The scheme of I-COACH with sparse chaotic point response is shown in Fig. 9. The SLM modulates light emitted from a point source, which comprises CPM with a diffractive lens of focal length *f*. The light modulated by CPM is projected on the sensor plane by the diffractive lens to satisfy the Fourier-transform relation between the CPM and sensor planes. By removing CPM from the SLM, the system acts as a direct imaging system. The image from the direct imaging system is further used as a reference compared to the proposed SCI-COACH, the chaotic continuous I-COACH, and the dot-based I-COACH.

The spatially incoherent, quasi-monochromatic source of light is used to critically illuminate the object. Hence, the system is treated as a spatially incoherent imaging system with linear space-invariant relations between the intensity patterns on the camera and on the object plane. The following mathematical analysis is based on the optical configuration of Fig. 9. Considering a point object located at \(\left( {\overline {{{r_s}}} , - {z_s}} \right)=\left( {{x_s},{y_s}, - {z_s}} \right)\)with an amplitude\(\sqrt {{I_s}}\), the complex amplitude just before SLM is given as,

\({I_k}\left( {{{\bar {r}}_{}};{{\bar {r}}_s},{z_s}} \right)=\sqrt {{I_s}} {C_0}L\left( {\frac{{{{\bar {r}}_s}}}{{{z_s}}}} \right)Q\left( {\frac{1}{{{z_s}}}} \right),\begin{array}{*{20}{c}} {}&{}&{} \end{array}\left( 1 \right)\)

where *L* and *Q* represent linear and quadratic phase functions, given by \(L\left( {\bar {s}/z} \right)=\exp \left[ {i2\pi {{\left( {\lambda z} \right)}^{ - 1}}\left( {{s_x}x+{s_y}y} \right)} \right]\)and \(Q\left( a \right)=\exp \left[ {i\pi a{\lambda ^{ - 1}}\left( {{x^2}+{y^2}} \right)} \right],\) respectively. *λ* is the wavelength, and *C*0 is a complex constant. The complex amplitude modulated by CPM is given by,

\({I_k}\left( {{{\bar {r}}_{}};{{\bar {r}}_s},{z_s}} \right)=\sqrt {{I_s}} {C_0}L\left( {\frac{{{{\bar {r}}_s}}}{{{z_s}}}} \right)Q\left( {\frac{1}{{{z_s}}}} \right)Q\left( { - \frac{1}{{{f_0}}}} \right)\exp \left[ {i{\Theta _k}\left( r \right)} \right],\begin{array}{*{20}{c}} {}&{}&{} \end{array}(2)\)

where \({\Theta _k}\left( r \right)\)is the *k**th* pseudorandom phase of the CPM calculated using the modified GSA. The complex amplitude at the image sensor is given as 2D convolution between Eq. (2) and quadratic phase function \(Q\left( {{1 \mathord{\left/ {\vphantom {1 {{z_h}}}} \right. \kern-0pt} {{z_h}}}} \right)\) for distance *z**h*. Therefore, the intensity pattern on the image sensor is given by,

\({I_k}\left( {{{\bar {r}}_0};{{\bar {r}}_s},{z_s}} \right)={\left| {\sqrt {{I_s}} {C_0}L\left( {\frac{{{{\bar {r}}_s}}}{{{z_s}}}} \right)Q\left( {\frac{1}{{{z_s}}}} \right)Q\left( { - \frac{1}{{{f_0}}}} \right)\exp \left[ {i{\Theta _k}\left( r \right)} \right]*Q\left( {\frac{1}{{{z_h}}}} \right)} \right|^2},\begin{array}{*{20}{c}} {}&{}&{} \end{array}(3)\)

where is a sign of 2D convolution and \({\bar {r}_0}=\left( {u,v} \right)\) is the transverse location vector on the sensor plane. The light diffracted from the CPM is Fourier transformed by the diffractive lens of focal length\({f_0}\), on the image sensor located at a distance of *z**h*.22

\({I_k}\left( {{{\bar {r}}_0};{{\bar {r}}_s},{z_s}} \right)={\left| {\nu \left[ {\frac{1}{{\lambda {z_h}}}} \right]\mathfrak{F}\left\{ {\sqrt {{I_s}} {C_0}L\left( {\frac{{{{\bar {r}}_s}}}{{{z_s}}}} \right)Q\left( \xi \right)\exp \left( {i{\Theta _k}\left( r \right)} \right)} \right\}} \right|^2}={I_k}\left( {{{\bar {r}}_0} - \frac{{{z_h}}}{{{z_s}}}{{\bar {r}}_s};0,{z_s}} \right)\begin{array}{*{20}{c}} {}&{}&{} \end{array}(4)\)

