Space-variant polarization-state manipulation with computer

Invited Paper

Space-variant polarization-state manipulation with computer-generated subwavelength gratings Erez Hasman*, Vladimir Kleiner, Gabriel Biener, Avi Niv Optical Engineering Laboratory, Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

ABSTRACT Novel methods for space-variant polarization-state manipulation using subwavelength metal and dielectric gratings are presented. By locally controlling the period and direction of the grating, we show that any desired polarization can be achieved. The methods presented in this paper are generic to any portion of the spectrum, and we present experimental demonstrations of our theory using CO2 laser radiation at a wavelength of 10.6µm. Moreover, we exploited our computer-generated subwavelength gratings to demonstrate a polarization Talbot self-imaging, as well as nondiffracting periodically space-variant polarization beams, and a unique method for real-time polarization measurement. We also present novel optical phase elements based on the space-domain Pancharatnam-Berry phase. Unlike diffractive and refractive elements, the phase is not introduced through optical path differences, but results from the geometrical phase that accompanies space-variant polarization manipulation. We intoduce and experimentally demonstrate PancharatnamBerry phase optical elements (PBOEs) based on computer-generated subwavelength gratings such as polarization beamsplitters, optical switches and spiral phase. Keywords: polarization, subwavelength gratings, ellipsometry and polarimetry, computer-generated holograms, Berry’s phase, Talbot effect.

1. INTRODUCTION The term subwavelength optical element (SWOE) refers to any optical element comprising typical structures that are smaller than the wavelength for which the element was designed. Subwavelength optical elements are usually onedimensional or two-dimensional periodic structures, such as the one dimensional periodic grating depicted in Fig. 1.1. When light is incident on a SWOE, no diffraction orders other than the zeroth order propagate, and the light displays approximately the same behavior as if it were travelling through a uniaxial or biaxial crystal characterized by effective refractive indices that depend on its structure. For any grating, the threshold period under which only the transmitted and the reflected zero orders are nonevanescent is,

Λ th =

λ n1 sin θ cos ξ + (n22

− n12 sin 2 ξ )1 / 2

,

(1.1)

where ξ is the azimuth angle relative to the grating stripes, θ is the angle of incidence, n1 and n2 are the refractive indices of the grating stripes and λ is the incident wavelength. SWOEs can be either globally periodic (space invariant) or locally periodic (space variant) and they have been applied to the fabrication of anti-reflection coatings [1], artificial refractive index distributions [2], optical filters [3], waveplates [4] and polarizers [5]. Their use can be dated back to Heinrich Hertz who in 1888 used subwavelength metal stripe gratings as polarizers to test the properties of radio waves [6]. *

[email protected]; phone 972 4 8292916; fax 972 4 8324533; www.technion.ac.il/optics.

Micromachining Technology for Micro-optics and Nano-optics, Eric G. Johnson, Editor, Proceedings of SPIE Vol. 4984 (2003) © 2003 SPIE · 0277-786X/03/$15.00

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In this introduction we present a brief review on periodic SWOEs. We review the theories used to explain these elements and calculate their effects. We discuss two distinct types of SWOEs, subwavelength metal stripe gratings and dielectric subwavelength gratings. We review some examples of their various applications and the methods by which they can be fabricated. Since the research presented in this paper relates mostly to one-dimensional periodic subwavelength SWOEs, the introduction will primarily relate to such elements, which we will refer to as subwavelength gratings.

y TM

TE

θ

Figure 1.1: Illustration of a one dimensional periodic binary subwavelength grating. The grating with period Λ comprises alternating stripes with refractive indexes n1 and n2 and widths t1 and t2 respectively. We define the duty cycle q as t1 /Λ. θ is the angle of incidence and ξ is the azimuthal angle. Note that the period is smaller than the incident wavelength λ.

ξ

x

n1 n2 t1

Λ

t2

z q=t1/Λ

1.1 The theory of subwavelength gratings Gratings play an important role in optics. Their use is at the foundation of fields such as spectroscopy and diffractive optics. Loosely speaking, when the variations of the surface relief or index modulation are slow compared to the wavelength, λ, the polarization of the incident wave can be neglected and approximate scalar theories can be used [7,8]. However, as the period of the grating decreases and becomes comparable or smaller than the wavelength, vectorial effects become more predominant and rigorous electromagnetic theories are needed. Unfortunately few rigorous analytical solutions are known, and therefore the calculation of diffraction from such gratings generally requires numerical methods [9,10]. The most commonly used method for these calculations is Rigorous Coupled Wave Analysis (RCWA) [11], which was formulated by Moharam and Gaylord. In this method the grating is approximated by a number of slices. Each slice within the grating region is treated as a thin grating, and the fields within the slice are expanded into a Fourier series using the Bloch-Floquet theorem. By equating Maxwellian boundary conditions on the fields at the interfaces between the slices, a countable set of coupled linear equations is found. Seeking the eigenvectors for these equations yields the fields within the grating. The reflection and transmission coefficients are then found from the amplitudes of the propagating orders in the regions before and after the grating. Since any continuous function can be approximated by a step function, continuos relief profiles can be modeled to any level of accuracy needed. An unfortunate setback of RCWA is that it converges slowly for metallic lamellar gratings. This problem was addressed by Lallanne and Morris [12] who reformulated the eigenproblem to achieve highly improved convergence rates, thereby extending the usefulness of RCWA related methods. However, despite the success of RCWA and other numerical approaches, they tend to be calculation-intensive and offer very little intuitive insight into subwavelegth grating problems, and therefore approximate methods are often sought. The simplest approximate model for subwavelength gratings is classical form birefringence [13]. This zero order approximation gives the effective refractive indices of a binary subwavelength grating with the geometry depicted in figure 1.1 as,

