the focal field E at the point x; y; z with respect to the focus can be described by the leftand right-hand circular polarization components ⇀
⇀
E x; y; z and E− x; y; z, which are related to the incident field as below [22]
8 RRR < E x; y; z A E0 x0 ; y0 exp i 2π λf xx0 yy0 zz0 Px0 ; y0 ; z0 dx0 dy0 dz0 : : E x; y; z A R R R E x ; y exp i 2π xx yy zz Px ; y ; z dx dy dz − 0− 0 0 0 0 0 0 0 0 0 0 0 λf
βx; y; z and a polarization angle of αx; y; z with respect to the x direction at the plane x; y. The polarization direction and phase factor of the incident beam are two free parameters that need to be optimized during the iterative procedure to generate the desired vectorial focus. It should be noted that such space-variant (phase and polarization) beams can be generated by our experimental setup. The focal field of a monochromatic light beam passing through an aplanatic lens is calculated by the 786
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2
3
Here, A is a multiplicative factor with its coordinate dependence on x; y; z (omitted for brevity). Px0 ; y0 ; z0 is the pupil function expressed by Px0 ; y0 ; z0 δ
q x20 y20 z20 − f
(4)
with f denoting the focal length of the lens and δ representing the Dirac function. The total field is
then obtained by the sum of the two components, i.e., ⇀
Ex; y; z E x; y; z1; −iT E− x; y; z1; iT (T represents the transpose of the matrix). Equation (3) establishes the relationship between the incident light and the focal field, and it can be calculated quickly by using the fast Fourier transform (FFT) algorithm. It should be noted that the triple integral in Eq. (3) enables the three-dimensional (3D) focus calculation as well. The 3D integral determines the field within the 3D region centered at the focus. Multiple transverse planes at different axial positions with respect to the focus can be calculated quickly by the two-dimensional (2D) FT because the δ function in Eq. (4) can reduce the number of integration variables from three down to two, as expressed by
controlling a set of prescribed components with special phase differences between them. For brevity, hereinafter we will omit the coordinate dependence of the amplitude distribution unless otherwise specified. The phase factors of the left- and right-hand circular polarization components of the incident light, δ and δ− , are taken as free parameters that need to be modified to create the desired field. We designed an iterative procedure to search for a set of optimal parameters δ and δ− that realize the desired field. The flow chart of the iterative algorithm is depicted in Fig. 2. The iterative process starts with specifying a set of randomly distributed phases to the circular components of the incident beam [i.e., the initial values of δ and δ− in Eq. (1) are uniform random numbers in (0, 2π)]. The diffraction
8 p RR f 2 −x20 y20 > > E x; y; z A E x ; y exp i2π z exp i 2π < −∞ 0 0 0 λf λf xx0 yy0 dx0 dy0 p : RR f 2 −x20 y20 > 2π > : E− x; y; z A −∞ E0− x0 ; y0 exp i2π z exp i λf xx0 yy0 dx0 dy0 λf
Our scheme simulates an iterative process between the forward and backward propagations that relate the incident field and the focal field, which is analogous to the Gerchberg–Saxton (GS) algorithm for retrieving the phase of a pair of light distributions. When applied to shaping a light beam, the GS algorithm is often used to calculate the phase pattern that light in one plane must have in order to form a desired intensity pattern in a second plane, the light distribution on that is related to that in the first plane by a propagating function such as the FT. With the help of the FFT algorithm, which can be implemented efficiently on a computer, the field in a plane at a given axial position can be obtained by a 2D FT. The proposed approach, with modification to the vectorial field, was extended to calculating the phase patterns of the polarization components of the incident light that shape the desired field distribution within the focal volume. The iterative scheme requires the calculation of a stack of planes at different axial positions [i.e., different values of z in Eq. (5)] with respect to the focus for the entire 3D complex amplitude distribution. It is noted that our method allows shaping of not only the light intensity but also the polarization state of the focal field. The purpose of this is to construct a focal volume that has a specific intensity distribution and a space-variant polarization distribution. It is known that the intensity of the optical field is determined by the absolute value of the polarization components, whilst the polarization state of the field depends on the phase difference between these components. Therefore, we aimed to achieve the desired optical field by
5
integrals in Eq. (5) are then calculated using the FFT, which yields the circular components of the focal field. Absolute values and phase differences of the calculated components in the focal field are replaced by the ones of the prescribed polarization components (i.e., A , A− , and γ in Fig. 2), while the global phase values are left unchanged (i.e., φ remains unchanged). New phase factors in the components of the incident beam are obtained through an inverse FFT by letting the reshaped focal field backward propagate to the entrance plane. We imposed a constraint on the renewal process such that the two circular components of the incident beampshould have the same and constant amplitudes E0 ∕ 2. The iterative search continues until a satisfactory focal field is found. In short, the iterative process optimizes three free parameters (i.e., the phase factors of the polarization components of the incident light, δ , and δ− , and the global phase of the focal field, φ ) in order to achieve the predetermined polarization components as well as the phase difference between them within the focal region, which are represented by A, A− , and γ, respectively, in Fig. 2. It should be emphasized that the above explained algorithm yields a space-variant distribution of locally linear polarization (i.e., a polarization state on the equator of the Poincaré sphere) because the circular basis representation is employed in the calculation [cf. Eq. (1)]. However, the algorithm can be modified into the linear polarization basis representation so that we can get a hybrid modulation of locally linear, circular, and elliptical polarization. In this way, the x and y linear components are used 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
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3. Numerical Simulation and Experimental Realization
Fig. 2. Flow chart for iteratively searching for the optimal polarization and phase modulation of the incident light that can yield the desired focal field.
