Complete shaping of optical vector beams - OSA Publishing

Complete shaping of optical vector beams Zhaozhong Chen, Tingting Zeng, Binjie Qian, and Jianping Ding* National Laboratory of Solid State Microstructures and School of physics, Nanjing University, Nanjing 210093, China * [email protected]

Abstract: We propose and experimentally demonstrate the complete and simultaneous modulation of the amplitude, phase and arbitrary state of polarization of optical beams. Based on a 4-f system including a spatial light modulator (SLM), two orthogonally polarized beams serving as the base vector components are produced by a computer generated hologram. The complex amplitude of orthogonal components is realized by a macropixel encoding technique purposely designed for phase-only SLMs. Vector beams can be created from the coaxial superposition of the two base beams. This enables us to design optical fields with arbitrarily structured amplitude, phase and polarization by using only one SLM, and thus provides an easyto-implement route for exploring the novel effects and expanding the functionality of vector beams with space-variant parameters. ©2015 Optical Society of America OCIS codes: (140.3300) Laser beam shaping; (260.5430) Polarization; (120.5060) Phase modulation; (230.6120) Spatial light modulators.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

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Received 29 Apr 2015; revised 23 Jun 2015; accepted 23 Jun 2015; published 29 Jun 2015 13 Jul 2015 | Vol. 23, No. 14 | DOI:10.1364/OE.23.017701 | OPTICS EXPRESS 17701

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1. Introduction The primary physical quantities describing the shape of a vector optical field include the amplitude, phase, and polarization. Vector optical fields with elaborately structured amplitude, phase and polarization distribution have proved useful for a wide variety of applications. Spatial intensity structure of optical field has been studied to build dynamic holographic tweezers system [1]. Vortex beams with spiral phases have been the subject of considerable interest in optical tweezers and quantum information processing, owing to their carrying orbital angular momentum [2, 3]. Shaping both the amplitude and the phase of optical field yielded non-diffracting beams such as the Airy beams and Bessel beams [4, 5]. Furthermore, tailoring the polarization structure of vector beams has drawn a lot of attention in recent years [6], due to their unique properties in various scientific and engineering realms, such as focus shaping [7, 8], optical trapping [9], surface plasmon sensing [10], and laser processing [11]. An even more interesting issue is the simultaneous structuring of the amplitude, phase, and polarization profiles of vector field for special purpose such as enabling a light beam to exert forces and torques on illuminated objects [12, 13]. These demands in optical science and engineering research prompt the everlasting quest to find novel methods of fully controlling the shape of light. A number of schemes for modulating optical beams have been proposed, among which the use of spatial light modulators (SLMs) is appealing in recent years because it has the advantage of providing dynamic and programmable modulations of, e.g., amplitude and phase [14, 15], phase and polarization [16, 17]. Complete control of an optical vector beams requires four independent modulation degrees of freedom, one for the amplitude, one for the phase retardation, and two for the polarization, which can be represented by a point on the surface of the Poincaré sphere (PS). However, due to most commercial SLMs can only or mostly modulate either the phase or the amplitude of light, it still remains a great challenge by using only single SLM to control the amplitude, phase and polarization of optical beams independently and simultaneously. There exist methods for achieving four-parameter modulation, by using two SLMs [18], by multiple-section configuration [19], or by double#239877 © 2015 OSA