nTE = qn12 + (1 − q )n 22 , 2

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(1.2)

nTM

2

n12 n 22 , = 2 qn 2 + (1 − q)n12

(1.3)

where TE denotes light polarized parallel to the grating stripes, TM denote light polarized perpendicular to the grating stripes, n1 and n2 denote the refractive indices of the materials that comprise the grating, and q=t1 /Λ is the duty cycle of the grating, i.e. the relative portion of the material with refractive index n1 within the grating. Thus the grating is replaced with an effective layer of uniaxial crystal. If the grating period is not binary, then it is approximated with a step-function, and the effective refractive indices for each step are calculated using Eqs. (1.2, 1.3). The structure is then replaced with a multilayer stack whose properties can be calculated using transfer matrix methods [14,15]. Some very important results regarding subwavelength gratings were presented by Rytov [16]. He showed that the effective refractive indices for a subwavelength grating could be found from a pair of transcendental equations,

( n1

2

( n1

2

− nTE 2 )1 / 2 tan[π λ (1 − q )(n12 − nTE 2 )1 / 2 ] = −(n2 2 − nTE 2 )1 / 2 tan[π λ q(n2 2 − nTE 2 )1 / 2 ] , Λ Λ − nTM 2 )1 / 2 tan[ π n1

2

λ

Λ

(1 − q )( n1

2

2 2 1/ 2 − nTM 2 ) 1 / 2 ] = − ( n 2 − n2TM ) tan[ π

n2

λ

Λ

q(n 2

2

− nTM 2 )1 / 2 ,

(1.4) (1.5)

where λ is the incident wavelength and Λ is the grating period. Developing these equations into Taylor series yields the second order approximations, (2) nTE

= {[nTE( 0) ]2 + 1 [ πΛ q(1 − q)]2 (n2 2 − n1 2 ) 2 }1 / 2 ,

(1.6)

(2) nTM

1 πΛ ( 0) 2 ] + [ = {[ nTM q (1 − q )]2 (

(1.7)

3 λ

3 λ

(0)

1 2 n2

−

1 2 ( 0) 6 (0 ) 2 1 / 2 ) [nTM ] [nTE ] } , 2 n1

( 0)

where nTM and nTE are the zero-order solutions of Eqs. (1.4,1.5) given in Eqs. (1.2, 1.3). Further research into form birefringence was performed by Bouchitte and Petit using homogenization techniques [17]. They rigorously proved that any refractive index distribution could be replaced by a stratified layer as long as the period of the grating tends to zero. The limitations of form birefringence in its various forms have been thoroughly examined both theoretically and experimentally. Brundrett et al. [18] compared the zero order and second order solutions acquired from Rytov’s equations with rigorous electromagnetic calculation of the effective indices for dielectric gratings. They showed that the transmission and reflection from a binary dielectric grating were correctly predicted to an accuracy of within 10% using the zero order approximation, even past the cut-on point where the first non-zero orders begin propagating in the substrate. However, McPhedran et al. [19] numerically set the limit of form birefringence for metal stripe gratings at a wavelength to period ratio of about 40:1, much smaller than the limit suggested by Brundrett for dielectric gratings. This result has been experimentally validated by Schnabel et al. [20] and by Lochbihler [21] who demonstrated the excitation of polaritons in metal stripe gratings, thereby connecting the deviations from form-birefringence with plasmon resonance [22]. 1.2 Applications involving subwavelength gratings As mentioned above, subwavelength gratings give rise to effective birefringence. Since the effective refractive indices of the grating depend on the gratings structure, these elements offer a large degree of flexibility in engineering artificial refractive index distributions for use as polarizers, waveplates or as substitutes for anti-reflection coatings and thin film multilayer filters. Subwavelength gratings offer the ability to artificially determine the refractive index of a layer for any portion of the spectrum thus relieving designers and engineers from the need to search for materials with the desired optical properties. Their lightweight and compactness makes subwavelength gratings extremely attractive for complex micro-optics systems. Unfortunately, the small features required in the realization of these elements make them Proc. of SPIE Vol. 4984