to construct the polarization distribution and thus the synthesized polarization states locate on a great circle crossing over the south and north poles of the Poincaré sphere.
In the subsequent numerical and experimental examples, all the incident light beams are specified as having the uniform intensity and the space-variant phase and polarization. The parameters used in the simulation process were chosen as: a 0.12 NA objective lens with a focal length of 200 mm and a laser light of wavelength 532 nm. The parameters of the focusing lens were chosen so that a CCD camera can detect a region-of-interest field in the focal plane, wherein a special polarization and intensity structure appears. This focusing geometry can afford a lateral resolution of λ∕2 NA 2.2 μm and a axial resolution of 2λ∕NA2 74 μm. A simulation of the results obtained by performing the above iterative procedure is presented in Fig. 3. An optimized incident light beam of wavelength 532 nm passing through a lens with a 0.12 NA creates, in the focal plane, three intensity rings each with different polarization states. The radii of the three rings were 20, 40, and 60 μm. The inner, middle, and outer rings were set to be right-hand circularly polarized, x linearly polarized, and left-hand circularly polarized, respectively. The synthesized incident beam can be generated by an experimental setup that produces vector beams with arbitrary space-variant modulations of both phase and polarization. In this setup, a holographic grating pattern displayed at a spatial light modulator (SLM) was designed such that it transformed the incident light into the beam modes with the desired amplitude and phase at the first positive and minus diffraction orders. After being filtered by a spatial filter, the two diffraction orders were transformed by each of the two wave-plates into the right- and left-hand circularly polarized lights.
Fig. 3. Simulation results of three intensity rings in the focal plane. The inner, middle, and outer rings have right-hand circular, x linear, and left-hand circular polarizations, respectively. (a) Intensity of the focal plane. (b) Phase difference of the right- and left-hand circular components. (c) and (d) Respective intensities of the right- and left-hand circular polarization components. 788
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Each of the two circularly polarized components acquired a specific phase factor from the diffraction of the specially designed holographic grating. Two circularly polarized components were then collinearly recombined by a grating to create the desired vector beams. The experimental setup takes advantage of a SLM (Holoeye Pluto with 1920 × 1080 pixels resolution) that is controlled by a computer, which enables the dynamic generation of the vector beams (refer to [23] for full details). The overall efficiency of the system was about 1.2% due to the loss of energy in the spatial filter, SLM, and grating. The light efficiency can be improved by using a more efficient modulation manner of SLM, e.g., by using the phase modulation rather than the amplitude modulation to write the grating pattern. In our proof-of-principle demonstration, however, the light efficiency was not the primary concern. In this paper, the experiment was performed to demonstrate the feasibility and capability of the phase and polarization modulation in the focus shaping. An experimental demonstration of the synthesized vector beam corresponding to Fig. 3 is presented in Fig. 4. A CCD camera having 1032 × 779 pixels with a pixel pitch of 4.65 μm of the vector beam in the focal plane. The leftmost pattern of Fig. 4 gives the intensity distribution in the focal plane and subsequent images were taken after a linear polarization analyzer. The arrows in the inset of each image indicate the polarization direction of the analyzer. In the rightmost pattern, the middle ring is faint because the x linear polarization within this ring region was extinguished by the vertical analyzer. In the above example, the polarization state remains uniform within each ring region. We now give another example of the spatially variant polarization
distribution in the focal field, as shown in Fig. 5. In this example, the inner and the outer rings are the right- and left-hand circular polarization, respectively, while the middle ring is set to the radial polarization. Radii of the three rings were 20, 30, and 40 μm. Phase difference of the right- and lefthand circular components in the focal plane is shown in the leftmost panel of Fig. 5, and experimentally measured intensity maps about the focused field are presented in the remaining panels. The intensity patterns were recorded after a linear analyzer oriented at 90, 0, 45, and 135°. Note that speckle-like patterns can be seen in the picture. In addition to the noise due to coherent illumination, we ascribe this phenomenon to the residue errors of the iterative search procedure, wherein not a complete exact construction of the desired focus but an optimal solution was achieved. Figure 6 presents the experimental results of another vectorial focal field that was configured to display playing card patterns. Four shapes corresponding to the suits of playing cards [spades (♠), hearts (♥), clubs (♣), and diamonds (♦)] and with different polarizations were assigned to the four quadrants of the focal plane. The four polarization states were horizontal linear, vertical linear, right-hand circular, and left-hand circular (polarization). A modification to our proposed algorithm was made in the next example, which demonstrates a generation of the polarization gradient in the focal field. The polarization distribution in this example was no longer limited to the polarization state corresponding to the equator on the Poincaré sphere, which is shown in the above examples. Instead, it spans a complete meridian, crossing from the south to the north pole of the sphere. Figure 7 shows a
