Spectral transfer from phase to intensity in Fresnel diffraction

We analyze theoretically and investigate experimentally the transfer of phase to intensity power spectra of spatial frequencies through free-space Fresnel diffraction. Depending on $\lambda z$ (where $\lambda$ is the wavelength and $z$ is the free-space propagation distance) and the phase-modulation strength $S$, we demonstrate that for multiscale and broad phase spectra critical behavior transmutes a quasilinear to a nonlinear diffractogram except for low frequencies. On the contrary, a single-scale and broad phase spectrum induces a critical transition in the diffractogram at low frequencies. In both cases, identifying critical behavior encoded in the intensity power spectra is of fundamental interest because it exhibits the limits of perturbative power counting but also guides resolution and contrast optimization in propagation-based, single-distance, phase-contrast imaging, given certain dose and coherence constraints.

Figure 1

Simulated parallel-beam and monochromatic forward propagation of the pure-phase wave field ${\psi }_0=\sqrt{I_0}exp\left(i\varphi \right)$. Here, ${\varphi }_{\mathrm{weak}}$ is the Lena phase map (MOBS) in the upper right corner of panel (a). This simulation employs a pixel size of $\mathrm{\Delta }x=1.1$ $\mu \mathrm{m}$, a photon energy of $E=20$ keV, and a propagation distance of $z=1\mathrm{m}$. (a) Main figure: $ln(1+|\mathcal{F}{g}_z|/|\mathcal{F}{g}_z{|}_{\text{max}})$, where the subscript “max” refers to the global maximum of $|\mathcal{F}{g}_z|$. (b) $\widetilde{|\mathcal{F}{g}_z|}(S,\sigma )/{\left|\mathcal{F}{g}_z\right|}_{\text{max}}\left(S\right)$; for the definition of angular average in $\widetilde{|\mathcal{F}{g}_z|}(S,\sigma )$ [see Eq. (8)]. Here, $|\mathcal{F}{g}_z{|}_{\max }\left(S\right)$ is defined as the maximum of $|\mathcal{F}{g}_z|\left(S\right)$ on disk $\sigma /\pi \le 1$ for $S=1...500$. The family of curves is ordered according to increasing values of $S$, the curve for $S=1$ lying lowest. Black, dashed line connects absolute minima ${\sigma }_{1,\min }\left(z\right)/\pi$ of $\widetilde{|\mathcal{F}{g}_z|}(S,\sigma )$. (c) ${\sigma }_{1,\min }\left(z\right)/\pi$ in dependence of $S$ extracted from a fit to the model of Eq. (9) for $\tau =2$ [see Eq. (10)] and $z=1\mathrm{m}$. (d) Transition line $S_c$ in the $S-z$ plane in dependence of $\tau$, compare with Fig. 1. Here, $S_c$ relates to the critical behavior of ${\sigma }_{1,\min }(z,\tau )/\pi$ with respect to the fit of $\widetilde{|\mathcal{F}{g}_z|}$ to the model of Eq. (9). To point out two orthogonal transitions between quasilinear (unprimed) and nonlinear (primed) contrast transfer, the pairs of points $P_S,P_{S^{\prime }}$ and $P_z,P_{z^{\prime }}$ are indicated.

Figure 2

Simulation of experimental forward propagation with $E=17.5\mathrm{keV}$, $\mathrm{\Delta }E/E=0.03$, $\mathrm{\Delta }x=1.6$ $\mu \mathrm{m}$, and variable $S$ and $z$. (a) Phase map ${\varphi }_{\mathrm{weak}}$. (b) $\widetilde{|\mathcal{F}{g}_z|}(S,\sigma )/{\left|\mathcal{F}{g}_z\right|}_{\max }\left(S\right)$, centered about ${\sigma }_{1,\min }/\pi =1$, for $S=1$ and $z=0.6$ m (quasilinear) and $S^{\prime }=900$ and $z=0.6$ m (nonlinear) [pair $P_S,P_{S^{\prime }}$ in Fig. 1], and for $S=450$ and $z=0.6\mathrm{m}$ (quasilinear) and $S=450$ and $z^{\prime }=0.4$ m (nonlinear) [pair $P_z,P_{z^{\prime }}$ in Fig. 1].