pass architecture [20], all of which required either cascaded systems or multiple impingements of light on SLM’s sections, resulting in complication of optical adjustment and more sensitivity to optic misalignments. In this work, we propose an approach to using just one single SLM for achieving the complete and simultaneous control of an optical beam over the amplitude, phase and arbitrary SoP. The SoP of generated beams can sweep across the entire surface of the PS. Based on a 4f system including a SLM, the vector beams with space-variant shape are created from a coaxial superposition of two orthogonally polarized components. Each polarized component carries structured amplitude profile and phase distributions which are produced from the SLM by employing a macro-pixel encoding scheme, resulting in four free modulation parameters. By doing so, the amplitude, phase and polarization distributions of vector beams with spacevariant shape can be tailored independently and simultaneously by electrically adjusting the SLM. The benefits of our scheme include the great flexibility in terms of the controllability over the space-variant amplitude, phase as well as SoP that can span the entire surface of the PS, and the capability of dynamic modulation. Therefore, our approach is promising in expanding the functionality of vector beams as well as in exploring their new applications. 2. Principle 2.1 Characterization of vector beams  In a plane (x, y) transverse to the direction z of propagation, a vector optical beam E ( x, y ) can always be expressed in the form of Jones vector by two orthogonal components as below [21],

  Ex ( x, y )   Ax ( x, y ) exp(iφx ( x, y ))  E ( x, y )=   =  ,  E y ( x, y )   Ay ( x, y ) exp(iφ y ( x, y )) 

(1)

where Ax ( x, y ) and Ay ( x, y ) are the amplitude distribution of x and y component, respectively, and φx ( x, y ) and φ y ( x, y ) are the phase factor of respective component. The Jones vector

defines a state of polarization (SoP) by the relative amplitude and phase of the two constituent polarization components. A SoP can also be geometrically represented as a point on the surface of the Poincaré sphere (PS), whose center is at the center of a Cartesian coordinate system defined by the Stokes parameters (S1, S2, S3) as below S1 = Ax2 − Ay2 , S 2 = 2 Ax Ay cos δ ,

(2)

S3 = 2 Ax Ay sin δ ,

where δ = φ y − φx . In this way, the SoP is determined by the latitude angle 2χ and longitude angle 2ψ of a point on the PS, as shown in Fig. 1, with 1 2

 S3  S0

χ = sin −1 

 , 

S  1 ψ = tan −1  2  , 2  S0 

where S0 = S12 + S22 + S32 =

-

π 4

≤χ≤

π 4

,

(3)

0 ≤ψ < π ,

Ax2 + Ay2 . It is obvious that the representation of a SoP needs

two free parameters. Furthermore, the full representation of a vector beam must also take into account its module (or intensity) and global phase. To make it clearer, we can rewrite Eq. (1) as below

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 cos α ( x, y )   E ( x , y ) = S 0 ( x, y ) e i β ( x , y )  , iδ ( x , y )  sin α ( x, y ) e 

(4)

where β ( x, y ) ( = φx ( x, y ) ) is the global phase and α ( x, y ) = tan −1 ( Ay ( x, y ) Ax ( x, y ) ) . It can be seen from Eq. (4) that four free parameters are needed to fully describe the vector beams, i.e. two for specifying the overall amplitude and phase and two for the SoP. Consequently, the complete manipulation of a vector beam requires an independent and simultaneous control over the four parameters. On the other hand, and perhaps more importantly, their control should be dynamic and space-variant. This requirement naturally invokes the use of liquid-crystal SLMs, which, working in a pixilated form, are the most prominent building blocks of many of today’s state-of-the-art electro-optical systems. However, to date commercial SLMs usually afford one modulation parameter (phase or amplitude, depending on its modulation mode) [22–24]. A means to overcome this issue is to employ the macro-pixel encoding scheme for a single parameter modulated (e.g., amplitudetype or phase-type) SLM, as outlined next.

Fig. 1. (a) Poincaré sphere and (b) different SoPs on Poincaré sphere.