Fig. 4. Experimentally generated vectorial field corresponding to Fig. 3. The inner, middle, and outer rings are right-hand circularly polarized, x linearly polarized, and left-hand circularly polarized, respectively. From left to right: intensity distribution recorded without an analyzer, and with an analyzer oriented at 0°, 45°, and 90° (as indicated by the arrow insets).
Fig. 5. Phase distribution and experimental results that perform the spatially variant polarization distribution of the focal field. Inner ring and outer ring show right- and left-hand circular polarization, respectively. Middle ring indicates a radially polarized beam. The arrows show the orientation of the analyzer. 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
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Fig. 6. Recorded focal intensities of the four playing card patterns described in the text. From left to right: no analyzer, horizontal linear, vertical linear, right-hand circular, and left-hand circular analyzers have been employed (as indicated by the arrows insets).
Fig. 7. Experimental demonstration of a polarization gradient distribution in the focal field. From left to right: no analyzer, right-hand circular, and left-hand circular analyzers have been applied (as indicated by the arrows insets).
ring-shaped focal field divided into two halves. The polarization state of the upper half of the ring varies gradually as the radius increases from left-hand circular, to linear, to right-hand circular polarization. Following a similar procedure, the polarization structure of the lower half varies gradually along the azimuthal direction. Such a focal field with polarization gradients may have applications in the optical manipulation of particles via the polarization modulation alone [24]. The examples presented so far are results of 2D shaping only within the focal plane. However, our method can also enable an iterative search between the incident plane and multiple planes within the focal volume. By doing so, the polarization and intensity distribution of the 3D focal volume can be
controlled. The following example demonstrates a configuration of the field distribution at different axial positions within the focal volume. Three transverse planes in the focal volume were assigned specific structures marked by the patterns of numbers 1, 2, 3, 4, 5, and 6, as shown in Fig. 8. The vertical linear polarization was assigned to “1, 2, and 3” and the horizontal linear polarization was assigned to “4, 5, and 6”. After focusing the optimized incident light by the lens, the intensities of a series of transverse planes was recorded. The first row in Fig. 8 corresponds to the resulting intensity distribution of the beams captured by the CCD with no analyzer. The second and third rows correspond to the intensity after a horizontally and vertically aligned analyzer was employed, respectively. This result demonstrates the potential of our approach for polarization multiplexing and intensity tailoring in a 3D focal volume. Note that the focusing condition at a low NA was used in this paper, which corresponds to the paraxial propagation of the light. On the other hand, tight focusing of light beams is needed in many applications. Under tight focusing, the vectorial property of the field will change when the light passes from the entrance plane of the objective to the exit plane. If the vectorial relation between fields of the entrance plane and exit plane is incorporated into the iterative search of the desired focus, our proposed algorithm could be extended to tight focusing with a high NA. Hence, further investigations are underway. 4. Conclusion
Fig. 8. Intensity distributions for the serial planes parallel to the focal plane. Top to bottom: experimental results with and without a polarization analyzer show that the light field in each image appeared in the specified position polarized as expected. The vertical linear polarization was assigned to “1, 2, and 3” and the horizontal linear polarization was assigned to “4, 5, and 6.” 790
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To conclude, we have demonstrated the possibility of generating a focal volume with the desired distribution of both the intensity and the polarization state. Our approach confirms that the modulation of both the polarization and the phase of incident light
can enable tailoring of the vectorial field. We believe that combining the modulation of both polarization and phase in such a way as demonstrated here would greatly facilitate the versatility and application of optical field manipulation. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11074116, 10934003, and 11274158) and the National Basic Research Program (973 Program) of China (Grant No. 2012CB921900). References 1. F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping Applications (Taylor & Francis, 2006). 2. M. Gu, Advanced Optical Imaging Theory (Springer, 2000). 3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). 4. N. Sanner, N. Huot, E. Audouard, C. Larat, J. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. 30, 1479–1481 (2005). 5. I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271, 40–47 (2007). 6. K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008). 7. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002). 8. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009) (and references therein). 9. S. N. Khonina and I. Golub, “Enlightening darkness to diffraction limit and beyond: comparison and optimization of different polarizations for dark spot generation,” J. Opt. Soc. Am. A 29, 1470–1474 (2012). 10. K. Hu, Z. Chenand, and J. Pu, “Generation of super-length optical needle by focusing hybridly polarized vector beams through a dielectric interface,” Opt. Lett. 37, 3303–3305 (2012). 11. S. Brasselet, “Polarization-resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011).
