Matrix Analysis of Warped Stretch Imaging

Abstract

Sensitive and fast optical imaging is needed for scientific instruments, machine vision, and biomedical diagnostics. Many of the fundamental challenges are addressed with time stretch imaging, which has been used for ultrafast continuous imaging for a diverse range of applications, such as biomarker-free cell classification, the monitoring of laser ablation, and the inspection of flat panel displays. With frame rates exceeding a million scans per second, the firehose of data generated by the time stretch camera requires optical data compression. Warped stretch imaging technology utilizes nonuniform spectrotemporal optical operations to compress the image in a single-shot real-time fashion. Here, we present a matrix analysis method for the evaluation of these systems and quantify important design parameters and the spatial resolution. The key principles of the system include (1) time/warped stretch transformation and (2) the spatial dispersion of ultrashort optical pulse, which are traced with simple computation of ray-pulse matrix. Furthermore, a mathematical model is constructed for the simulation of imaging operations while considering the optical and electrical response of the system. The proposed analysis method was applied to an example time stretch imaging system via simulation and validated with experimental data.

Figure 1

Principles of time/warped stretch imaging. (a) Time/warped stretch imaging employs frequency-to-time and frequency-to-space mapping of an ultrashort optical pulse to perform a line scan. Here, to clarify the method, we use a one-dimensional spatial disperser, which leads to a line scan per pulse. A 4 × 4 ray-pulse matrix computation of vector (space (x), slope (θ), time (t) and frequency (f)) specifies the optical mapping relationship. Colored circles represent uniformly spaced frequency components (beam-pulselets) and rainbow gradient lines represent their continuous distribution in both space and time domains. (b) Uniform temporal sampling under this space-to-time relationship determines the spatial sampling position on the image space. Yellow dots on the fingerprint sample image account for the warped pixel distribution of a given imaging system. (c) The corresponding temporal scan signal from the sampled points is acquired by a single-pixel photodetector and an analog-to-digital converter (ADC). The three pronounced peaks in the scan signal correspond to the fingerprint ridges in the center of the line scan. The scan signal is reconstructed by remapping the temporal data-stream back to the original image space. Regions with higher temporal dispersion are effectively assigned more samples (central region), while the part of the waveform that is not highly time-stretched corresponds to fewer imaging pixels (peripheral regions).

Figure 2

Demonstration of modified matrix analysis for laser pulse compressor. Red, blue and green line represent optical paths for each frequency component. (a) Schematic of laser pulse compressor. The center wavelength of the pulse is 810 nm, and the incident angle of the pulse at the first diffraction grating is 63 degrees (Littrow angle), and the separation distance between the diffraction grating pair is 1 cm. (b) Group delay, (c) group-delay dispersion, and (d) third-order dispersion of the laser pulse compressor evaluated by analytical formula (blue solid line), modified matrix (red circle) and original Kostenbauder matrix (grey dashed line).

Figure 3

Matrix analysis of time stretch imaging system. (a) Schematic of time stretch imaging system (laser scanner-type). Ultrashort optical pulses with a center wavelength of 810 nm and an optical bandwidth of 20 nm is generated by a Ti:Sapphire mode-locked laser. The group delay dispersion of the dispersive fiber is −650 ps/nm and the groove density of the diffraction grating pair is 2200 lines/mm. The light blue and light green bounded graphs represent the spectro-spatio-temporal maps of the optical pulse after the dispersive fiber and after the diffraction grating, respectively. The temporal position is t = −z/c (z: propagation direction and c: velocity of light) (b) Frequency-to-time mapping function $t_{\omega }(\omega )$ after the dispersive fiber. (c) Frequency-to-space relationship x _ω(ω) and (d) time-to-space mapping function x _t(t) after the diffraction grating. Different types of spatial dispersion configuration are shown in (e) and (f). Either transmission or reflection type diffraction gratings can be used in any of these configurations.

Figure 4

Reconstruction and numerical simulation of time stretch imaging system from ref. . (a) A 0.5 mm wide slit is placed immediately before the target sample and translated transversely along the scanning beam at intervals of 0.5 mm. The total number of scans is 49. The FOV of the system is limited to 24.5 mm, since the time stretched pulse width is limited by 11 ns to avoid overlapping of the consecutive pulses. (b) Recorded scan data from the digitizer. (c) Reconstructed image from simulation. (d) Reconstructed image from the experimental scan data. The inset shows a comparison of the reconstructed image from the simulation (solid red) and from experimental data (solid blue). The 24th scan number is indicated by the dashed grey line.

Figure 5

LTF and MTF spatial resolution analysis of time stretch imaging system in ref. . The temporal response of the detection system was assumed to have a rise-time of 750 ps, and a sampling rate of 20 GS. (a) The spatial resolution of a single line scan. Grating: diffraction grating pair; Fiber: dispersive fiber; EB: electrical bandwidth; SR: sampling rate; Total: overall spatial resolution as defined by the FWHM of the LSF. (b) Three-dimensional color plot of MTF. The black-dashed line represents the MTF cut-off frequency at half-maximum. (c) MTF as evaluated for x = 2, 13, 24 mm each corresponding to the white-dashed lines in (b).

Figure 6

The effects of incidence angle on the space-to-time mapping and spatial resolution. (a) The space-to-time mapping at three different incident angles 55.2° (“1”, solid red), 56.7° (“2”, solid blue), 63° (“3”, solid green). (b) The reconstructed images from numerical simulation at the three incidence angles. The 5-mm period picket fence pattern used as the sample is shown in the background for reference. (c,d, and e) Spatial resolution profiles at the three incidence angles. The FOV is 45.1 mm, 28.2 mm, and 10.9 mm at each respective angle, respectively.