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extremely difficult to fabricate for the visible and infra-red portions of the electromagnetic spectrum, and therefore such elements are not widely available. An exception to this rule is the use of subwavelength metal stripe gratings as polarizers for the far infra-red. When light polarized parallel to the subwavelength metal stripes is on the grating, it induces a dipole in the wires, which cancels out the wave transmitted through the grid, and the wave is reflected. When the polarization is perpendicular to the wires, no such dipole can be induced, and the wave is transmitted. Metal stripe gratings are commercially available for infrared and microwave radiation [23]. The gratings are usually realized on dielectric substrates such as ZnSe or Ge, with a period of around 300nm. Typical extinction ratios are around 1:100 at a wavelength of about 10µm and about 1:10 at wavelength around 1.5µm. This demonstrates the strong dependence of the polarizing properties of metal stripe gratings on wavelength-period ratio as discussed above. This received further verification in the work of Honkanen et al. [24] who showed that at periods comparable with the wavelength, the polarization properties of the grating were reversed, and that mainly TE polarized light was transmitted. Experiments involving metal stripe polarizers for the visible regime have been performed for example Schnabel et al [20] realized 35nm thick chromium stripes on a glass substrate with periods down to 190nm using electron beam lithography. Their polarizers yielded a polarization ratio of around 5, with 60% TM transmission. Their calculations showed that by increasing the thickness of the wires they could increase the extinction ratio while reducing TM transmission, thereby confirming the strong dependence of the performance of subwavelength gratings on their structure. Zeitner et al. demonstrated the diversity of metal stripe gratings by realizing polarization-dependent amplitude elements using metal-stripe grating pixels [25]. These unique elements transmitted different images depending on whether the illuminating beam was polarized in the x or y direction. Although subwavelength dielectric gratings are not commercially available, they have been the subject of much research, and their many applications have been demonstrated. Raguin and Morris [26] demonstrated the fabrication of an anti-reflective surface for CO2 laser radiation at a wavelength of 10.6µm. Their design was based on pyramid-like structures thereby yielding an effective layer with a graded index. The use of subwavelength gratings as retarders has also been extensively studied. This is especially important in the mid infra-red and far infra-red where appropriate retarders are scarce. For example Cescato et al. [27] studied holographic quarter waveplates using effective medium theory and Brundrett et al. [18] used RCWA and optimization procedures to design waveplates for CO2 laser radiation using subwavelength gratings on a GaAs substrate. More recently Bokor et al. [28] demonstrated the possibility of using subwavelength gratings as achromatic retarders, which are elements that are difficult to realize using naturally birefringent crystals. The diversity of subwavelength dielectric gratings has also been demonstrated in the work of Lopez and Craighead [29] who demonstrated a multilayer grating on fused silica that acted as a quarter waveplate for normal incidence and as a polarizing beam splitter for oblique incidence. Additional work on using dielectric gratings as polarizing selective mirrors was performed by Brundrett et al. [30]. Work combining metal stripe gratings with dielectric gratings was performed by Deguzman and Nordin [31] who fabricated a circular polarizer by combining a metal stripe polarizer with a subwavelength dielectric grating in the same element. Nordin et al. [32] also suggested the use of a micropolarizer array incorporating such a circular polarizer for real time polarimetry in the 3-5µm region. Similarly Doumuki and Tamada [33] incorporated a wire grid polarizer on a GaAs photodiode thereby demonstrating the feasibility of integrating subwavelength structures into micro electro-optical devices. 1.3 Realization of subwavelength gratings As mentioned above, the main difficulty in realizing subwavelength structures is the small feature size. Therefore their realization requires advanced and often creative photolithographic techniques. Roughly speaking, there are three methods for the realization of these elements: direct writing [34], indirect writing [35] and interference writing [36]. In indirect writing, first a mask of the element is made. This mask is usually a glass substrate, on which the relief pattern of the element has been placed. This is usually done by first coating the glass with a metal, coated by photoresist. The pattern is then developed onto the photoresist, using either a laser or electron beam, and then etched, leaving the desired pattern on the mask. The element can then be realized by imaging the mask onto the a photoresist coated substrate. The photoresist on the substrate is then developed and a copy of the element can be made. If the pattern is not binary, then it is necessary to make separate masks for the different layers. The main advantage of this 174

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method is that it enables relatively cheap reproduction of a single element. This technique was used by Deguzman and Nordin [31] to fabricate a circular polarizer for the mid infra-red The second method used is direct writing. No mask is made, and instead a laser (usually UV), or electron beam, are used in order to imprint the pattern directly onto the substrate. The substrate is first coated with photoresist, and the pattern is written directly onto this coated substrate. This is then developed, and we receive a single copy of the element. This system offers high resolution, especially when electron beams are used. It is widely used in academic research of subwavelength gratings for the visible region [29, 34], however production-time is long and the fabrication is very expensive. Therefore this method is not widely used for commercial production. Interference recording is the most commonly used method for the fabrication of homogenous 1-D gratings [30, 32]. The subwavelength lines are produced by interference of ultra-violet or blue laser light, leading to periods of around 200nm. This technique is very useful in the formation of space-invariant gratings, and simple space-variant structures can be achieved by incorporating simple computer-originated phase masks into the interferometer. However it ‘s application to the formation of intricate space-variant subwavelength gratings is limited. There are also differences in the fabrication of metal and dielectric SWOEs. Metal SWOEs are usually fabricated using lift-off [33]. After the pattern has been transferred to the photoresist-coated substrate by either direct or indirect writing, the photoresist is developed. The substrate is then coated with metal, and the photoresist removed. Thus, the metal remains only in the areas that were clear of photoresist after development. This technique is very useful in the realization of thin metal stripes with sub-micron features. Since the metal stripes in SWOE need not be much thicker than the skin depth, this technique is well suited for the realization of space-variant metal stripe SWOEs for the IR and visible spectra. On the other hand, dielectric gratings are realized using etching techniques. Since the features of the SWOEs are very small, and since the depth-feature size aspect ratio is usually large, it is important to choose a technique that has a large degree of anistropy. Dry etching is usually more suitable than wet-etching. In particular reactive ion etching is especially useful [29,31,32]. After the photoresist on the substrate is developed, the element is placed in a vacuum chamber and subject to bombardment of a mixture of plasma. The characteristics of the plasma depend on the choice of etching technique. The areas on the substrate which were coated by photoresist remain untouched, whereas the areas that were exposed to the plasma are etched away. In this way a relief pattern is achieved on the substrate and the grating is realized. 1.4 The outline Subwavelength gratings have opened up new methods for forming beams with sophisticated phase and polarization distributions. In this paper we present a novel method for designing and realizing nonuniformly polarized beams using computer-generated space-variant subwavelength gratings. Our design is based on determining the local period and direction of the grating at each point, forming space varying polarizers or waveplates that convert uniformly polarized light into any desired space-variant polarization. We realized the gratings for CO2 laser radiation at a wavelength of 10.6 micron on GaAs and ZnSe substrates utilizing advanced photolithographic and etching techniques[37]. Linear polarized light with axial symmetry is demonstrated. Moreover, we exploited our space-variant subwavelength grating to demonstrate a polarization Talbot self-imaging, as well as nondiffracting periodically space-variant polarization beams, and also introducing a unique method for rapid polarization measurement based on subwavelength grating. We also present novel optical phase elements based on the space-domain Pancharatnam-Berry phase. Unlike diffractive and refractive elements, the phase is not introduced through optical path differences, but results from the geometrical phase that accompanies space-variant polarization manipulation. We introduce and experimentally demonstrate continuous Pancharatnam-Berry phase optical elements (PBOEs) based on subwavelength gratings such as polarization beam-spiltters and optical switches. We also discuss a theoretical analysis and an experimental demonstration of multilevel discrete Pancharatnam-Berry phase diffractive optics (Multilevel-PBOE). The formation of a multilevel geometrical phase is done by discrete orientation of the local subwavelength grating. We realized a quantized geometrical blazed phase of polarization diffraction grating, as well as polarization dependent focusing lens, and spiral phase elements for infra-red radiation. The PBOEs will enable a variety of applications in modern optics introducing novel approaches in nanooptics. Proc. of SPIE Vol. 4984

