Canonical Clocks and Hidden Geometric Freedom in Self-Imaging

Abstract

Self-imaging represents a core hallmark of paraxial (quadratic)-wave evolution; yet, across its many realizations and generalizations over the past two centuries, the uniformity of recurrence planes along the propagation axis has been considered fundamental. However, by reframing the general phenomenon of self-imaging within its natural symplectic framework, we show that all self-imaging effects are necessarily tied to uniformly periodic recurrences in the canonical evolution coordinate -- metaplectic time -- and that the correspondence of that coordinate to the physical propagation axis represents an unexplored degree of freedom, which can be engineered arbitrarily by the initial transverse phase structure. Using a single programmable spatial light modulator, we demonstrate the construction of Talbot carpets characterized by recurrence spacings that accelerate and decelerate along the propagation axis, as well as those that follow polynomial, exponential, and sinusoidal axial trajectories, all of which preserve exact reconstruction in the canonical metaplectic time. These results establish metaplectic time as the fundamental invariant of self-imaging and reveal a regime of controllable axial dynamics previously thought to be fixed.

Figures (4)

• Figure 1: Measurements of accelerating Talbot effect. Measured intensity $I(x,y=0,z)$ with $d = 85$$\mu$m and $\lambda = 635$ nm for an effective focal length of (a) $f = \infty$, (b) $100$ mm, and (c) $75$ mm. White dotted lines mark the self-imaging planes for each case.
• Figure 2: Measurements of decelerating Talbot effect. Measured intensity $I(x,y=0,z)$ with $d = 60$$\mu$m and $\lambda = 635$ nm for (a) $f = \infty$, (b) $-100$ mm, and (c) $-75$ mm. White dotted lines mark the self-imaging planes for each case.
• Figure 3: Measurements and calculations of tailored Talbot effect. Intensity measurements of $I(x,y=0,z)$ in the first column and calculated in the second column for varying conditions with initial period of $d = 70$$\mu$m for (a) linear Talbot effect, (b) quadratic acceleration with $z_k = z_Tk+\delta z \ k^2$ condition with $\delta z/z_T =-0.08$, (c) cubic acceleration with $z_k = z_Tk+\delta z \ k^3$ with $\delta z/z_T = -0.02$, (d) fractional acceleration with $z_k = z_Tk+\delta z \ k^{5/4}$ with $\delta z/z_T = -0.325$, (e) exponential acceleration with $z_k = z_Tk+\delta z \ e^{\alpha k}$ with $\delta z/z_T = 0.015$ and $\alpha = 1$ and (f) sinusoidal acceleration with $z_k = z_Tk+\delta z \ \sin(2\pi k/\alpha)$ with $\delta z/z_T = 0.15$ and $\alpha = 1$ where $k$ is the numbered self-imaging plane.
• Figure 4: Plot of arbitrary Talbot self-imaging planes. Plots of the measurements in Fig. \ref{['Fig:Arbitrary']} of the half self-imaging planes for the quadratic (black empty circle), cubic (green square), fractional (blue diamond), exponential (orange x), and sinusoidal (red full circle).