From an engineering standpoint, it will be helpful to have a working relation between phase mask and PCF given in terms of the parameters and . In general, an analytic form for the SRW cannot be found easily, especially since the phase mask can take in arbitrary nonanalytic forms. However, for slowly varying phase mask distribution such as a circle, it can be approximated by a sum of Gaussian functions. Consider an input beam such as a laser beam with a Gaussian profile . When the laser beam passes through a phase mask containing the target pattern , it is modulated as . For a circular phase mask, the modulation takes the form of , and the input field becomes . The first lens takes the Fourier transform of and is multiplied by the transfer function of the PCF. The circular -phase-shifting region of the PCF has a radius of corresponding to frequency cutoff , and thus the transfer function is given by . The PCF can be seen as a low-pass filter where the low-spatial frequency components are -phase shifted. The field is once again Fourier transformed by the second lens and the output is given by Display Formula
(2)The term represents the SRW. For an input circular phase mask, will have a circ function in the phase term and be multiplied with a Gaussian profile, and the will involve convolution of Gaussian and jinc functions. To simplify the calculation, the jinc is approximated with a Gaussian function. For , which transforms into , the Gaussian approximation is given by , where . The parameter is chosen to match the central part of the jinc. The convolution is now between two Gaussian functions and simply results in a broadened Gaussian function. However, this does not account for the negative values in the original jinc function, and the amplitude should be corrected. The correct central amplitude can be analytically obtained6 as . This correction is done for every instance of Fourier transform of a bounded Gaussian function. Finally, we arrive at the following approximate equation for the GPC output: Display Formula
(3)where , and . Assuming the input field has a unit on-axis amplitude, we also require the SRW to have a value equal to one. Thus, the “darkness” condition can be written as . For the circular -phase shifted mask that we used, the input amplitude is ; thus, Display Formula
(4)To see whether Eq. (4) will lead to the creation of a dark area, we compute for the efficiency, which we define as the power within the area defined by the phase pattern divided by the incident power. Dark nodal areas are therefore those resulting in low efficiency. An efficiency map is shown in Fig. 3, where an overlay of Eq. (4) shows that on-axis amplitude matching is sufficient to identify combinations of parameters that will result in darkness. The parameters that will result in bright GPC can also be seen. The insets in Fig. 3 show the calculated GPC outputs for selected pairs. Interestingly, there are parameters where the on-axis SRW amplitude is zero and the Gaussian input reappears.