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2. SPACE-VARIANT POLARIZATION STATE MANIPULATION USING SUBWAVELENGTH METAL GRATING Subwavelength metal stripe gratings are usually used as homogenous space-variant polarizers. Sometimes, however, a different polarization state is required at each location. Such nonuniform space-variant polarization state is useful for polarization coding of data in optical communication, optical computers and neural networks, optical encryption, tight focusing, imaging polarimetry and for particle trapping and acceleration. Here, we demonstrate a novel method for designing, analyzing and realizing computer generated space-variant metal stripe polarization elements [37-39]. The design method is based on determining the local direction and period of subwavelength metal stripe grating using vectorial optics to obtain any desired continuous polarization change, hence, completely suppressing any diffraction arising from polarization discontinuity. Analysis of the element can then be performed using an original method combining Coupled Wave Analysis (RCWA) and Jones Calculus, in which the element is represented as a spacevarying Jones Matrix, defined by the local period and orientation of the grating. Gratings are typically defined by a grating vector, perpendicular to the grating stripes. The grating vector can be expressed as, r K g = K 0 cos( β )xˆ + K 0 sin( β ) yˆ , (2.1)

ˆ , yˆ are the unit vectors along the where K0 is the spatial frequency of the grating, β is the direction of the vector and x x-axis and the y-axis respectively. A space varying grating can therefore be described by the vector, r K g = K 0 ( x, y ) cos( β ( x, y ))xˆ + K 0 ( x, y ) sin( β ( x, y ))yˆ , (2.2) for which the local period and direction vary as a function of x and y. In order for such a grating to be physically r r K g 0 , or more explicitly, realizable in a continuous way, K g should be a conserving vector i.e.

∇×

∂K o ∂y

 ∂β  ∂K o = ∂x  ∂y 

cos( β ) − K 0 sin( β ) 

 ∂β    ∂x 

sin( β ) + K 0 cos( β ) 

,

=

(2.3)

which is a necessary restraint on Ko(x, y) for a continuous grating with a local groove direction β(x, y) to exist. Once the r grating vector is determined, the grating function φ (x, y), can be found by integrating K g along any arbitrary path in r the x-y plane so that φ K g .

∇ =

We have applied our method to the design of a space-variant polarization element, which enables the transformation of circularly polarized light into a wave with a direction of polarization that is a linear function of the x coordinate. The element was fabricated on a GaAs [37] and ZnSe wafers using photolithographic processing. Figure 2.1 shows the

Figure 2.1. Illustration of the magnified geometry of a computer-generated space-variant polarizer for which the orientation of the transmission axis rotates in the x-direction from 00 to 1800.

magnified geometry of such a computer-generated mask with the resulting transmission axis varying in the x-direction from 0° to180° . The continuity of the grating can be clearly observed. Figure 2.2 shows the experimental measurements of the azimuthal angle for circularly polarized CO2 laser transmitted through the ZnSe polarization state manipulation element. The experimental result which were based on a complete space-variant Stokes parameters measurement, revealed 98.6 percent overall polarization purity, taking into account the azimuthal and ellipticity deviations. In our recent experiments pure azimuthal as well as radial polarization state transformations have been obtained [40]. Figure 2.3 shows the magnified geometry of the computer generated space-variant radial polarization 176

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element. The results include full space variant polarization measurements and calculations (Fig.2.3), and show high quality azimuthal and radial polarized beams.

Figure 2.2: Experimental measurement of the 2-D space-variant polarization orientations

50 40 30 20 10 0 -10 -20 -30 -40

50

(a)

(b)

Figure 2.3: Magnified geometry of the grating for converting cirularly polarized light into radial polarization (a), and experimental measurement of the local azimuthal angle (b).