Figure 3

ESRF experiment with $E=17.5\mathrm{keV}$, $\mathrm{\Delta }E/E=0.03$, $\mathrm{\Delta }x=1.6$ $\mu \mathrm{m}$, and variable $S$. Sample consists of two regions creating weak (scratched) and strong (Manhattan) phase variations (see Appendix pp3). (a) $ln\left(1+|\mathcal{F}{g}_z|/|\mathcal{F}{g}_z{|}_{\max }\right)$ at $z=3.6\mathrm{m}$ from scratched (left) and Manhattan (right). (b) $|\widetilde{\mathcal{F}{g}_z|}/{\left|\mathcal{F}{g}_z\right|}_{\max }$ at $z=3.6$ m from scratched (upper curve) and Manhattan (lower curve), both defining the pair $P_S,P_{S^{\prime }}$. (c) $|\widetilde{\mathcal{F}{g}_z|}/{\left|\mathcal{F}{g}_z\right|}_{\max }$ from scratched for $z=0.625$ m, $z=0.6$ m, $z=0.575$ m, $z=0.55$ m, and $z^{\prime }=0.525$ m (pair $P_z,P_{z^{\prime }}$).

Figure 4

Calculated spectra of normalized intensity contrast based on forward propagation of Gaussian-phase wave field (SOBS) [see Eq. (2)]. (a) $\mathcal{F}{g}_z/\left(4\pi w\right)$ as a function of $\sigma /\pi$ for $S=1.0$ and $F_w=0.02$, 0.01, and 0.005. Arrows indicate two additional zeros ${\sigma }_{*}$ $(S=1.0$, $F_w=0.01$ and 0.005) (dashed and dotted curves). According to Eq. (19), $S=1.0$ is subcritical for $F_w=0.02$ (no additional zero ${\sigma }_{*}$ in the solid curve). (b) $\mathcal{F}{g}_z/\left(4\pi w\right)$ as a function of $\sigma /\pi$ for $F_w=0.01$ and $S=0.5$, 1.0, and 2.0. Again, the arrows indicate two additional zeros ${\sigma }_{*}$ $(S=1.0$ and 2.0, $F_w=0.01)$ (dashed and dotted curves). $F_w=0.01$ is subcritical for $S=0.5$ (no additional zero ${\sigma }_{*}$ in the solid curve).

Figure 5

Critical behavior of additional zero ${\sigma }_{*}$: $S$ dependence. (a) ${\sigma }_{*}(S,F_w=0.01,0.02,0.03)/\pi$ (violet $+$, red $\times$, black $*$) and ${\sigma }_s$ [solid curve, see Eq. (16)]. Note that ${\sigma }_{*}$ approaches ${\sigma }_s={\sigma }_{*}(S,F_w=0)$ earlier for lower values of $F_w$. (b) Plot of $ln{\sigma }_{*}$ over $ln\left|S-S\left(F_w\right)\right|$ for $F_w=0.01$.

Figure 6

Critical behavior of additional zero ${\sigma }_{*}$: $F_w$ dependence. (a) ${\sigma }_{*}(S=1.0,2.0,3.0,F_w)/\pi$ (violet $+$, red $\times$, black $*$). (b) Plot of $ln{\sigma }_{*}$ over $ln\left|F_w-F_w\left(S\right)\right|$ for $S=1.0$.

Figure 7

$\widetilde{|\mathcal{F}{g}_z|}(S,\sigma )/\widetilde{|\mathcal{F}{g}_z|}(S=1,{\sigma }_1)$ as a function of $S$ at ${\sigma }_{1/2},{\sigma }_{3/2}$ when upscaling the phase map ${\varphi }_{\text{weak}}$ of Fig. 1.

Figure 8

Family of curves $S_c(z,\tau )$, defined by the maxima in $S$ of $G(S,z,\tau )$ [see Eq. (B1)], for $0.5\le \tau \le 2$ with an equidistant spacing of $\mathrm{\Delta }\tau =0.1$. Simulation based on the phase map of Fig. 1 of the main text.

Figure 9

Plot of function $G(S,z,\tau =0.5)$ in Eq. (B1) for $S=1,...,1000$ and $z=0.4,...,1.6$ m, as computed from the Lena phase map ${\varphi }_{\mathrm{weak}}$ indicated in Fig. 1, employing pixel size $\mathrm{\Delta }x=1.1$ $\mu \mathrm{m}$ and $E=20$ keV.

Figure 10

Experiment on a single-line harmonic at ID19 of ESRF. Parameter values are $E=17.5$ keV, $\mathrm{\Delta }E/E=0.03$, $\mathrm{\Delta }x=1.6$ $\mu \mathrm{m}$, and variable $S$ and $z$. (a) 3D printed sample including holding shaft (left) and magnified scratched (1) and Manhattan (2) regions. (b) Intensity $I_0$ of wave field ${\psi }_0$ (upper half) and propagated intensity ($z=3.6$ m, lower half), both representing the Manhattan region.