12. A. F. Abouraddy and K. C. Toussaint, Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006). 13. O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009). 14. W. T. Tang, E. Y. S. Yew, and C. J. R. Sheppard, “Polarization conversion in confocal microscopy with radially polarized illumination,” Opt. Lett. 34, 2147–2149 (2009). 15. M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650–2656 (2006). 16. H. P. Urbach and S. F. Pereira, “Field in focus with a maximum longitudinal electric component,” Phys. Rev. Lett. 100, 123904 (2008). 17. B. R. Boruah, “Lateral resolution enhancement in confocal microscopy by vectorial aperture engineering,” Appl. Opt. 49, 701–707 (2010). 18. W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12, 045707 (2010). 19. J. Lin, O. G. Rodríguez-Herrera, F. Kenny, D. Lara, and J. C. Dainty, “Fast vectorial calculation of the volumetric focused field distribution by using a three-dimensional Fourier transform,” Opt. Express 20, 1060–1069 (2012). 20. H. Chen, Z. Zheng, B. F. Zhang, J. Ding, and H. T. Wang, “Polarization structuring of focused field through polarization-only modulation of incident beam,” Opt. Lett. 35, 2825–2827 (2010). 21. B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using fast Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009). 22. J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. 36, 1341–1343 (2011). 23. H. Chen, J. Hao, B. F. Zhang, J. Xu, J. Ding, and H. T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. 36, 3179–3181 (2011). 24. O. Angelsky, A. Bekshaev, P. Maksimyak, A. Maksimyak, S. Hanson, and C. Zenkova, “Orbital rotation without orbital angular momentum: mechanical action of the spin part of the internal energy flow in light beams,” Opt. Express 20, 3563–3571 (2012).
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We propose a method to generate a vectorial focal field with reconfigurable distributions for both the intensity and polarization state. The three-dimensional focal volume was configured by modulating the phase and polarization of the incident light. The incident light yielding the desired field was determined based on an iterative scheme involving vectorial diffraction calculations and fast Fourier transforms. Optical experiments on vectorial field shaping were performed to validate the feasibility of our method. This method may have applications in optical tweezers, such as for realizing the optical manipulation of particles via polarization modulation in addition to phase control. © 2014 Optical Society of America OCIS codes: (050.1960) Diffraction theory; (260.5430) Polarization; (120.5060) Phase modulation. http://dx.doi.org/10.1364/AO.53.000785
1. Introduction
Focus shaping is applicable in a variety of fields ranging from laser-based material processing to microscopic imaging [1–6]. The common procedure for focus shaping is to configure the phase and/or amplitude profiles of an incident beam so that it yields the desired irradiance distribution in the focal volume. Recently, vector beams with space-variant polarization have drawn much attention as they are able to significantly influence the focusing behavior of light [7–10]. Microscopic optics can greatly benefit from the polarization engineering of light beams in this way [11–13], and consequently, there is a growing interest in not only adjusting the overall intensity distribution, but also in manipulating the polarization state distribution in the focal volume [14–19]. In a previous paper, we reported on the use of polarization-only modulation of incident beams to implement focus shaping [20]. The study had demonstrated the functionality of polarization 1559-128X/14/040785-07$15.00/0 © 2014 Optical Society of America
manipulation. On the other hand, it is well known that the tailoring of the wavefront (or phase profile) of a light beam is preferred for focus shaping because, normally, the phase plays the most important role in controlling the propagation behavior of the light field. Therefore, it is natural to expect that the combined modulation of the polarization and phase would provide more elaborate and versatile structuring of the focus field. This work demonstrates the possibility of simultaneous tailoring of both polarization and phase. Moreover, the previous work could only achieve a centro-symmetric irradiance pattern with respect to the optical axis across the transverse plane in the focal volume. This was due to the fact that the polarization-only modulation employed in the previous work allowed only real-valued modulation to the incident beams. The synthesized incident beams had no phase profile and thus yielded a Hermitian distribution in the focal field. In this paper, we build on this work and incorporate phase modulation in addition to polarization modulation into the vector beams so that they can generate the desired intensity distribution and polarization state in the focal volume. We propose an iterative 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
