Ptychographic Iterative Engine (Gradients and Hessians)

PIE - Gradient

PIE ultimately assumes independent Object and Probe. This is respected in these derivatives.

\(S_{mk} = O_k\cdot P_{mk}\)

\(P_{mk} = G(A_{mk})\)

With respect to Object

Same for all nonlinear methods

\(\frac{dL_m}{dO_k^*} = -1\cdot P_{mk}^*\cdot \Delta S_{mk}\)

With respect to Probe and Probe-Pulse

\(\frac{dL_m}{dA_k^*} = \left(\frac{dA_{mj}}{dA_k}\right)^*\left(\frac{dP_{mj}}{dA_{mj}}\right)^*\frac{dL_m}{dP_{mj}^*}\)

\(\frac{dL_m}{dP_{mk}^*} = -1\cdot O_k^*\cdot \Delta S_{mk}\)

\(\frac{dA_{mj}}{dA_k} = \sum_{n}D_{kn}D_{nj}\cdot e^{1j\phi_{mn}}\qquad\) (in the exponent \(j\) refers to the imaginary unit)

\(\frac{dP_{mj}}{dA_{mj}}\) depends on the nonlinear method.

SHG

\(\frac{dP_{mj}}{dA_{mj}} = 1\)
THG

\(\frac{dP_{mj}}{dA_{mj}} = 2A_{mj}\)
PG

\(\frac{dP_{mj}}{dA_{mj}} = A_{mj}^*\)
SD

\(\frac{dP_{mj}}{dA_{mj}^*} = 2A_{mj}^*\qquad\) (the other wirtinger derivative is zero)
nth-HG

\(\frac{dP_{mj}}{dA_{mj}} = (n-1)\cdot A_{mj}^{n-2}\)

For SD the wirtinger derivative of probe with respect to the modified probe pulse is zero. Thus one has to use a different chain-rule composition:

\(\frac{dL_m}{dA_k^*} = \left(\frac{dA_{mj}}{dA_k}\right)^*\frac{dP_{mj}}{dA_{mj}^*}\frac{dL_m}{dP_{mj}}\)

\(\frac{dL_m}{dP_{mj}} = \left(\frac{dL_m}{dP_{mj}^*}\right)^*\)

With respect to Object (Chirp-Scan)

In a Chirp-Scan the object itself is modified along woth the probe.

\(S_{mk} = O_{mk}\cdot P_{mk}\)

\(\frac{dL_m}{dO_k^*} = \left(\frac{dO_{mj}}{dO_k}\right)^*\frac{dL_m}{dO_{mj}^*}\)

\(\frac{dL_m}{dO_{mk}^*} = -1\cdot P_{mk}^*\cdot \Delta S_{mk}\)

\(\frac{dO_{mj}}{dO_k} = \sum_{n}D_{kn}D_{nj}\cdot e^{1j\phi_{mn}}\qquad\) (in the exponent \(j\) refers to the imaginary unit)

PIE - Pseudo Hessian

With respect to Object

Same for all nonlinear methods

\(\frac{d|S_{mn}|}{dO_k^*} = \frac{S_{mn}}{2|S_{mn}|} \cdot D_{kn} P_{mk}^*\)

\(\frac{d}{dO_j}\left(\frac{d|S_{mn}|}{dO_k}\right)^* = \left(\frac{d}{dO_j^*}\left(\frac{d|S_{mn}|}{dO_k^*}\right)^*\right)^* = \frac{1}{4}D_{kn}D_{nj} P_{mk}^*P_{mj}\cdot\frac{1}{|S_{mn}|}\)

With respect to Probe and Probe-Pulse

\(\frac{d|S_{mn}|}{dA_k^*} = \left(\frac{dA_{mj}}{dA_k}\right)^*\left(\frac{dP_{mj}}{dA_{mj}}\right)^*\frac{d|S_{mn}|}{dP_{mj}^*}\)

\(\frac{d|S_{mn}|}{dP_{mk}^*} = \frac{S_{mn}}{2|S_{mn}|} \cdot D_{kn} O_k^*\)

\(\frac{dP_{mj}}{dA_{mj}}\) depends on the nonlinear method. Same as in gradient.

\(\frac{dA_{mj}}{dA_k} = \sum_{n}D_{kn}D_{nj}\cdot e^{1j\phi_{mn}}\qquad\) (in the exponent \(j\) refers to the imaginary unit)

\(\frac{d}{dA_j}\left(\frac{d|S_{mn}|}{dA_k}\right)^* = \frac{dA_{mu}}{dA_k}\left(\frac{dA_{mi}}{dA_j}\right)^*\frac{dP_{mu}}{dA_{mu}}\left(\frac{dP_{mi}}{dA_{mi}}\right)^*\frac{d}{dP_{mu}}\left(\frac{d|S_{mn}|}{dP_{mi}}\right)^*\)

\(\frac{d}{dP_{mj}}\left(\frac{d|S_{mn}|}{dP_{mk}}\right)^* = \left(\frac{d}{dP_{mj}}\left(\frac{d|S_{mn}|}{dP_{mk}}\right)^*\right)^* = \frac{1}{4}D_{kn}D_{nj} O_k^*O_j\cdot\frac{1}{|S_{mn}|}\)

For SD the wirtinger derivative of probe with respect to the modified probe pulse is zero. Thus one has to use a different chain-rule composition:

\(\frac{d}{dA_j}\left(\frac{d|S_{mn}|}{dA_k}\right)^* = \frac{dA_{mu}}{dA_k}\left(\frac{dA_{mi}}{dA_j}\right)^*\frac{dP_{mu}^*}{dA_{mu}}\left(\frac{dP_{mi}^*}{dA_{mi}}\right)^*\frac{d}{dP_{mu}^*}\left(\frac{d|S_{mn}|}{dP_{mi}^*}\right)^*\)

\(\frac{d|S_{mn}|}{dP_{mi}^*}=\left(\frac{d|S_{mn}|}{dP_{mi}}\right)^*\)

With respect to Object (Chirp-Scan)

\(\frac{d|S_{mn}|}{dA_k^*} = \left(\frac{dA_{mj}}{dA_j}\right)^*\frac{d|S_{mn}|}{dA_{mj}^*}\)

\(\frac{d|S_{mn}|}{dA_{mk}^*} = \frac{S_{mn}}{2|S_{mn}|} \cdot D_{kn} P_{mk}^*\)

\(\frac{d}{dA_j}\left(\frac{d|S_{mn}|}{dA_k}\right)^* = \frac{dA_{mu}}{dA_k}\left(\frac{dA_{mi}}{dA_j}\right)^*\frac{d}{dA_{mu}}\left(\frac{d|S_{mn}|}{dA_{mi}}\right)^*\)

\(\frac{d}{dA_{mj}}\left(\frac{d|S_{mn}|}{dA_{mk}}\right)^* = \left(\frac{d}{dA_{mj}^*}\left(\frac{d|S_{mn}|}{dA_{mk}^*}\right)^*\right)^* = \frac{1}{4}D_{kn}D_{nj} P_{mk}^*P_{mj}\cdot\frac{1}{|S_{mn}|}\)