2.2 Macro-pixel encoding method for complex amplitude Our approach is based on the principle of superimposing orthogonally polarized beams with spatially varying complex amplitude distributions. The complex amplitude is created from a pixelated SLM that is written with a computer-generated hologram pattern. As most of commercial SLMs can modulate only the phase of optical field, encoding the complex amplitude of base vector beams with the help of phase-only computer-generated hologram (CGH) technique is a feasible choice [25–27]. Assume a complex amplitude Amn exp(iφmn ) to be carried by the light field coming from the (m, n) th macro-pixel of a SLM. This complex amplitude can be represented by a sum of two complex quantities having a constant 1 2 module a but different phases φmn and φmn , i.e. 1 2 Amn exp(iφmn ) = a exp(iφmn ) + a exp(iφmn )

(5)

This complex amplitude decomposition can be illustrated in the complex plane as in shown Fig. 2. Obviously, we have a = max ( Amn ) ) 2 , because the maximal value of Amn is 2a when the two constituent phasors are in phase.

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Fig. 2. Phasor diagram of decomposing a complex amplitude Amn exp(iφmn ) into two constituent phasors ( a exp(iφmn ) and a exp(iφmn ) ). 1

2

For a macro-pixel (m, n) in the SLM plane there are different ways to arrange φmn1 and φmn 2 in its sub-pixels, e. g. one-dimensional symmetrical side-by-side structure [25] or twodimensional embedding structure [26]. Taking into consideration the two-dimensional symmetry and modulation accuracy, we choose the centrosymmetric scheme as shown in Fig. 3. In this structure, a macro-pixel is composed of four sub-pixels; then the complex amplitude Amn exp(iφmn ) is constructed at the macro-pixel (m, n) by its four sub-pixels in the phase-only form. After this encoding procedure we get a phase-only CGH that have the ability to encode arbitrary phase and amplitude distributions.

Fig. 3. Symmetric structure of the (m, n)-th macro-pixel synthesized from four sub-pixels with phase-only terms

1 φmn

and

2 φmn .

2.3 Creation of arbitrary vector beams The experimental arrangement for creating arbitrary vector beams is similar to that of our previous works [16], which is shown in Fig. 4. The collimated linear polarized light illuminates the SLM on which a two-dimensional holographic grating (HG) is displayed. In current work, the gratings along two orthogonal directions carry their respective offset phase which is encoded though the macro-pixel structure shown in Fig. 3. The transmission function of the SLM with the size of 2N × 2N sub-pixels is expressed in the pixelated form by,

{

}

1 1  j j  + cos  2π f 0 n ( 2Δ ) + ϕ yi ,, mn   , t i , j (m, n) =  + cos  2π f 0 m ( 2Δ ) + ϕ xi ,, mn 2 4  

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(6)


where f 0 is the spatial frequency of HG, Δ is the pixel pitch, and (i, j ) ∈ [1, 2] denote the sub-

pixel indices in the (m, n) th macro-pixel ( (m, n) ∈ [1, N ] ). In the above grating transmittance j j and ϕ y,i , mn , as expression, we set the offset phases in the x- and y-directional grating, ϕ xi ,, mn j ϕ xi ,, mn = φx1, mnδ ij + φx2, mn (1 − δ ij ) ,

j = φ 1y , mnδ ij + φ y2, mn (1 − δ ij ) , ϕ yi ,, mn

(7)

where δij denoting the Kronecker delta, i. e. δij = 1 when i = j and δij = 0 when i≠j. A 4-f system consisting of a pair of lens is arranged behind the SLM. After the SLM a linear polarized light is diffracted into various diffraction orders, among which only the two + 1st orders are allowed to pass through a spatial filter located at the focal plane of the first lens. By means of macro-pixel encoding illustrated in the foregoing, the two + 1st orders (in x- and y-direction) diffraction light leaving from the SLM are endowed with complex amplitudes, Ax , mn exp ( iφx , mn ) and Ay , mn exp ( iφ y , mn ) , respectively, that are required by the two orthogonal polarization components in Eq. (1). Two wave plates (half wave plate - HWP or quarter wave plates - QWP) covert the two passing orders into two mutually orthogonal polarization components (linear or circular), which serve as a pair of base vector beams for the subsequent superposition process. The two base vector beams are collinearly recombined at the rear focal plane of the second lens by a Ronchi grating whose period is 84 μm and matches with that of the HG. In this way, the fields incident on the Ronchi grating are coming from the two + 1st orders of the HG on the SLM, forming the output beam which is the superposition of the two base components. The two HWPs are used to produce a pair of linear polarization components, while the two QWPs are used for the circular components. The Ronchi grating may be replaced by other elements, such as Wollaston prisms, to realize the recombining of light, but the alignment complexity of optical path will be increased.