785
scheme for realizing specific polarization and intensity configurations in the focal volume by optimizing the modulation of both the polarization and phase of the incident beams. 2. Principles
Consider the geometry shown in Fig. 1. An incoming light beam with an elaborately structured phase profile and polarization state is focused by a lens to produce the desired distribution in the focal volume. The proposed scheme enables light beams to be spacevariantly modulated in both phase and polarization, which is realized in an optimization procedure for achieving the prescribed focal field. We started with an inhomogeneously polarized incident beam across whose cross section a space-variant linear polarization and an arbitrary phase profile were distributed. For simplicity, we assumed a uniform amplitude distribution across the entrance plane of the focusing lens (although our approach allows for amplitude modulations of the beam as well). We employed a circular polarization basis to express the incident vector beam, as follows:
1 x;y;z
1 E eiδ− Ei x; y; z p0 eiδ x;y;z −i i 2 1 0 δ x;y;z−δ− x;y;z cos 2 δ x;y;zδ− x;y;z @ 2 E0 ei A − x;y;z sin δ x;y;z−δ 2 cos αx; y; z ; (1) E0 eiβx;y;z sin αx; y; z
⇀
where δ x; y; z and δ− x; y; z are the phase factors imposed on the left- and right-hand circular polarization components, respectively, E0 is the amplitude of the incident light, αx; y; z δ x; y; z − δ− x; y; z∕2, and βx;y;z δ x;y;z δ− x;y;z∕2. The last equality of Eq. (1) indicates that the incident beam has a global phase factor represented by
Fig. 1. Focusing of incident light beams with space-variant phase and polarization.
vectorial diffraction integral that can be written in terms of a Fourier transform (FT) [20–22]. For the sake of simplicity, we restricted our attention to the paraxial focusing in which the longitudinal component of the focal field can be omitted. Following Eq. (1), calculations of the field distribution were performed by decomposing the circular polarization into a pair of orthogonal basis vectors, the left- and righthand circular polarization components. Letting the left- and right-hand circular polarization compo⇀
nents of the incident field be denoted by Ei x0 ; y0 ⇀
and Ei− x0 ; y0 , respectively, h i 8⇀ 1 < Ei x0 ; y0 E0 x0 ; y0 −i h1i ; : ⇀ Ei− x0 ; y0 E0− x0 ; y0 i
p and where E0 x0 ; yp 0 E0 ∕ 2 expiδ x0 ; y0 E0− x0 ; y0 E0 ∕ 2 expiδ− x0 ; y0 denote the complex amplitudes of polarization components. ⇀
Accordingly, the focal field E at the point x; y; z with respect to the focus can be described by the leftand right-hand circular polarization components ⇀
⇀
E x; y; z and E− x; y; z, which are related to the incident field as below [22]
8 RRR < E x; y; z A E0 x0 ; y0 exp i 2π λf xx0 yy0 zz0 Px0 ; y0 ; z0 dx0 dy0 dz0 : : E x; y; z A R R R E x ; y exp i 2π xx yy zz Px ; y ; z dx dy dz − 0− 0 0 0 0 0 0 0 0 0 0 0 λf
βx; y; z and a polarization angle of αx; y; z with respect to the x direction at the plane x; y. The polarization direction and phase factor of the incident beam are two free parameters that need to be optimized during the iterative procedure to generate the desired vectorial focus. It should be noted that such space-variant (phase and polarization) beams can be generated by our experimental setup. The focal field of a monochromatic light beam passing through an aplanatic lens is calculated by the 786
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2
3
Here, A is a multiplicative factor with its coordinate dependence on x; y; z (omitted for brevity). Px0 ; y0 ; z0 is the pupil function expressed by Px0 ; y0 ; z0 δ
q x20 y20 z20 − f
(4)
with f denoting the focal length of the lens and δ representing the Dirac function. The total field is
then obtained by the sum of the two components, i.e., ⇀
Ex; y; z E x; y; z1; −iT E− x; y; z1; iT (T represents the transpose of the matrix). Equation (3) establishes the relationship between the incident light and the focal field, and it can be calculated quickly by using the fast Fourier transform (FFT) algorithm. It should be noted that the triple integral in Eq. (3) enables the three-dimensional (3D) focus calculation as well. The 3D integral determines the field within the 3D region centered at the focus. Multiple transverse planes at different axial positions with respect to the focus can be calculated quickly by the two-dimensional (2D) FT because the δ function in Eq. (4) can reduce the number of integration variables from three down to two, as expressed by