3. VECTORIAL VORTEX STRUCTURES FORMED BY COMPUTER-GENERATED SPACEVARIANT DIELECTRIC SUBWAVELENGTH GRATINGS Optical singularities appear at points or lines where the phase or amplitude of the wave is undefined or changes abruptly. In particular vortices are spiral phase ramps around a singularity. If the polarization is space varying (i.e. transversely inhomogeneous), then disclinations, which are points or lines of singularity in the patterns or directions of the transverse field, can arise. An example is the center of a beam with radial polarization. In this part we present a method for the formation of radially and azimuthally polarized light utilizing computergenerated space-variant subwavelength dielectric gratings [41-43]. By correctly determining the direction, period and depth of the grating, the desired polarization is obtained. Furthermore the continuity of our grating ensures the continuity of the transmitted field, thus suppressing diffraction effects arising from discontinuity. The gratings were realized on GaAs substrates using photolithographic processes and dry etching. Our gratings are compact, lightweight, flexible in design and have high transmission efficiency. We discuss the design and realization processes, and show that the resulting beams (both radially and azimuthally polarized), carry angular momentum expressible through a Proc. of SPIE Vol. 4984

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topological charge. The topological charge is introduced through a geometrical space-variant Pancharatnam-Berry phase that results from the space-variant polarization manipulation [39]. We support this discussion with experimental results with CO2 laser radiation at a wavelength of 10.6 µm. Figure 3.1 shows polarization measurements of the beam transmitted through the computer-generated subwavelength dielectric grating when the incident beam had right hand circular polarization (a-d), and when the incident beam had left 40

(b)

(a)

(c)

(d) 30 20 10 0 10 20 30 40

(e)

(h)

(g)

(f)

Figure 3.1 Experimental intensity distributions for the radial (a-d) and azimuthal (e-h) polarizations directly after the grating; (a,e) after passing through a polarizer oriented vertically (b,f) a polarizer oriented at 45º, (c,g) a polarizer oriented horizontaly; (d,h) the measured local azimuthal angle of the beams.

hand circular polarization (e-h). The first three images show the intensity after passing the light through a polarizer oriented horizontally (a,e), diagonally at 45° (b,f) and vertically (c,g), and graphs (d,h) show the local azimuthal angles of the resulting beams, as calculated from the Stokes parameters derived from the measurements. The element yields radially and azimuthally polarized beams, depending on the incident polarization, thereby emphasizing the versatility of this method.

ld = 0

(a) ld = +1

(c) ld= +2

(b) ld= -1

(d) ld= -2

(e)

(e)

`

(c)

(d)

Figure 3.2 The instantaneous real part of the electric field vectors for the radially polarized beam formed by spacevariant grating as well as the measured and calculated far-field intensity crossections (a); the fields and far-field intensity crossections when spiral phase elements exp(il dθ ) are inserted into the beam’s path (b-e).

Figure 3.2(a) shows the instantaneous real part of the electric field vectors for the radially polarized beam formed by our space-variant grating as well as the measured and calculated far-field intensity. The distribution of the electric field results from the Pancharatnam-Berry phase. We verified the geometrical phase of the beam by inserting spiral phase 178

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elements of the form exp(il d θ ) ( l d is an integer). Figures 3.2(b-e) show the beam that is created when in addition to the grating, the wave is transmitted through such elements. We note the distinct far-field images of the beams, resulting from their different phase structures. Calculation of the phase of the beams in Fig. 3.2 reveals vortex-like phase structure [42], and their angular momentum can be expressed by a topological charge analogous to the topological charge of scalar vortices. 4. POLARIZATION TALBOT SELF-IMAGING BY USE OF SUBWAVELENGTH GRATINGS The Talbot effect is a well-known interference phenomenon in which coherent illumination of a periodic structure gives rise to a series of self-images at well-defined planes. This effect has many applications to fields such as wave-front sensing, spectrometry, and Talbot laser resonators. We demonstrated the formation of continuous space-variant polarized fields by using computer-generated subwavelength dielectric gratings [44,45]. By correctly controlling the Subwavelength grating

y x

R

x

R

z

L y x

Polarization distribution

Figure 4.1. Formation of propagation-invariant fields by space-variant subwavelength grating. The insets show the geometry of the grating (top), as well as the vector-field formed by it (bottom).

local orientation and periodicity of the grating, one can achieve any desired space-variant polarization. We demonstrate a Talbot effect involving a unique type of polarization-diffraction grating that comprises a periodic space-variant wave plate for which the orientation of the fast axis varies linearly in the x direction. We show that for any incident polarization the resultant field undergoes self-imaging and fractional Talbot effects that involve polarization, intensity, and phase. Furthermore, we show that the Talbot effect for incident circular polarization yields a one-dimensional nondiffracting beam that conserves its space-varing polarization and uniform intensity as it propagates. Propagation-invariant vector fields could be useful for applications such as volume holography and metrology and for the study of ion diffraction from spatially varying polarized fields. If we assume a space-varying periodic subwavelength grating, and illuminate the grating with an right-circularly polarized plane wave, R , at an angle, resulting in nondiffracting vector fields. Figure 4.1 illustrates nondiffrating periodically space-variant polarization beams by use of subwavelength gratings. The transmitted beam comprises two orders, a zero order that maintains the original polarization, and does not undergo any phase modification and a diffracted order, L , whose polarization has switched helicity, and whose phase is modified at each point. The two orders travel in different directions. We show that interference of the two orders in the region where they overlap results in a propagation-invariant beam with constant intensity and transversely periodically varying polarization. We believe that the vectorial Talbot and nondiffracting effects can be applied to the improvement of many existing Talbot-effect-based applications and point the way to the use of some novel ideas. Proc. of SPIE Vol. 4984

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5. SPATIAL FOURIER-TRANSFORM POLARIMETRY WITH SPACE-VARIANT SUBWAVELENGTH DIELECTRIC GRATINGS Polarization measurements are important for a large range of applications such as ellipsometry, bio-imaging, and polarization mode dispersion compensation in optical communications. A commonly used method is to measure the time dependent signal when a beam is transmitted through a rotating quarter waveplate followed by an analyzer. However this method is time consuming, making it unsuitable for real-time polarization measurements. Here, we present a space-domain analogy of the rotating quarter waveplate method. Our method is based on a spacevarying waveplate, which we realized as a computer-generated space-variant subwavelength dielectric grating [46,47]. Subwavelength metal polarizer

Space-variant subwavelength grating

Laser

Image acquisition

Polarization sensitive medium

0.5µm

Figure 5.1 Illustration of the concept of real-time spatial Fourier transform polarimetry using subwavelength dielectric gratings. The inset shows a scanning electronic microscope image of the grating profile.