Fig. 4. Schematic of the experimental setup.

As illustrated in Fig. 3, the module and phase of arbitrary complex amplitude of the light at one macro-pixel can be simultaneously modulated in term of phase-only encoding from four sub-pixels. Consequently, four parameters of the two orthogonal polarization components, Ax , Ay , φx and φ y , can be independently tailored. In this way, an arbitrary

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spatially variant polarization beam that spans all possible states of polarization, which is recently referred to the full-Poincare beam [28], can be obtained. Meanwhile, both the overall amplitude and phase of the vector beam can also be modulated (cf. Equation (4)). In short, by using the single SLM, we achieve the independent and dynamic manipulation of the amplitude, phase and SoP of full-Poincare vector beams. 3 Experimental results and discussions

Fig. 5. Experimental results of optical vortex field with circular and square intensity distribution.

For validating the feasibility of the proposed approach we present examples demonstrating the capability of simultaneously manipulating the amplitude, phase and SoP of vector beams. First, we design an experiment to demonstrate the encoding ability of complex amplitude through the macro-pixel method. It is well known that an optical vortex yields, in the far field, a dark hollow encompassed by a bright ring whose radius is proportional to the vortex’s topological charge (TC) [29]. However, if the amplitude and the phase distribution of optical vortices is be purposely modulated at the same time, the size of bright intensity ring will not change with the TC of vortex phase, forming the so-called perfect vortex [30]. In our experiment we demonstrate that the intensity distribution of optical vortex field can be designed arbitrarily. We design two special optical vortices, each of which has a far field intensity distribution with a shape of circle or square. The TC values are l = 5 , l = 10 and l = 15 for each vortex. The steps to get the desired vortices in experiments are: (1) to calculate the Fourier transform of the desired field to obtain the initial field; (2) to encode the complex amplitude of the initial field using macro-pixel method, forming a holographic function; (3) to display the holographic pattern on the SLM and to record the reconstructed field in the rear focal plane of a lens with the focal length of 40 cm. The experimental results of perfect vortices are shown in Fig. 5. We can see that the intensity distribution does indeed not change with the TC values. In order to get insight into the phase profile of the created field, the interference pattern between the designed field and a reference beam with plain phase are also given in Fig. 5.

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Second, we want to verify the adaptability of the proposed method in modulating the SoPs of optical beams by presenting experimental results on full-Poincare beams. Figure 6(a) shows the polarization structure in the beam’s cross-section plane (x, y), wherein the distributions of the latitude angle 2 χ ( x, y ) and the longitude angle 2ψ ( x, y ) is space-variant. From the top to the bottom of the cross-section, the SoP varies gradually from the right circular polarization to the right elliptical polarization, linear polarization, left elliptical polarization, and finally left circular polarization, and correspondingly the latitude angle changes from π/2 to –π/2. In addition, from the left to the right of the cross-section, the longitude angle changes from 0 to 2π. The simulated results about the Stokes parameters of the optical field are shown in Figs. 6(b)-6(d). Figure 6(e) gives the recorded intensity. Figures 6(f)-6(h) show the measured results of the Stokes parameters, which agree well with the simulated results. This example validates the proposed method can create vector beams with the space-variant SoP that covers the entire surface of the PS.

Fig. 6. Theoretical (first row) and experimental (second row) results of space-variant polarization field to be designed: (a) SoPs in the cross-section, (b-d) simulated values of the Stokes parameters, (e) recorded intensity (or S0), and (f-h) measured values of the Stokes parameters of S1, S2 and S3.

