electronic-liquid Crystal Crystal Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals Lorenzo Marrucci Dipartimento di Scienze Fisiche Università di Napoli Federico II & CNR-INFM Coherentia R&D center ITALY
electronic-liquid Crystal Crystal A hystorical line within the field of the optics of liquid crystals (since 1980): Setting liquid crystal molecules in rotation by the action of light (giant optical nonlinearity) What about the reverse effect? Setting a beam of light in rotation by the action of liquid crystals?
electronic-liquid Crystal Crystal Introduction: rotating light beams Can a beam of light rotate upon itself, while propagating? More technically: can a beam of light carry internal angular momentum?
The standard answer: Circular polarizations: (Left and Right) Field expression: (paraxial approximation, complex notation) Angular momentum carried by the light: Circularly polarized light L y E(t) 1 Er (,) t = i E0(, x y, z) e 0 L z = +ħ per photon z x ( ωt) R y E(t) z x L z = ħ per photon But in 1992, L. Allen et al. pointed out that actually there is another way ikz 1 Er (,) t = i E0 (, x y, z) e 0 ( ωt) Notice: this is spin angular momentum (vector nature of the field) electronic-liquid Crystal ikz
Helical modes of light [L. Allen, M.W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992)] Field expression: (using cylindrical coordinates r, ϕ, z) Er (,) t = E(, r z) e e (An important example: Laguerre-Gaussian modes) 0 Helicity helical phase factor: e im ϕ ( t) imϕ ikz ω m = 0, ± 1, ± 2, ± 3 Notice: the polarization remains arbitrary (may be linear or circular) electronic-liquid Crystal Let us see the main properties of these modes
Wavefront shape: Helical modes of light m = 0 electronic-liquid Crystal Crystal m = +1 m = 1 m = +2 m = 2
Helical modes of light electronic-liquid Crystal Crystal Doughnut intensity profile: (except for m = 0) Energy transport: Optical vortex (topological defect) Helical rays winding around the beam axis
Helical modes of light Angular momentum: L z = mħ per photon Notice: this is orbital angular momentum (spatial distribution of field) Nevertheless, it is still internal angular momentum (independent of origin) electronic-liquid Crystal Crystal But, is this rotation of light real, or just a formal mathematical property?
Detection of light rotation: transfer to matter electronic-liquid Crystal Crystal Circular-polarized and/or helical beam of light An absorbing particle is set in rotation The angular momentum is transferred from light to matter and the difference between spin and orbital angular momentum?
Detection of light rotation: transfer to matter electronic-liquid Crystal Crystal Circular-polarized and/or helical beam of light Spin angular momentum of light makes particle spin around its own axis (internal angular momentum) Orbital angular momentum of light makes particle rotate around beam axis (external angular momentum) but what happens if particles are transparent?
Two-ways exchange of optical angular momentum with matter electronic-liquid Crystal Crystal Important: can also be used for generating rotating light beams anisotropic (birefringent) medium (e.g., wave-plates) inhomogenous medium (e.g., phase plates) acts on the polarization (spin) acts on the wavefront (orbital)
Two-ways exchange of optical angular momentum with matter electronic-liquid Crystal Crystal The case of a strongly inhomogeneous medium: holography Fork-like hologram Order -1: helical mode with m = -1 Order 0: no change Order 1: helical mode with m = 1
electronic-liquid Crystal Crystal And what if the medium is both anisotropic and inhomogenous? enter Liquid Crystals
Rotating droplets of dye-doped liquid crystals by circularly polarized light electronic-liquid Crystal Crystal [C. Manzo, D. Paparo, L. Marrucci, I. Jánossy, Phys. Rev. E 73, 051707 (2006)] Our main result here: dye doped droplets rotate as fast as pure droplets: no dye-induced total optical torque enhancement! But some droplets did not rotate. Why?