controlling a set of prescribed components with special phase differences between them. For brevity, hereinafter we will omit the coordinate dependence of the amplitude distribution unless otherwise specified. The phase factors of the left- and right-hand circular polarization components of the incident light, δ and δ− , are taken as free parameters that need to be modified to create the desired field. We designed an iterative procedure to search for a set of optimal parameters δ and δ− that realize the desired field. The flow chart of the iterative algorithm is depicted in Fig. 2. The iterative process starts with specifying a set of randomly distributed phases to the circular components of the incident beam [i.e., the initial values of δ and δ− in Eq. (1) are uniform random numbers in (0, 2π)]. The diffraction
8 p RR f 2 −x20 y20 > > E x; y; z A E x ; y exp i2π z exp i 2π < −∞ 0 0 0 λf λf xx0 yy0 dx0 dy0 p : RR f 2 −x20 y20 > 2π > : E− x; y; z A −∞ E0− x0 ; y0 exp i2π z exp i λf xx0 yy0 dx0 dy0 λf
Our scheme simulates an iterative process between the forward and backward propagations that relate the incident field and the focal field, which is analogous to the Gerchberg–Saxton (GS) algorithm for retrieving the phase of a pair of light distributions. When applied to shaping a light beam, the GS algorithm is often used to calculate the phase pattern that light in one plane must have in order to form a desired intensity pattern in a second plane, the light distribution on that is related to that in the first plane by a propagating function such as the FT. With the help of the FFT algorithm, which can be implemented efficiently on a computer, the field in a plane at a given axial position can be obtained by a 2D FT. The proposed approach, with modification to the vectorial field, was extended to calculating the phase patterns of the polarization components of the incident light that shape the desired field distribution within the focal volume. The iterative scheme requires the calculation of a stack of planes at different axial positions [i.e., different values of z in Eq. (5)] with respect to the focus for the entire 3D complex amplitude distribution. It is noted that our method allows shaping of not only the light intensity but also the polarization state of the focal field. The purpose of this is to construct a focal volume that has a specific intensity distribution and a space-variant polarization distribution. It is known that the intensity of the optical field is determined by the absolute value of the polarization components, whilst the polarization state of the field depends on the phase difference between these components. Therefore, we aimed to achieve the desired optical field by
5
integrals in Eq. (5) are then calculated using the FFT, which yields the circular components of the focal field. Absolute values and phase differences of the calculated components in the focal field are replaced by the ones of the prescribed polarization components (i.e., A , A− , and γ in Fig. 2), while the global phase values are left unchanged (i.e., φ remains unchanged). New phase factors in the components of the incident beam are obtained through an inverse FFT by letting the reshaped focal field backward propagate to the entrance plane. We imposed a constraint on the renewal process such that the two circular components of the incident beampshould have the same and constant amplitudes E0 ∕ 2. The iterative search continues until a satisfactory focal field is found. In short, the iterative process optimizes three free parameters (i.e., the phase factors of the polarization components of the incident light, δ , and δ− , and the global phase of the focal field, φ ) in order to achieve the predetermined polarization components as well as the phase difference between them within the focal region, which are represented by A, A− , and γ, respectively, in Fig. 2. It should be emphasized that the above explained algorithm yields a space-variant distribution of locally linear polarization (i.e., a polarization state on the equator of the Poincaré sphere) because the circular basis representation is employed in the calculation [cf. Eq. (1)]. However, the algorithm can be modified into the linear polarization basis representation so that we can get a hybrid modulation of locally linear, circular, and elliptical polarization. In this way, the x and y linear components are used 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
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3. Numerical Simulation and Experimental Realization
Fig. 2. Flow chart for iteratively searching for the optimal polarization and phase modulation of the incident light that can yield the desired focal field.
to construct the polarization distribution and thus the synthesized polarization states locate on a great circle crossing over the south and north poles of the Poincaré sphere.