By performing a spatial Fourier analysis of the intensity transmitted through this polarization grating followed by an analyzer, we can measure all four Stokes parameter simultaneously, thereby enabling real-time polarimetry of polarized and partially polarized beams. We discuss the theory behind our method, and describe the design and realization processes of our grating. We present experimental measurements for CO2 laser radiation at a wavelength of 10.6 µm. Figure 5.1 is a schematic representation of our concept. Polarized light from the laser is incident on a polarization sensitive medium (e.g. biological tissue, optical fiber, ellipsometric sample etc.). The beam is then transmitted through a space-variant subwavelength dielectric grating, followed by an analyzer. The resulting space-variant intensity is measured with a camera, and a Fourier analysis of the recorded space-variant intensity is performed, yielding the polarization of the beam in real-time. We realized the system of Fig. 5.1 with a CO2 laser that emitted linearly polarized light at a wavelength 10.6µm. The space-variant subwavelength grating was realized on a GaAs substrate, and the design ensured the continuity of the grating. We replaced the polarization sensitive medium with a quarter waveplate that could be rotated. The images were captured by a Spiricon Pyrocam I, at a rate of 48 Hz. Figure 5.2(a) shows the measured and predicted Stokes parameters of the resulting beam on the Poincare sphere. There is good agreement between measurements and the anticipated results. The measurements yielded an average error of 0.017rad in the measured azimuthal angle and an average error of 0.025rad in the measured ellipticity angle. The errors stemmed mainly from non-uniformity in the grating fabrication and from fluctuations in the laser’s polarization. Figure 5.2(b) shows measurements of the degree of polarization (DOP) for incident partially polarized light. In order to simulate the partially polarized beam, we replaced the polarization sensitive medium of Fig. 5.1, with a rotating half180

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wave plate, and increased the integration time of the camera. Calculating the degree of polarization (DOP) yields that it

Ω Ω , where Ω is the angle that the HWP is rotated during the integration time. There was a

is equal to (sin 2 ) / 2

good agreement between theory and measurement indicating that our method enables full polarization characterization.

S3 (a)

(b) 1.0

calculated measured

0.8 0.6 DOP

S1

0.4

S2 0.2 0.0 0 Polarization

90

180

270

Ω [°]

360

predicted measured

Figure 5.2 (a) Measured and predicted Stokes parameters for incident polarized light, as depicted on the Poincare sphere. (b) Measured and predicted degree of polarization for incident partially polarized light.

6. PANCHARATNAM-BERRY PHASE OPTICAL ELEMENTS The Pancharatnam-Berry phase is a geometrical phase associated with the polarization of light [48,49]. When the polarization of a beam is made to traverse a closed loop on the Poincare sphere, the final state differs from the initial state by a phase factor equal to half the area encompassed by the loop on the sphere. Recently, we demonstrated a Pancharatnam-Berry phase that accompanied space-variant polarization state manipulations in the space domain [39]. When circular polarization was converted into radial polarization, our calculations and experiments indicated a geometrical phase that left a clear signature on the far-field image of the beam [42]. We demonstrate novel optical phase elements based on the space-domain Pancharatnam-Berry phase. Unlike diffractive and refractive elements, the phase is not introduced through optical path differences, but results from the geometrical phase that accompanies space-variant polarization manipulation. We analyze Pancharatnam-Berry phase optical elements (PBOEs) that consist of space varying, (transversely inhomogeneous) waveplates with constant retardation, and space-varying fast axis orientation. We realized such PBOEs for CO2 laser radiation at a wavelength of 10.6 µm using computer-generated space-variant subwavelength gratings [50]. Figure 6.1 illustrates the concept of PBOEs by use of the Poincare sphere. Circularly polarized light is incident on a waveplate with a space varying fast axis whose orientation is denoted by θ(x,y). The resulting polarization is spacevarying since it depends on the local orientation of the waveplate. Thus, the beam at different points traverses different paths on the Poincare sphere, resulting in a space-variant phase-front modification originating from the PancharatnamBerry phase. For incident circular polarization, the resulting beam has two components, the zero order and the diffracted orders. The amount of energy in each component depends on the retardation of the grating. The zero order has the same polarization as the original beam and undergoes no phase modification. Whereas, the diffracted orders have a

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polarization orthogonal to that of the incident beam and undergo a phase modification equal to 2θ(x,y), which we call the Diffractive Geometrical Phase (DGP). The DGP can be utilized to form novel optical elements [50-52].

| R〉

r

| E (0)〉

Ω | Er (θ )

Figure 6.1. The concept of space-variant Pancharatnam-Berry phase elements illustrated on the Poincare sphere .

〉

2χ 2ψ

φp

= Ω/2

− i exp( −i 2θ ) |

L〉

Figure 6.2 illustrates the geometry of a Pancharatnam-Berry phase diffraction grating for CO2 laser radiation realized using subwavelength dielectric gratings. The retardation of the grating was φ=π, leading to 100% efficiency in the diffracted order.