Non-rotating droplets: why? Bipolar droplets. Almost homogeneous birefringence. Can be rotated by light. Radial droplets. Inhomogeneous and (locally) anisotropic. Cannot be rotated by light. But radial droplets anyway modify the light polarization and therefore should exchange (spin) angular momentum with light. [Istvan Jánossy, private discussion] Why don t they rotate? electronic-liquid Crystal Answering this question led us to the results presented in the following
electronic-liquid Crystal Crystal A simple inhomogenous anisotropic optical medium: The q-plate
q-plates structure: patterned half-wave plates electronic-liquid Crystal Crystal Example with radial pattern (similar to radial droplets): The plate thickness and birefringence must be chosen so as to have uniform half-wave retardation α gives the angle between the local optical axis n and a reference axis at any position (x, y) or (r, ϕ) in the xy plane y The optical axis orientation in the plate is patterned x α ( xy, ) = α( r, ϕ) = ϕ y r ϕ n α x
q-plates structure: patterned half-wave plates electronic-liquid Crystal Crystal General pattern: 0 Three examples: q = ½ (α 0 = 0) α ( xy, ) = α( r, ϕ) = qϕ+ α ( r) q = 1 (α 0 = 0) with q integer or half-integer There is a topological defect in the center q = 1 (α 0 = π/2) Important: q = 1 yields rotational-symmetric patterns
q-plates optical effect q-plate Jones matrix: cos 2 α( x, y) sin 2 α( x, y) M( xy, ) = sin 2 α( x, y) cos 2 α( x, y) Let us now apply it to an input left-circular polarized plane wave: 1 cos2α + i sin2α 1 i2 α ( x, y) M( xy, ) E0( rz, ) = E0( rz, ) e E0( rz, ) i = icos 2α + sin 2α i A half-wave plate switches the circular polarization handedness The output polarization is uniform righthanded circular Pancharatnam-Berry geometrical phase (unrelated with optical path length) The wavefront has acquired a positiondependent phase retardation! Φ (, xy) = 2(, α x y) electronic-liquid Crystal
q-plates optical effect Wavefront phase factor: i2 ( x, y ) e α α ( xy, ) = α( r, ϕ) = qϕ+ α ( r) 0 Output wavefront is helical! Helicity: m= 2 q If the input polarization is right-circular, i i and the sign of the output helicity m is inverted: m= ± 2 q Polarization controlled helical mode generation! ( ) 0 im ϕ 0 = e e = e e i 2 q ϕ i2α i2 α ( r electronic-liquid Crystal Crystal
Case q = 1: photon angular momentum balance electronic-liquid Crystal Crystal Left-circular input: Right-circular input: Spin: L z = +ħ Orbital: L z = 0 Total: L z = +ħ Spin: L z = ħ Orbital: L z = 0 Total: L z = ħ q- plate Spin: L z = ħ Orbital: L z = 2ħ Total: L z = +ħ Spin: L z = +ħ Orbital: L z = 2ħ Total: L z = ħ Spin-to-orbital conversion of optical angular momentum!
In general: photon angular momentum balance electronic-liquid Crystal Crystal Spin: L z = ±ħ Orbital: L z = mħ Total: L z = (m±1)ħ For q 1, L z = ±2(q 1)ħ 0 For q = 1, L z = 0 This is why radial droplets don t rotate! Spin: L z = ħ Orbital: L z = mħ ± 2qħ Total: L z = [m±(2q 1)]ħ Torque on the q-plate No torque on the medium (medium is only a coupler between spin and orbital angular momentum of light)
electronic-liquid Crystal Crystal Spin-to-orbital optical angular momentum conversion: The experiment In collaboration with: Carlo Manzo and Domenico Paparo [Phys. Rev. Lett. 96, 163905 (2006), highlighted in Physical Review Focus]
electronic-liquid Crystal Crystal The cell between crossed polarizers: Making a liquid crystal q-plate 1) Circular rubbing of one substrate (with planar anchoring) 2) Assemble the cell with thickness chosen for having half-wave retardation (only approximate) q = 1 geometry Nematic liquid crystal
The output light intensity profile electronic-liquid Crystal Crystal Optical vortex! The doughnut profile is obtained!
Measuring the output wavefront Mach-Zender interferometer (version 1): Laser (He-Ne) Polarizing beam-splitter λ/4 q-plate λ/4 Polarizing beam-splitter Plane-wave reference (with a small angle) Screen or CCD electronic-liquid Crystal Crystal
electronic-liquid Crystal Crystal Measuring the output wavefront Left-circular input Right-circular input Disclination-like defect in the interference pattern: optical vortex It switches sign with input polarization handedness!
