Controlling Light with Geometric~Phase Holograms |

Michael J. Escuti, Jihwan Kim and Michael W. Kudenov |

^{Controlling Light with Geometric~Phase Holograms}

**Michael J. Escuti, Jihwan Kim and Michael W. Kudenov**

**? ** Improved fabrication techniques are creating a new generation of gratings, lenses, and other elements that are physically thin and optically thick.

**? ** From above pict: **Amsterdam’s “Rainbow Station” project, by artist Daan Roosegaarde. The optical design and implementation were led by Frans Snik and Michiel Rodenhuis of Leiden University, using custom geometric~phase holograms from ImagineOptix Co. [© Studio Roosegaarde]**

**? ** During the International Year of Light (IYL) in 2015, visitors to the historic Central train station of Amsterdam, Netherlands, in the hour after sunset could witness a unique site: the station’s platform~spanning arch, lit by a curved rainbow (opening spread). The “Rainbow Station” project was the brainchild of the Dutch artist Daan Roosegaarde. And the result — a carefully sculpted, arched, richly colored dispersion profile, with minimal leakage of the projected light — was the product of an array of thin optical elements that adjusted the light using a relatively little~known effect called geometric phase.

**? ** Until recently, elements taking advantage of this phenomenon — geometric~phase holograms — have been plagued by shortcomings in efficiency, light leakage, and spectral range that have limited their practical use. But improved fabrication technologies are surmounting this hurdle. The result could be a new generation of ultra~thin, versatile gratings, lenses and other optical elements that are already finding use in exoplanet studies, and that could extend to a range of other applications.

**? ** A geometric~phase shift arises as a kind of “memory” of the evolution of a lightwave through an anisotropic parameter space.

**What is geometric phase?**

**? ** Commonly, a wavefront is controlled by adjusting optical path length (OPL), defined for an isotropic plate as the product of the wave’s speed (dependent on the material’s refractive index) and its physical propagation distance through the medium. An example is the spatially varying OPL caused by a lens’ curved surface. The phase shift that results from such OPL variations is sometimes called a dynamic~phase shift, as these parameters directly affect the wave’s propagation time through a medium.

**? ** A geometric~phase shift, by contrast, arises as a kind of “memory” of the evolution of a lightwave through an anisotropic parameter space. Notably, this phase shift depends only on the geometry of the pathway through the anisotropy transforming the lightwave. The most important sources of such transformations are molecular anisotropy and nanostructures causing anisotropic scattering, in which the phase shift of the transmitted or reflected lightwave is directly proportional to the orientation of an effective optic axis, the shape of anisotropic scattering particles, or both. Because this differs remarkably from traditional refraction, it is sometimes referred to as anomalous refraction or reflection.

**? ** Geometric phase has three particularly remarkable and useful features. First, the geometric~phase~shift magnitude, δ, is solely related to geometrical parameters of the medium creating it. In some implementations, δ is proportional to twice the orientation angle of the local optic axis, ϕ; in others, it depends on both the shape and orientation of an anisotropic scattering particle. Either way, any desired spatially varying phase shift can in principle be embodied in an inhomogeneous anisotropy map. This creates the possibility, as discussed below, of extremely thin optical elements; furthermore, the geometric~phase shift is wavelength independent, notably unlike a dynamic~phase shift (see diagrams).

**? ** Second, there is no physical difference between phase shifts of 0 and 2πp, for any integer p, because the embodiment is in orientation, shape, or both. Hence, a relative geometric~phase shift can be both continuously smooth and unbounded, for example, not restricted to 2π (see sidebar). Third, the geometric~phase shift has a sign corresponding to each of two orthogonal polarizations — that is, δ = ±2ϕ, assuming we neglect a dynamic~phase contribution that is constant.

**From geometric phase to GPHs**

**? ** Geometric phase was first described in terms of interference theory by S. Pancharatnam in 1956, and subsequently expanded upon in quantum-mechanical terms by Michael V. Berry in 1984 — indeed, it is often referred to as Pancharatnam–Berry phase. Since then, as the optical community has increasingly taken an interest in the phenomenon, researchers have given numerous names to inhomogeneous geometric~phase elements: certain metasurface, polarization or vector holograms; polarization kinoforms; anisotropic aperture antennas; cycloidal diffractive waveplates; and space~variant Pancharatnam–Berry phase optical elements, among others. All operate based on geometric phase, however, and are bound by the same principles. Hence, we refer to all of these elements by a common name: geometric~phase holograms (GPHs).