The third example is to produce the higher-order PS (HOPS) beams [31, 32]. A SoP on the HOPS can be expressed by the combination of a pair of orthogonal basis vectors,  l + A R  E =AL L (8) R −l ,  l and R  l are the orthogonal vortices with the where AL and AR are the amplitude factors, L opposite TC, defined by 1 − ilϕ 1  l = 1 eilϕ  1  , R  L e   (9)   l = 2 2  −i  i When l = −1 , the SoPs on the HOPS are shown in Fig. 7(a). The north and south poles of this higher-order PS represent right and left circular polarization vortices with the TCs of –1 and + 1, respectively. In order to demonstrate the ability of creating different HOPS SoPs at the same time, we divide the field cross-section into four regions, denoted by A, B, C and D, shown in Fig. 7(b). The respective SoPs shown in Fig. 7(b) are represented by the points denoted by the corresponding letters on the higher-order PS in Fig. 7(a). The SoP in A corresponds to the point (0, 0) on the higher-order PS, and that in B corresponds to the point (π/2, π/4). The field in area C and D are right and left circular polarized base vector respectively; in addition, the intensities in the three concentric regions are prescribed to be 1, 2 and 4 from the outer to the inner. The complex amplitudes in the respective regions can be represented by

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  1 + 0.707 L 1 , E A =0.707 R   1 + 0.541exp(iπ / 2) L 1 , E B =1.307 R  1 , E C = 2R   −1 E D = 2L

(10)

Fig. 7. (a) Higher-order PS for the TC of l = −1, (b) SoP of the field to be produced: four regions denoted by A, B, C and D carrying the SoPs marked by the corresponding letters on the HOPS. The outer annulus (A), middle annulus (B), and inner ring-shaped zone (C and D) are endowed with different intensities; the respective values are 1, 2 and 4.

Fig. 8. Theoretical and experimental results of the higher-order PS field illustrated in Fig. 7(b): (a) theoretical intensity I in the cross-section, (b-d) simulated values of the Stokes parameters, (e) recorded intensity I in the cross-section, (f-h) measured values of the Stokes parameters, (i) and (k) experimental interference patterns between the generated field with right and left circularly polarized plane waves, (j) and (l) theoretical phases carried by right and left circularly polarized components.

The SoP of the generated field is examined by the measurement of Stoke parameters and compared with numerical simulation, as shown in Figs. 8(a)-8(h). The phase structure embedded within the basis vector of the field is depicted in Figs. 8(i)-8(l), and experimentally validated by the interference pattern. The interference pattern is obtained by interfering the generated field with the circularly polarized reference wave carrying plain phase, shown in Fig. 8(i) for the right circular polarization and Fig. 8(k) for the left circular polarization, respectively. The fork fringes in Figs. 8(i) and 8(k) confirm that the two orthogonal vector vortices hold equal but opposite TCs of 1.

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The presented experimental results validate that not only the amplitude and phase but also arbitrary SoP can be manipulated in the space-variant manner simultaneously. Due to the limited diffraction efficiency of the SLM and the grating, the total efficiency of our experimental system is about 2%. However, our method enables a complete and independent control over the amplitude, phase and SoP of light by using only one SLM, and thus provides an easy-to-implement route for exploring the novel effects and expanding the functionality of vector beams with space-variant parameters. 4. Summary

In summary, we have proposed a reliable and flexible method for controlling amplitude, phase and polarization distribution of optical vector beams simultaneously. With the help of macropixel encoding method, arbitrarily structured optical beam can be dynamically created in the 4-f system including one SLM. Engineering of vector beams will facilitate a lot of applications of complex light, such as the focus shaping, high-resolution imaging and optical tweezers. Acknowledgment

This work was supported by the National Natural Science Foundation of China (11074116, 11274158 and 11474156).

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