Measuring the output wavefront Mach-Zender interferometer (version 2): Laser (He-Ne) Polarizing beam-splitter λ/4 q-plate λ/4 Polarizing beam-splitter Spherical-wave reference Screen or CCD electronic-liquid Crystal Crystal
electronic-liquid Crystal Crystal Measuring the output wavefront Left-circular input Right-circular input Double spiral interference pattern Helical wavefront with m = ±2 Spin-to-orbital optical angular momentum conversion experimentally demonstrated
electronic-liquid Crystal Crystal Some further prospects for q-plates
Helical modes generation for applications No more spiral phase plates and fork-like holographic elements Only q-plates! q-plate advantages: theoretical 100% conversion efficiency simple way to switch the helicity (input polarization multiplexing) single beam conversion, with no deviation like in holograms q-plate (current) disadvantages: not a pure Laguerre-Gauss mode (superposition of radial modes) electronic-liquid Crystal
electronic-liquid Crystal Crystal Helical modes generation for applications Fast-switchable generation of helical modes for (multi-valued) free-space optical communication: [L. Marrucci, C. Manzo, D. Paparo, Appl. Phys. Lett. 88, 221102 (2006)]
Optical quantum computation with helical modes electronic-liquid Crystal Crystal The q-plate as a quantum gate (controlled shift): With a linear-polarized input: 1 V = ( L + R ) 2 L, m R, m+ 2q R, m L, m 2q 1 V, m (, 2, 2 ) 2 R m+ q + L m q Entangled (Bell) state of spin and orbital angular momentum of the same photon
electronic-liquid Crystal Crystal Concept generalization: Pancharatnam-Berry phase optical elements (PBOE) for wavefront shaping [R. Bhandari, Phys. Rep. 281, 1 64 (1997)] [Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, Opt. Lett. 27, 1141 (2002)] [L. Marrucci, C. Manzo, D. Paparo, Appl. Phys. Lett. 88, 221102 (2006)]
Patterned half-wave plates (like q-plates ) Jones matrix: cos 2 α( x, y) sin 2 α( x, y) M( xy, ) = sin 2 α( x, y) cos 2 α( x, y) Apply it to an input left-circular polarized plane wave: 1 cos2α + i sin2α 1 i2 α ( x, y) M( xy, ) E0( rz, ) = E0( rz, ) e E0( rz, ) i = icos 2α + sin 2α i Pancharatnam-Berry geometrical phase Wavefront acquires a position-dependent phase retardation Φ (, xy) = 2(, α x y) With suitable patterning of the plate, we may generate wavefronts of any prescribed shape electronic-liquid Crystal
Example: a PBOE lens Optical axis pattern: α(, r ϕ) = cr 2 This pattern could be made with computercontrolled microrubbing or with photo-alignment This lens will be focusing or defocusing depending on the input circular polarization handedness: fast polarization multiplexing The lens thickness will be uniform and very thin (few microns). Similar to Fresnel lens, but without optical discontinuities electronic-liquid Crystal
PBOE and polarization holography electronic-liquid Crystal Crystal Input signal wavefront (circularly polarized) Polarization hologram Reference wavefront (with opposite circular polarization) Develop it into a cell with half-wave retardation PBOE which reconstructs the signal wavefront or its conjugate (with 100% efficiency, single order output)
Conclusions: A beam of light can carry internal angular momentum of two species: spin and orbital. When carrying a well defined quantum value of orbital angular momentum per photon, the beam is in a helical mode. We invented a way to convert the variation of spin angular momentum into orbital angular momentum. The embodiment of this invention was based on patterned liquid crystals. This process allows generating helical modes controlled by the input polarization. Fast switching among helical modes is thus possible. This approach to generating helical modes is promising for applications such as optical communication and quantum computation The associated concept of Pancharatnam-Berry phase can be exploited for making new optical elements for arbitrary wavefront shaping. Patterned liquid crystals offer a promising approach to this technology. electronic-liquid Crystal