**? ** We define GPHs as those inhomogeneous optical elements that impose a purely geometric~phase shift, and in which isotropic absorption and dynamic phase are constant. Such elements output at most three distinct waves — a primary (“+”) wave, a conjugate (“–”) wave, and a leakage (“0”) wave (see diagrams) — each of which experiences a different geometric~phase shift (or, in the case of the leakage wave, no phase shift). Regardless of the input wave polarization, the primary and conjugate waves have fixed (usually circular), mutually orthogonal polarizations, while the leakage wave retains the input polarization.

**? ** Depending on the GPH’s physical characteristics and the input light’s polarization, different relative fractions of input power will couple to the three possible output waves, quantified as the respective conversion efficiencies η+, η–, and η0. Three surprising characteristics of this coupling hint at some powerful advantages of GPHs.

**? ** Potentially high efficiency. The theoretical conversion efficiencies follow the relation (1 – A) = (η+ + η– + η0), where A is the net absorbance of the geometric~phase layer, averaged over all phase values from 0 to 2π — with the power lost to absorption proportionally lost from all three output waves. This means that if the net absorbance and leakage efficiency are low, then the maximum efficiency of the primary wave, the conjugate wave, or both may approach 100 percent.

**? ** Selectable phase shift. The primary and conjugate efficiencies in most implementations follow the relation η± = (1 – A – η0)(1 ∓ S′3)/2, where S′3 is a normalized Stokes parameter of the input light. Thus, when the input is circularly polarized, only one of the primary or conjugate waves may have power coupled into it, and the orthogonal polarization couples into the other wave. Consequently, the + and – geometric~phase shifts in GPHs may be selected simply by adjusting the input polarization — which, in a given element such as a GP lens, can bring a two~for~one benefit not seen in conventional holograms.

**? ** From achromatic to chromatic. The spectral behavior is largely independent of the phase profile, and is determined almost entirely by the properties of the materials or structures creating the anisotropy. Thus the efficiency spectra may range from very achromatic to highly chromatic.

(Excerpt) **? https://www.osa-opn.org/home/articles/volume_27/february_2016/features/controlling_light_with_geometric-phase_holograms/**

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Controlling Light with Geometric~Phase Holograms |

