FINCH creates holograms in a single beam system as a result of interference between a plane wave and a spherical wave originating from every object point. In our previous reports we created a random constant phase mask so that with a phase-only SLM, the plane wave from an infinity corrected microscope objective could be directed to the camera along with the spherical wave created by the SLM. The use of a constant phase mask presents certain disadvantages in that it requires half the pixels on the SLM and also degrades the resolution of the mask which creates the spherical wave. Because only one linear polarization state on the liquid crystal based SLM can change the phase of incoming light, half of the randomly polarized fluorescent light striking the device can have quadratic phase modulation whereas the other half is shifted by a constant phase, as shown in
. However, the sensitivity of the SLM to a specific linear polarization also makes it possible to use the portion of the light not affected by the SLM to deliver the plane wave as shown in , and discussed earlier and below.

The following analysis refers to the system scheme shown in , where it is assumed that the object is an infinitesimal point and therefore the result of this analysis is considered as a point spread function (PSF). For an arbitrary object point at

, in a working distance

*z*_{s} before the objective, where

, the complex amplitude beyond the first polarizer, just before the SLM, is

where it is assumed that the polarizer axis is tilted in a

*ϕ*
_{1} angle to the

*x* axis,

*f*_{o} is the focal length of the objective,

*d*
_{1} is the distance between the objective and the SLM and

*A*_{x}, A_{y} are the constant amplitudes in the

*x, y* axes, respectively. The asterisk denotes a two dimensional convolution and are unit vectors in the

*x, y* directions, respectively. For the sake of shortening, the quadratic phase function is designated by the function

*Q*, such that

the function

*L* stands for a the linear phase function, such that,

and

is a complex constant dependent on the source point's location. The SLM modulates the light in only a single linear polarization and in our case, without loss of generality, this axis is chosen to be

*x*. The light polarized in

*y* direction is reflected from the SLM with only a constant phase shift. Therefore the complex amplitude on the output plane of the SLM is,

where

*B*_{Q} and

*B*_{M} are complex constants.

*θ* is one of the three angles used in the phase shift procedure in order to eliminate the bias term and the twin image [

6,

7]. The complex amplitude after passing the second polarizer, with axis angle of

*ϕ*
_{2} to the

*x* axis, has linear polarization in the direction of the polarizer axis. Therefore we can abandon the vector notation and express the complex amplitude beyond the second polarizer, on the CCD plane, as

where

*z*_{h} is the distance between the SLM and the CCD. The intensity of the recorded hologram is,

Following the calculation of

Eq. (7), the intensity on the CCD plane is,

where

*A*_{o},

*C*
_{2},

*C*
_{3} are constants and

, the reconstruction distance of the object point, is given by

The transverse location of the reconstructed object point is,

Equation (8) is the typical expression of an on-line Fresnel hologram of a single point and therefore

*I*_{p}(

*x*
_{2},

*y*
_{2}) is the PSF of the recording part of the FINCH. To avoid the problem of the twin image, one of the interference terms, (the second or third terms) in

Eq. (8) should be isolated by the phase-shifting procedure [

10,

11]. Reconstructing this term by Fresnel back propagation yields the image of the point at a distance

*z*_{r} from the hologram given by

Eq. (9), and at a transverse location

given by

Eq. (10). The sign '±' in

Eq. (9) indicates the possibility to reconstruct from the hologram either the virtual or the real image depending on which term, second or third, is chosen from

Eq. (8). The polarization angles

*ϕ*
_{1} and

*ϕ*
_{2} are chosen in order to maximize the interference terms [the second and third terms in

Eq. (8)]. Their precise values depend on the values of the constants |

*B*_{Q}| and |

*B*_{M}|. In this study we choose their values empirically by picking the angles that yield the best reconstructed image.

Based on

Eq. (10), the transverse magnification of this FINCH system is

In this stage we can simplify

Eqs. (8) – (11) by choosing the working distance to be

*z*_{s} = f_{o}, as was indeed chosen in the present experiment. In this case

*f*_{e}→∞, and therefore

*f*
_{1}
*= -f*_{d},

*z*_{r} = ± (

*z*_{h}-f_{d}) and

.

The minimal resolved object size observed by reconstructing the FINCH hologram is dictated by either the input or output apertures according to the following equation

where

and

*D*_{H} are the diameters of the SLM, and the recorded hologram, respectively.

*NA*_{in} and

*NA*_{out} are the numerical apertures of the system input and output, respectively. The

*NA*_{in} is independently determined by the objective and cannot be changed by the design of the FINCH system. However the product

*NA*_{out}M_{T} is dependent on the system parameters and our goal should be to keep this product equal or larger than

*NA*_{in} in order not to reduce the resolution determined by the input aperture. Therefore, referring to

Eq. (12), an optimal FINCH system satisfies the inequality,

In this inequality all the parameters are well defined besides the diameter of the hologram. This size is dependent on the overall size of the reconstructed image. Based on simple geometrical considerations the diameter of the hologram is,

where

*a* is the ratio between the image and the SLM sizes.

*a* ranges between almost zero for an image of a point, to 1 for a full frame image. Substituting

Eq. (14) and quantities

*z*_{r} = |

*f*_{d}-z_{h}| and

into

Eq. (13) yields

The only free parameter in this analysis that does not influence other performances of the system is

*f*_{d}. Therefore by calculating the inequality in

Eq. (13) we find the optimal

*f*_{d} in sense of best image resolution. The solution of

Eq. (15) is

. In this study and in Ref [

11], we used the complete field of view, and therefore we assume

*a =* 1. Consequently the focal length of the diffractive lens should be equal or smaller than twice the distance between the SLM and the CCD, or in a formal way,

. Because the CCD chip is not ideal as a medium for hologram recording, practically it is optimal to display the image as far as possible from the CCD chip. Therefore we find that

*f*_{d} = 2

*z*_{h} is the optimal choice for the length of the focal length of the diffractive lens.