Fig. 6.2. The geometry of the subwavelength PancharatnamBerry diffraction grating.

Figure 6.3 shows the DGP when the incident beam has left hand circular polarization, as well as the experimental measurement of the far-field intensity. The DGP resembles a blazed grating and all the energy is located in the first order to the left. When the incident polarization is right hand circular polarization the DGP is blazed in the opposite direction, and the energy is diffracted to the first order on the right. We have also realized other PBOEs such as spiral Pancharatnam-Berry phase elements, indicating the ability to form smart phase elements based on geometrical phase. We also introduced and experimentally demonstrate multilevel discrete Pancharatnam-Berry phase diffractive optics (Multilevel-PBOE) [53]. The formation of a multilevel geometrical phase is done by discrete orientation of the local subwavelength grating having uniform periodicity. We realized a quantized blazed geometrical phase of polarization diffraction grating, as well as polarization dependent focusing lens for CO2 laser radiation at a wavelength of 10.6 micron on GaAs substrates, utilizing advanced photolithographic and etching techniques. The measured diffraction efficiency was 94.5% ± 1% as predicted, indicating that high diffraction efficiency can be attained utilizing single binary computer-generated mask. The PBOEs will advance a variety of applications in modern optics introducing novel approaches in nanooptics.

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(a)

(b)

Figure 6.3. The Diffractive Geometrical Phase (a) and measured far-field intensity for incident lefthand and right-hand circular polarizations (b).

CONCLUSIONS We developed a novel method for designing and realizing nonuniformly polarized beams using computer-generated space-variant subwavelength gratings. We demonstrated that by locally controlling the direction and period of the grating any desired space-variant polarization state manipulation could be obtained. We also introduced space-variant Pancharatnam-Berry phase optical elements based on computer-generated subwavelength gratings. Unlike conventional elements, PBOEs are not based on optical path difference but on geometric phase modification resulting from spacevariant polarization manipulation. The elements are polarization dependent thereby enabling multi-purpose optical elements, suitable for applications such as optical switching, optical interconnects and beam splitting.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

E. B. Grann, M. G. Moharam and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of twodimensional subwavelength binary gratings”, J. Opt. Soc. Am. A 11, 2695-2703, 1994. J. N. Mait, D. W. Prather and M. S. Mirotznik, “Design of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory”, J. Opt. Soc. Am. A 16, 1157-1167, 1999. I. Puscasu, D. Spencer and G. Boreman, “Refractive-index and element-spacing effects on the spectral behavior of infrared frequency-selective surfaces”, Appl. Opt. 39, 1570-1574, 2000. D. L. Brundrett, E. N. Glytiss and T. K. Gaylord, “Subwavelength transmission grating retarders for use at 10.6 µm”, Appl. Opt. 35, 6195-6202, 1996. G. R. Bird and M. Parrish Jr, J. Opt. Soc. Am. 50, 886-891, 1960. H. Hertz, Electric Waves, Macmillan and Company Ltd., 1893. J. W. Goodman, “An introduction to Fourier optics”, 2nd edition, Mcgraw Hill, New-York 1996. E. Hasman, N. Davidson and A. A. Friesem, “Efficient multilevel phase holograms for CO2 lasers”, Opt. Lett. 16, 423-425,1991.

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9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31. 32. 33. 34. 35. 36. 37.