Michael J. Escuti, Jihwan Kim and Michael W. Kudenov |

^{Controlling Light with Geometric~Phase Holograms}Michael J. Escuti, Jihwan Kim and Michael W. Kudenov?Improved fabrication techniques are creating a new generation of gratings, lenses, and other elements that are physically thin and optically thick.?From above pict:Amsterdam’s “Rainbow Station” project, by artist Daan Roosegaarde. The optical design and implementation were led by Frans Snik and Michiel Rodenhuis of Leiden University, using custom geometric~phase holograms from ImagineOptix Co. [© Studio Roosegaarde]?During the International Year of Light (IYL) in 2015, visitors to the historic Central train station of Amsterdam, Netherlands, in the hour after sunset could witness a unique site: the station’s platform~spanning arch, lit by a curved rainbow (opening spread). The “Rainbow Station” project was the brainchild of the Dutch artist Daan Roosegaarde. And the result — a carefully sculpted, arched, richly colored dispersion profile, with minimal leakage of the projected light — was the product of an array of thin optical elements that adjusted the light using a relatively little~known effect called geometric phase.?Until recently, elements taking advantage of this phenomenon — geometric~phase holograms — have been plagued by shortcomings in efficiency, light leakage, and spectral range that have limited their practical use. But improved fabrication technologies are surmounting this hurdle. The result could be a new generation of ultra~thin, versatile gratings, lenses and other optical elements that are already finding use in exoplanet studies, and that could extend to a range of other applications.?A geometric~phase shift arises as a kind of “memory” of the evolution of a lightwave through an anisotropic parameter space.What is geometric phase??Commonly, a wavefront is controlled by adjusting optical path length (OPL), defined for an isotropic plate as the product of the wave’s speed (dependent on the material’s refractive index) and its physical propagation distance through the medium. An example is the spatially varying OPL caused by a lens’ curved surface. The phase shift that results from such OPL variations is sometimes called a dynamic~phase shift, as these parameters directly affect the wave’s propagation time through a medium.?A geometric~phase shift, by contrast, arises as a kind of “memory” of the evolution of a lightwave through an anisotropic parameter space. Notably, this phase shift depends only on the geometry of the pathway through the anisotropy transforming the lightwave. The most important sources of such transformations are molecular anisotropy and nanostructures causing anisotropic scattering, in which the phase shift of the transmitted or reflected lightwave is directly proportional to the orientation of an effective optic axis, the shape of anisotropic scattering particles, or both. Because this differs remarkably from traditional refraction, it is sometimes referred to as anomalous refraction or reflection.?Geometric phase has three particularly remarkable and useful features. First, the geometric~phase~shift magnitude, δ, is solely related to geometrical parameters of the medium creating it. In some implementations, δ is proportional to twice the orientation angle of the local optic axis, ϕ; in others, it depends on both the shape and orientation of an anisotropic scattering particle. Either way, any desired spatially varying phase shift can in principle be embodied in an inhomogeneous anisotropy map. This creates the possibility, as discussed below, of extremely thin optical elements; furthermore, the geometric~phase shift is wavelength independent, notably unlike a dynamic~phase shift (see diagrams).?Second, there is no physical difference between phase shifts of 0 and 2πp, for any integer p, because the embodiment is in orientation, shape, or both. Hence, a relative geometric~phase shift can be both continuously smooth and unbounded, for example, not restricted to 2π (see sidebar). Third, the geometric~phase shift has a sign corresponding to each of two orthogonal polarizations — that is, δ = ±2ϕ, assuming we neglect a dynamic~phase contribution that is constant.From geometric phase to GPHs?Geometric phase was first described in terms of interference theory by S. Pancharatnam in 1956, and subsequently expanded upon in quantum-mechanical terms by Michael V. Berry in 1984 — indeed, it is often referred to as Pancharatnam–Berry phase. Since then, as the optical community has increasingly taken an interest in the phenomenon, researchers have given numerous names to inhomogeneous geometric~phase elements: certain metasurface, polarization or vector holograms; polarization kinoforms; anisotropic aperture antennas; cycloidal diffractive waveplates; and space~variant Pancharatnam–Berry phase optical elements, among others. All operate based on geometric phase, however, and are bound by the same principles. Hence, we refer to all of these elements by a common name: geometric~phase holograms (GPHs).?We define GPHs as those inhomogeneous optical elements that impose a purely geometric~phase shift, and in which isotropic absorption and dynamic phase are constant. Such elements output at most three distinct waves — a primary (“+”) wave, a conjugate (“–”) wave, and a leakage (“0”) wave (see diagrams) — each of which experiences a different geometric~phase shift (or, in the case of the leakage wave, no phase shift). Regardless of the input wave polarization, the primary and conjugate waves have fixed (usually circular), mutually orthogonal polarizations, while the leakage wave retains the input polarization.?Depending on the GPH’s physical characteristics and the input light’s polarization, different relative fractions of input power will couple to the three possible output waves, quantified as the respective conversion efficiencies η+, η–, and η0. Three surprising characteristics of this coupling hint at some powerful advantages of GPHs.?Potentially high efficiency. The theoretical conversion efficiencies follow the relation (1 – A) = (η+ + η– + η0), where A is the net absorbance of the geometric~phase layer, averaged over all phase values from 0 to 2π — with the power lost to absorption proportionally lost from all three output waves. This means that if the net absorbance and leakage efficiency are low, then the maximum efficiency of the primary wave, the conjugate wave, or both may approach 100 percent.?Selectable phase shift. The primary and conjugate efficiencies in most implementations follow the relation η± = (1 – A – η0)(1 ∓ S′3)/2, where S′3 is a normalized Stokes parameter of the input light. Thus, when the input is circularly polarized, only one of the primary or conjugate waves may have power coupled into it, and the orthogonal polarization couples into the other wave. Consequently, the + and – geometric~phase shifts in GPHs may be selected simply by adjusting the input polarization — which, in a given element such as a GP lens, can bring a two~for~one benefit not seen in conventional holograms.?From achromatic to chromatic. The spectral behavior is largely independent of the phase profile, and is determined almost entirely by the properties of the materials or structures creating the anisotropy. Thus the efficiency spectra may range from very achromatic to highly chromatic.(Excerpt)

? https://www.osa-opn.org/home/articles/volume_27/february_2016/features/controlling_light_with_geometric-phase_holograms/??????????????????????????????????????