where \(\xi ={{\left( {{f_0}{z_s}+{f_0}{z_h} - {z_s}{z_h}} \right)} \mathord{\left/ {\vphantom {{\left( {{f_0}{z_s}+{f_0}{z_h} - {z_s}{z_h}} \right)} {{f_0}{z_s}{z_h}}}} \right. \kern-0pt} {{f_0}{z_s}{z_h}}},\)\(\mathfrak{F}\)is the 2D Fourier transform operator, and \(\nu\)is the scaling operator defined by \(\nu \left[ a \right]f\left( x \right)=f\left( {ax} \right)\). The object intensity response on the sensor plane is a shifted version [by\(\left( {{z_h}/{z_s}} \right){\bar {r}_s}\)] of the intensity response of a point object at\({\bar {r}_s}=\left( {0,0} \right)\).

A 2D object at a distance *z**s* from the SLM can be considered as a collection of *N* uncorrelated point objects given as \(o\left( {{{\bar {r}}_s}} \right)=\sum\limits_{j}^{N} {{a_j}\delta \left( {{{\bar {r}}_s} - {{\bar {r}}_j}} \right)}\), where \({a_j}\) is the intensity of the *j**th* object point at \({\bar {r}_j}\). The object is illuminated by an incoherent quasi-monochromatic light, and therefore there is no interference between the individual point responses due to the spatial incoherence of the object light. The overall intensity distribution on the sensor plane is a sum of the point responses given by \({I_{OBJ}}\left( {{{\bar {r}}_0};{z_s}} \right)=\sum\limits_{j} {{a_j}} {I_k}\left( {{{\bar {r}}_0} - \left( {{z_h}/{z_s}} \right){{\bar {r}}_j};0,{z_s}} \right)\). \({I_{OBJ}}\left( {{{\bar {r}}_0};{z_s}} \right)\) and \({I_k}\left( {{{\bar {r}}_0};{z_s}} \right)\)are both positive real functions with dominant bias terms. Therefore, images reconstructed by a cross-correlation between \({I_k}\left( {{{\bar {r}}_0};{z_s}} \right)\) and \({I_{OBJ}}\left( {{{\bar {r}}_0};{z_s}} \right)\) yield an undesired background distribution. Since the average intensity of any response generated with any CPM is approximately the same, the difference distribution between any two intensity responses almost lacks bias. Therefore, in order to minimize the background distribution, both \({H_{PSH}}\left( {{{\bar {r}}_0};{z_s}} \right)\) and \({H_{OBJ}}\left( {{{\bar {r}}_0};{z_s}} \right)\) become bipolar by capturing two shots with different CPMs, as follows,

\(\begin{gathered} {H_{PSH}}\left( {{{\bar {r}}_0};{z_s}} \right)={I_1}\left( {{{\bar {r}}_0};{z_s}} \right) - {I_2}\left( {{{\bar {r}}_0};{z_s}} \right) \hfill \\ {H_{OBJ}}\left( {{{\bar {r}}_0};{z_s}} \right)={I_{OBJ,1}}\left( {{{\bar {r}}_0};{z_s}} \right) - {I_{OBJ,2}}\left( {{{\bar {r}}_0};{z_s}} \right) \hfill \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array}=\sum\limits_{j} {{a_j}} {I_1}\left( {{{\bar {r}}_0} - \frac{{{z_h}}}{{{z_s}}}{{\bar {r}}_j};0,{z_s}} \right) - \sum\limits_{j} {{a_j}} {I_2}\left( {{{\bar {r}}_0} - \frac{{{z_h}}}{{{z_s}}}{{\bar {r}}_j};0,{z_s}} \right) \hfill \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array}=\sum\limits_{j} {{a_j}} {H_{PSH}}\left( {{{\bar {r}}_0} - \frac{{{z_h}}}{{{z_s}}}{{\bar {r}}_j};0,{z_s}} \right)\begin{array}{*{20}{c}} {}&{}&{} \end{array}\begin{array}{*{20}{c}} {}&{}&{} \end{array}\begin{array}{*{20}{c}} {}&{}&{} \end{array}(5) \hfill \\ \end{gathered}\)

Therefore, two intensity response patterns are recorded for both the object and the point source using two different CPMs synthesized with different initial random phases.