In the subsequent numerical and experimental examples, all the incident light beams are specified as having the uniform intensity and the space-variant phase and polarization. The parameters used in the simulation process were chosen as: a 0.12 NA objective lens with a focal length of 200 mm and a laser light of wavelength 532 nm. The parameters of the focusing lens were chosen so that a CCD camera can detect a region-of-interest field in the focal plane, wherein a special polarization and intensity structure appears. This focusing geometry can afford a lateral resolution of λ∕2 NA 2.2 μm and a axial resolution of 2λ∕NA2 74 μm. A simulation of the results obtained by performing the above iterative procedure is presented in Fig. 3. An optimized incident light beam of wavelength 532 nm passing through a lens with a 0.12 NA creates, in the focal plane, three intensity rings each with different polarization states. The radii of the three rings were 20, 40, and 60 μm. The inner, middle, and outer rings were set to be right-hand circularly polarized, x linearly polarized, and left-hand circularly polarized, respectively. The synthesized incident beam can be generated by an experimental setup that produces vector beams with arbitrary space-variant modulations of both phase and polarization. In this setup, a holographic grating pattern displayed at a spatial light modulator (SLM) was designed such that it transformed the incident light into the beam modes with the desired amplitude and phase at the first positive and minus diffraction orders. After being filtered by a spatial filter, the two diffraction orders were transformed by each of the two wave-plates into the right- and left-hand circularly polarized lights.
Fig. 3. Simulation results of three intensity rings in the focal plane. The inner, middle, and outer rings have right-hand circular, x linear, and left-hand circular polarizations, respectively. (a) Intensity of the focal plane. (b) Phase difference of the right- and left-hand circular components. (c) and (d) Respective intensities of the right- and left-hand circular polarization components. 788
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Each of the two circularly polarized components acquired a specific phase factor from the diffraction of the specially designed holographic grating. Two circularly polarized components were then collinearly recombined by a grating to create the desired vector beams. The experimental setup takes advantage of a SLM (Holoeye Pluto with 1920 × 1080 pixels resolution) that is controlled by a computer, which enables the dynamic generation of the vector beams (refer to [23] for full details). The overall efficiency of the system was about 1.2% due to the loss of energy in the spatial filter, SLM, and grating. The light efficiency can be improved by using a more efficient modulation manner of SLM, e.g., by using the phase modulation rather than the amplitude modulation to write the grating pattern. In our proof-of-principle demonstration, however, the light efficiency was not the primary concern. In this paper, the experiment was performed to demonstrate the feasibility and capability of the phase and polarization modulation in the focus shaping. An experimental demonstration of the synthesized vector beam corresponding to Fig. 3 is presented in Fig. 4. A CCD camera having 1032 × 779 pixels with a pixel pitch of 4.65 μm of the vector beam in the focal plane. The leftmost pattern of Fig. 4 gives the intensity distribution in the focal plane and subsequent images were taken after a linear polarization analyzer. The arrows in the inset of each image indicate the polarization direction of the analyzer. In the rightmost pattern, the middle ring is faint because the x linear polarization within this ring region was extinguished by the vertical analyzer. In the above example, the polarization state remains uniform within each ring region. We now give another example of the spatially variant polarization
distribution in the focal field, as shown in Fig. 5. In this example, the inner and the outer rings are the right- and left-hand circular polarization, respectively, while the middle ring is set to the radial polarization. Radii of the three rings were 20, 30, and 40 μm. Phase difference of the right- and lefthand circular components in the focal plane is shown in the leftmost panel of Fig. 5, and experimentally measured intensity maps about the focused field are presented in the remaining panels. The intensity patterns were recorded after a linear analyzer oriented at 90, 0, 45, and 135°. Note that speckle-like patterns can be seen in the picture. In addition to the noise due to coherent illumination, we ascribe this phenomenon to the residue errors of the iterative search procedure, wherein not a complete exact construction of the desired focus but an optimal solution was achieved. Figure 6 presents the experimental results of another vectorial focal field that was configured to display playing card patterns. Four shapes corresponding to the suits of playing cards [spades (♠), hearts (♥), clubs (♣), and diamonds (♦)] and with different polarizations were assigned to the four quadrants of the focal plane. The four polarization states were horizontal linear, vertical linear, right-hand circular, and left-hand circular (polarization). A modification to our proposed algorithm was made in the next example, which demonstrates a generation of the polarization gradient in the focal field. The polarization distribution in this example was no longer limited to the polarization state corresponding to the equator on the Poincaré sphere, which is shown in the above examples. Instead, it spans a complete meridian, crossing from the south to the north pole of the sphere. Figure 7 shows a
Fig. 4. Experimentally generated vectorial field corresponding to Fig. 3. The inner, middle, and outer rings are right-hand circularly polarized, x linearly polarized, and left-hand circularly polarized, respectively. From left to right: intensity distribution recorded without an analyzer, and with an analyzer oriented at 0°, 45°, and 90° (as indicated by the arrow insets).