184

W. A. Challener, “Vector diffraction of a grating with conformal thin films”, J. Opt. Soc. Am. A 13, 18591869, 1996. B. Guizal and D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating”, Opt. Commun. 165, 16, 1999. M. G. Moharam and T.K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings” J. Opt. Soc. Am. A 3 ,1780-1787, 1986. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization”, J. Opt. Soc. Am. A 13, 779-784, 1996. M. Born and E. Wolf, Principles of Optics, 7th Edition, Cambridge University Press, 1999. H. A. Mcleod, Thin Film Optical Filters, 2nd Edition, Mcgraw Hill, New York, 1989. P. Yeh, “Electromagnetic propagation in birefringent layered media”, J. Opt. Soc. Am 69, 742-756, 1979. S.M. Rytov, Soviet Physics JETP 2, 466-475, 1956. G. Bouchitte and R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings” Electromagnetics 5, 17-36, 1985. D. L. Brundrett, E. N. Glytiss and T. K. Gaylord, “Homogeneous layer models for high-spatial-frequency dielectric surface-relief gratings: conical diffraction and antireflection designs”, Appl. Opt. 33, 2695-2706, 1996. R. C. McPhedran, L. C. Botton, M. S. Craig, M. Neviere and D. Maystre, “Lossy lamellar gratings in the quasistatic limit”, Optica Acta 29, 289-312, 1982. B. Schnabel, E. B. Kley and F. Wyrowski, “Study on polarizing visible light by subwavelength-period metal-stripe gratings”, Opt. Eng. 38, 220-226, 1999. H. Lochbihler, “Surface polaritons on gold-wire gratings”, Phys. Rev. B 50, 4795-4801, 1994. U. Schroter and D. Heitmann, “Surface-plasmon-enhanced transmission through metallic gratings”, Phys Rev. B 58, 15419-15421, 1998. See for example Molectron Detector Inc. Catalogue. M. Honkanen, V. Kuittinen, J. Lautanen, J. Turunen, B. Schnabel, F. Wyrowski, “Inverse metal-stripe polarizers”, Appl. Phys. B 68, 81-85, 1999. U. D. Zeitner, B. Schnabel, E. B. Kley and F. Wyrowski, “Polarization multiplexing of diffractive elements with metal-stripe grating pixels”, Appl. Opt. 38, 2177-2181, 1999. D. H. Raguin and G. M. Morris, “Antireflection structured surfaces for the infrared spectral region”, Appl. Opt. 32, 1154-1161, 1993. L. H. Cescato, E. Gluch and N. Streibl, “Holographic quarterwave plates”, Appl. Opt. 29, 3286-3290, 1990. N. Bokor, R. Schechter, N. Davidson, A. A. Friesem and E. Hasman, “Achromatic phase retarder by slanted illumination of a dielectric grating with period comparable with the wavelength”, Appl. Opt. 40, 2076-2080, 2001. A. Lopez and H. G. Craighead, “Wave-plate polarizing beam splitter based on a form-birefringent multilayer grating”, Opt. Lett. 23, 1627-1629, 1998. D. L. Brundrett, T. K. Gaylord and E. N. Glytsis, “Polarizing mirror/absorber for visible wavelengths based on a silicon subwavelength grating: design and fabrication”, Appl. Opt. 37, 2534-2541, 1998. P. C. Deguzman and G. P. Nordin, “Stacked subwavelength gratings as circular polarization filters”, Appl. Opt. 31,, 5731-5737, 2001. G. P. Nordin, J. F. Meier, P. C. Deguzman and M. W. Jones, “Micropolarizer array for infrared imaging polarimetry”, J. Opt. Soc. Am A 16, 1168-1174, 1999. T. Doumuki and H. Tamada, “An aluminum-wire grid polarizer fabricated on a gallium-arsenide photodiode”, Appl. Phys. Lett. 71, 686-688, 1997. M. E. Warren, R. E. Smith, G. A. Vawter and J. R. Wendt, “High-efficiency subwavelength diffractive optical element in GaAs for 975 nm”, Opt. Lett. 20, 1441-1443, 1995. M. J. Bowden, in Introduction to Microlithography, 2nd Editon, 19-136, Editors L. F. Thompson, C. G. Wilson and M. J. Bowden, American Chemical Society, 1994. R. C. Enger and S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations”, Appl. Opt. 22, 3220-3228, 1983. Z. Bomzon, V. Kleiner and E. Hasman, “Computer-generated space-variant polarization elements with subwavelength metal stripes”, Opt. Lett. 26, 33-35, 2001.

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38. Z. Bomzon, V. Kleiner and E. Hasman, “Space-variant polarization state manipulation with computergenerated subwavelength metal stripe gratings”, Opt. Commun. 192, 169-181, 2001. 39. Z. Bomzon, V. Kleiner and E Hasman, “Pancharatnam-Berry phase in space-variant polarization-state manipulations with subwavelength gratings”, Opt. Lett. 26 , 1424-1426, 2001. 40. Z. Bomzon, V. Kleiner, E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings”, Appl. Phys. Lett. 79, 1587-1589, 2001. 41. Z. Bomzon, G. Biener, V. Kleiner and E. Hasman, “Space-variant polarization-state manipulation with computer-generated subwavelength gratings”, Optics & Photonics News 12, 33, 2001. 42. Z. Bozmon, V. Kleiner, E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings”, Opt. Lett. 27, 285-287, 2002. 43. A. Niv, G. Biener, V. Kleiner, and E. Hasman,”Formation of linear polarized light with axial symmetry using space-variant subwavelength gratings”, submitted to Opt. Lett., 2002. 44. Z. Bomzon, A. Niv, G. Biener, V. Kleiner, E. Hasman, “Polarization Talbot self-imaging with computergenerated space-variant subwavelength dielectric gratings”, Appl. Opt. 41, 5218-5222, 2002. 45. Z. Bomzon, A. Niv, G. Biener, V. Kleiner, E. Hasman, “Non-diffracting periodically space-variant polarization beams with subwavelength gratings”, Appl. Phys. Lett. 80, 3685-3687, 2002. 46. Z. Bomzon, G. Biener, V. Kleiner and E. Hasman, “Spatial Fourier-transform polarimetry using space-variant subwavelength metal-stripe polarizers”, Opt. Lett. 26, 1711-1713, 2001. 47. Z. Bomzon, G. Biener, V. Kleiner and E. Hasman, “Real-time analysis of partially polarized light with a spacevariant subwavelength dielectric grating”, Opt. Lett. 27, 188-190, 2002. 48. S. Pancharatnam, “Generalized theory of interference, and its applications”, Proc. Ind. Acad. Sci. 44, 247-262, 1956. 49. M.Berry, “The adiabatic phase and Pancharatnam's phase for polarized light”, J. Mod. Opt. 34, 1401-1407, 1987. 50. Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings”, Opt. Lett. 27, 1141-1143, 2002. 51. E. Hasman, Z. Bomzon, A. Niv, G. Biener, and V. Kleiner, “Polarization beam-splitters and optical switches based on space-variant computer-generated subwavelength quasi-periodic structures”, Opt. Commun. 209, 4554, 2002. 52. G. Biener, A. Niv, V. Kleiner, and E. Hasman,"Formation of helical beams by use of Pancharatnam-Berry phase optical elements", Opt. Lett. 27, 1875-1877, 2002. 53. E. Hasman, V. Kleiner, G. Biener, and A. Niv, “Multilevel Pancharatnam-Berry phase diffractive optics with space-variant subwavelength gratings”, submitted to Appl. Phys. Lett., 2002.

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