The role of the relay lenses

In a standard HOT system, the DOE is positioned in the input plane of a telescope whose role is to project the hologram onto the input pupil of the objective lens. In so doing, however, it also rotates the light's polarization in a position-dependent manner. The image of the DOE in the objective's input pupil thus differs from the ideal DOE in a way that affects trapping performance. Such spatially varying polarization was not considered in previous studies of the forces exerted on spheres by strongly focused light fields (9,10). We account for this effect by making use of recent results obtained with the Debye-Wolf formalism (21).

Because the telescope forms an image in the far field, each pixel in the DOE may be treated as a magnetic dipole. The contribution to the vector potential from the $j$ -th pixel thus has the form (21)

$$A_{j,\mu}^{\text{D}}(\vec{r}) = - \frac{a_j}{2 \pi c} \, \exp\lef... ..._j \right\vert \right)}{ \left\vert \vec{r} - \vec{r}_j \right\vert } \, p_\mu,$$ (8)

where $k = 2 \pi / \lambda$ is the wavenumber of light of wavelength $\lambda$ , and where $a_j$ and $\varphi_j$ are the phase and amplitude at the $j$ -th pixel, respectively. The polarization of the ray propagating along

$$\hat{\boldsymbol{s}}_1 = (\sin\theta_1 \cos\phi_1, \sin\theta_1 \sin\phi_1, \cos\theta_1).$$ (9)

is given by

$$\vec{p} = \hat{\boldsymbol{s}}_1 \times (\hat{\boldsymbol{z}} \times \hat{\boldsymbol{\varepsilon}}),$$ (10)

where $\hat{\boldsymbol{\varepsilon}}$ is the polarization of the incident light. The total vector potential at point $\vec{r}$ is then

$$A_\mu^{\text{D}}(\vec{r}) = \sum_{j=1}^N A_{j,\mu}^{\text{D}}(\vec{r})$$ (11)

for a DOE consisting of $N$ pixels.

The image in the objective's input pupil is a superposition of contributions from each of the DOE's pixels. Referring to Fig. 1, a ray propagating from $\vec{r}_1$ on the DOE in the direction $\hat{\boldsymbol{s}}_1$ arrives at $\vec{r}_2$ in the image plane in the direction $\hat{\boldsymbol{s}}_2$ . The angles of departure and arrival are related by the Abbe sine condition, $f_1 \sin\theta_1 = f_2 \sin \theta_2$ , and by continuity, $\phi_1 = \phi_2 + \pi$ . All such contributions can be expressed as a superposition of plane waves through the Debye-Wolf integral (21),

$$A_\mu^{\text{O}}(\vec{r}_2) = \int_{\Omega_2} B_\mu(\hat{\boldsym... ...2) \, \exp\left(i k \hat{\boldsymbol{s}}_2 \cdot \vec{r}_2\right) \, d\Omega_2,$$ (12)

where the complex amplitude of the plane wave propagating in direction $\hat{\boldsymbol{s}}_2$ is

$$B_\mu(\hat{\boldsymbol{s}}_2) = \frac{f_2}{2 \pi c} \, G_\mu^\nu(... ...arphi_j\right) \, \exp\left(i k \hat{\boldsymbol{s}}_1 \cdot \vec{r}_j \right).$$ (13)

The geometric operator $\mathbb{G}(\hat{\boldsymbol{s}})$ accounts for rotation of the light's polarization as it propagates through the telescope and may be represented as (21)

$$\mathbb{G}(\hat{\boldsymbol{s}}_2) = \sqrt{\frac{\cos\theta_2}{\c... ...hi_1) \mathbb{L}(\pi - \theta_2) \mathbb{L}(\pi - \theta_1) \mathbb{R}(\phi_1),$$ (14)

in terms of the generalized Jones matrices

$$\mathbb{R}(\phi) = \left( \begin{matrix}\cos\phi & \sin\phi & 0 \\ -\sin\phi & \cos\phi & 0 \\ 0 & 0 & 1 \end{matrix} \right) \quad \mathbb{L}(\theta) = \left( \begin{matrix}\cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{matrix} \right).$$ (15)