Fig. 5. Phase distribution and experimental results that perform the spatially variant polarization distribution of the focal field. Inner ring and outer ring show right- and left-hand circular polarization, respectively. Middle ring indicates a radially polarized beam. The arrows show the orientation of the analyzer. 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
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Fig. 6. Recorded focal intensities of the four playing card patterns described in the text. From left to right: no analyzer, horizontal linear, vertical linear, right-hand circular, and left-hand circular analyzers have been employed (as indicated by the arrows insets).
Fig. 7. Experimental demonstration of a polarization gradient distribution in the focal field. From left to right: no analyzer, right-hand circular, and left-hand circular analyzers have been applied (as indicated by the arrows insets).
ring-shaped focal field divided into two halves. The polarization state of the upper half of the ring varies gradually as the radius increases from left-hand circular, to linear, to right-hand circular polarization. Following a similar procedure, the polarization structure of the lower half varies gradually along the azimuthal direction. Such a focal field with polarization gradients may have applications in the optical manipulation of particles via the polarization modulation alone [24]. The examples presented so far are results of 2D shaping only within the focal plane. However, our method can also enable an iterative search between the incident plane and multiple planes within the focal volume. By doing so, the polarization and intensity distribution of the 3D focal volume can be
controlled. The following example demonstrates a configuration of the field distribution at different axial positions within the focal volume. Three transverse planes in the focal volume were assigned specific structures marked by the patterns of numbers 1, 2, 3, 4, 5, and 6, as shown in Fig. 8. The vertical linear polarization was assigned to “1, 2, and 3” and the horizontal linear polarization was assigned to “4, 5, and 6”. After focusing the optimized incident light by the lens, the intensities of a series of transverse planes was recorded. The first row in Fig. 8 corresponds to the resulting intensity distribution of the beams captured by the CCD with no analyzer. The second and third rows correspond to the intensity after a horizontally and vertically aligned analyzer was employed, respectively. This result demonstrates the potential of our approach for polarization multiplexing and intensity tailoring in a 3D focal volume. Note that the focusing condition at a low NA was used in this paper, which corresponds to the paraxial propagation of the light. On the other hand, tight focusing of light beams is needed in many applications. Under tight focusing, the vectorial property of the field will change when the light passes from the entrance plane of the objective to the exit plane. If the vectorial relation between fields of the entrance plane and exit plane is incorporated into the iterative search of the desired focus, our proposed algorithm could be extended to tight focusing with a high NA. Hence, further investigations are underway. 4. Conclusion
Fig. 8. Intensity distributions for the serial planes parallel to the focal plane. Top to bottom: experimental results with and without a polarization analyzer show that the light field in each image appeared in the specified position polarized as expected. The vertical linear polarization was assigned to “1, 2, and 3” and the horizontal linear polarization was assigned to “4, 5, and 6.” 790
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To conclude, we have demonstrated the possibility of generating a focal volume with the desired distribution of both the intensity and the polarization state. Our approach confirms that the modulation of both the polarization and the phase of incident light
can enable tailoring of the vectorial field. We believe that combining the modulation of both polarization and phase in such a way as demonstrated here would greatly facilitate the versatility and application of optical field manipulation. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11074116, 10934003, and 11274158) and the National Basic Research Program (973 Program) of China (Grant No. 2012CB921900). References 1. F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping Applications (Taylor & Francis, 2006). 2. M. Gu, Advanced Optical Imaging Theory (Springer, 2000). 3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). 4. N. Sanner, N. Huot, E. Audouard, C. Larat, J. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. 30, 1479–1481 (2005). 5. I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271, 40–47 (2007). 6. K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008). 7. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002). 8. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009) (and references therein). 9. S. N. Khonina and I. Golub, “Enlightening darkness to diffraction limit and beyond: comparison and optimization of different polarizations for dark spot generation,” J. Opt. Soc. Am. A 29, 1470–1474 (2012). 10. K. Hu, Z. Chenand, and J. Pu, “Generation of super-length optical needle by focusing hybridly polarized vector beams through a dielectric interface,” Opt. Lett. 37, 3303–3305 (2012). 11. S. Brasselet, “Polarization-resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011).
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