The image is reconstructed by cross-correlating between \({H_{PSH}}\left( {{{\bar {r}}_0};{z_s}} \right)\) and \({H_{OBJ}}\left( {{{\bar {r}}_0};{z_s}} \right)\) by POF as follows,

\(\begin{gathered} R({{\bar {r}}_t})={H_{OBJ}}\left( {{{\bar {r}}_0};{z_s}} \right) \otimes H{'_{PSH}}\left( {{{\bar {r}}_0} - {{\bar {r}}_t};{z_s}} \right)={\mathcal{F}^{ - 1}}\left\{ {\mathcal{F}\left\{ {{H_{OBJ}}} \right\}{\mathcal{F}^*}\left\{ {H{'_{PSH}}} \right\}} \right\} \hfill \\ \begin{array}{*{20}{c}} {}&{} \end{array}={\mathcal{F}^{ - 1}}\left\{ {\mathcal{F}\left\{ {\sum\limits_{j} {{a_j}} {H_{PSH}}\left( {{{\bar {r}}_0} - \frac{{{z_h}}}{{{z_s}}}{{\bar {r}}_j};{z_s}} \right)} \right\}{\mathcal{F}^*}\left\{ {H{'_{PSH}}} \right\}} \right\} \hfill \\ \begin{array}{*{20}{c}} {}&{} \end{array}={\mathcal{F}^{ - 1}}\left\{ {\sum\limits_{j} {{a_j}} \mathcal{F}\left\{ {{H_{PSH}}\left( {{{\bar {r}}_0} - \frac{{{z_h}}}{{{z_s}}}{{\bar {r}}_j};{z_s}} \right)} \right\}\exp \left[ { - i\arg \left( {\mathcal{F}\left\{ {{H_{PSH}}} \right\}} \right)} \right]} \right\} \hfill \\ \begin{array}{*{20}{c}} {}&{} \end{array}={\mathcal{F}^{ - 1}}\left\{ {\sum\limits_{j} {{a_j}} \left| {\mathcal{F}\left\{ {{H_{PSH}}\left( {{{\bar {r}}_0}_{{}};{z_s}} \right)} \right\}} \right|\exp \left[ { - i\frac{{{z_h}}}{{{z_s}}}{{\bar {r}}_j} \cdot \bar {\rho }} \right]} \right\} \hfill \\ \begin{array}{*{20}{c}} {}&{} \end{array}=\sum\limits_{j} {{a_j}} \Lambda \left( {{{\bar {r}}_t} - \frac{{{z_h}}}{{{z_s}}}{{\bar {r}}_s};{z_s}} \right) \approx o\left( {\frac{{\bar {r}}}{{{M_T}}}} \right),\begin{array}{*{20}{c}} {}&{}&{} \end{array}\begin{array}{*{20}{c}} {}&{}&{} \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{}&{}&{} \end{array}(6) \hfill \\ \end{gathered}\)

where Λ is a *δ*-like function ~1 at (0,0) and ~0 elsewhere, \(\bar {\rho }\)is the position vector of the spatial frequency plane, and \({M}_{T}={z}_{h}/{z}_{s} .\)It must be noted that the image is reconstructed using a cross-correlation. Therefore, the transverse resolution is dictated by the transverse correlation length, determined by the width and length of the smallest spot that can be recorded on the sensor plane by the SLM with an active area of a *D* diameter (assuming that the active area is the smallest aperture in the system). Therefore, the transverse and axial resolutions are approximately 1.22*λz**s*/*D* and 8*λ*(*z**s*/*D*)2, respectively, matching with the resolution values of the regular incoherent imaging system with a similar numerical aperture (NA).

## Synthesis of the phase mask

The CPMs are synthesized in the computer using a modified version of the GSA shown schematically in Fig. 10. Two parameters dictate the nature of the CPMs; the first is the size of each island, and the second is the number of islands. In the case of the sparse dot pattern, it was found that 6 dots give the optimal reconstruction results in terms of SNR and visibility, and hence for the sparse chaotic island, 6 islands were chosen. The arrangement of the islands is arbitrary such that five islands are placed in a pentagon shape, and one island is in the center. By changing the size of the islands, different CPMs are tested, and the optimal CPM is used for the experimental study.

In the GSA of Fig. 10, an initial random phase mask is Fourier transformed from the CPM plane to the sensor plane. The magnitude distribution is replaced with the chosen pattern of chaotic islands on the sensor plane, where the phase distribution remains unchanged. The resulting complex amplitude is inversely Fourier transformed to the CPM plane, and the magnitude distribution is replaced with the uniform magnitude. This iterative process continues till the generated intensity profile on the sensor converges to satisfy the constraints. We follow the same procedure for the annular aperture, but in this case, the aperture starts with a ring-shaped phase mask instead of the uniform disk. The generated CPM is displayed on the SLM with a diffractive lens to satisfy the Fourier relation between the CPM and sensor planes. In the case of annular aperture, to focus unwanted light away from the sensor, we displayed a diffractive optical element (DOE) containing a quadratic phase function with a linear phase function. This DOE is displayed in the internal area of the SLM surrounded by the annular CPM. Therefore, only the light passing through the annular CPM arrives at the sensor. The synthesis of the complete phase mask displayed on the SLM for the full aperture system and for the partial aperture system is shown schematically in Fig. 11.