^{1,a)}

### Abstract

Equations to enable determination of the helicity (angular momentum orientation) of photofragments resulting from single-photon dissociation of an isotropic sample of molecules are presented. The symmetry of the photofragment distribution is illustrated by three-dimensional vector plots of the expectation values of projections of the fragment total angular momentum. Equations describing circular polarization of light in the spherical tensor basis are presented. Methods for the optical measurement of angular momentum orientation are discussed, including determination of the helicity of circularly polarized light by a quarter-wave plate or single Fresnel rhomb.

The author thanks David Van Baak (Calvin College), Alex Brown (University of Alberta), Dmitry Budker (UC, Berkeley), Matt Costen (Heriot-Watt University), Marcelo de Miranda (University of Leeds), Arthur Suits (Wayne State University), and Oleg Vasyutinskii (Ioffe Institute, St. Petersburg) for varied and useful discussions. The author also thanks The Royal Society for financial support and a University Research Fellowship.

I. INTRODUCTION

II. CIRCULAR POLARIZATION

III. PHOTOFRAGMENT ORIENTATION

A. Circularly polarized photolysis

B. Linearly polarized photolysis

IV. EXPERIMENTAL DETERMINATION OF HELICITY

V. CALCULATION OF SENSITIVITY OF POLARIZED DETECTION SCHEMES

VI. CONCLUDING REMARKS

### Key Topics

- Polarization
- 58.0
- Photodissociation
- 34.0
- Angular momentum
- 17.0
- Photons
- 14.0
- Tensor methods
- 8.0

## Figures

Plot illustrating the laboratory distribution resulting from the molecule-frame orientation parameter , shown as arrows. The laboratory axis is marked in (a), this is the propagation direction of the circularly polarized photolysis radiation. The arrows represent vectors of expectation values for total angular momentum . The vector plot is superposed onto an orbital plot of the spatial anisotropy (direction of travel) of the photofragments, Eq. (8), with . If the photolysis radiation is taken as being left circular polarized, then (a) illustrates , and (b) illustrates . Alternatively, if we take , (a) could result from photolysis by left-circular polarized light, and (b) from right-circular polarized light.

Plot illustrating the laboratory distribution resulting from the molecule-frame orientation parameter , shown as arrows. The laboratory axis is marked in (a), this is the propagation direction of the circularly polarized photolysis radiation. The arrows represent vectors of expectation values for total angular momentum . The vector plot is superposed onto an orbital plot of the spatial anisotropy (direction of travel) of the photofragments, Eq. (8), with . If the photolysis radiation is taken as being left circular polarized, then (a) illustrates , and (b) illustrates . Alternatively, if we take , (a) could result from photolysis by left-circular polarized light, and (b) from right-circular polarized light.

As for Fig. 1, but illustrating the laboratory distribution resulting from the molecule-frame orientation parameter . In this plot the spatial anisotropy is characterized by . For left-circular polarized photolysis radiation, (a) shows , and (b) shows .

As for Fig. 1, but illustrating the laboratory distribution resulting from the molecule-frame orientation parameter . In this plot the spatial anisotropy is characterized by . For left-circular polarized photolysis radiation, (a) shows , and (b) shows .

This figure is similar to Fig. 1, but illustrates the laboratory distribution resulting from the molecule-frame orientation parameter . The laboratory axis is marked in (a), this lies along the electric field of the *linearly* polarized photolysis radiation. In this plot the spatial anisotropy is characterized by (i.e., an equal mix of ‖ and ⊥). Here (a) shows [corresponding to, (Refs. 53 and 58)], and (b) shows [corresponds to ].

This figure is similar to Fig. 1, but illustrates the laboratory distribution resulting from the molecule-frame orientation parameter . The laboratory axis is marked in (a), this lies along the electric field of the *linearly* polarized photolysis radiation. In this plot the spatial anisotropy is characterized by (i.e., an equal mix of ‖ and ⊥). Here (a) shows [corresponding to, (Refs. 53 and 58)], and (b) shows [corresponds to ].

Schematic figure of a zero-order quarter-wave plate, consisting of two cylindrical slabs of uniaxial crystal with thicknesses and , as marked. Also marked are the double-headed slow and fast axes for both slabs, and the right-handed Cartesian frame . Light is input at the input side, travelling in the direction (i.e., towards ), and exits at the output side. See text for discussion.

Schematic figure of a zero-order quarter-wave plate, consisting of two cylindrical slabs of uniaxial crystal with thicknesses and , as marked. Also marked are the double-headed slow and fast axes for both slabs, and the right-handed Cartesian frame . Light is input at the input side, travelling in the direction (i.e., towards ), and exits at the output side. See text for discussion.

Schematic figure of the single Fresnel rhomb. See text for detailed discussion. A perspective view is shown on the left, and a side-on view is shown on the right. The right-handed Cartesian frame is shown. Light is input at the input face, travelling in the direction (i.e., towards ), and exits at the output face. The rhomb is oriented so that light is displaced along the direction, and is not displaced along the direction. The angle of incidence at the first reflection is shown. The symbol dot-in-circle (☉) represents a vector pointing toward the reader, out of the page; the symbol cross-in-circle (⊗) represents a vector pointing away from the reader, into the page.

Schematic figure of the single Fresnel rhomb. See text for detailed discussion. A perspective view is shown on the left, and a side-on view is shown on the right. The right-handed Cartesian frame is shown. Light is input at the input face, travelling in the direction (i.e., towards ), and exits at the output face. The rhomb is oriented so that light is displaced along the direction, and is not displaced along the direction. The angle of incidence at the first reflection is shown. The symbol dot-in-circle (☉) represents a vector pointing toward the reader, out of the page; the symbol cross-in-circle (⊗) represents a vector pointing away from the reader, into the page.

## Tables

Values of the contracted polarization tensor in the photon frame for linearly and circularly polarized lights (Ref. 1). Recall that in the photon frame, the direction is the propagation direction of the light for circular polarization (left or right), and lies along the electric field for linear polarization.

Values of the contracted polarization tensor in the photon frame for linearly and circularly polarized lights (Ref. 1). Recall that in the photon frame, the direction is the propagation direction of the light for circular polarization (left or right), and lies along the electric field for linear polarization.

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