Fig. 1. (a) Physical mechanism of SPMI’s defocus. Numerical simulations of PSF and corresponding OTF with σ=1,  2,  3,  4. PSF is an image of the ideal dot source transmitted from an objective lens. The PSF is transformed into the OTF, representing the optics system’s low-pass filter characteristic. The high-frequency amplitudes decrease as σ increases. The microscopic image is blurred due to this high-frequency band limit. (b) Blur model in objective focus. WD, working distance. The objective DOF and WD decrease with increasing NA. The reduced DOF of the objective leads to increased difficulty in focusing accurately. (c) and (d) Schematic of the autofocus SPMI system. PD, photodetector; G, grating; O2, objective lens 2; T, target; O1, objective lens 1; LC, light screen; DLP, digital light processing projector. (c) Grating modulation as an autofocus mark. (d) O2 and grating are removed, and the corresponding out-of-focus is solved.

Fig. 2. Experimental comparison of the two autofocus methods for SPMI. FSPS, four-step phase-shifting. The frequency coefficient for autofocus is measured with PD signals after four-step phase-shifting patterns modulation. (a) The procedure of dual modulation autofocus. The grating frequency calibration is necessary for determining which patterns to modulate the target. (b) The procedure of grating-free modulation autofocus. The arbitrary high-frequency patterns are theoretically feasible for grating-free SPMI autofocus.

Fig. 3. SPMI experiments with 10×, 20×, 40× objectives. DM, dual modulation; EOG, energy of gradient; LM, linear motor; GF, grating frequency. The measured Fourier spectra and reconstructed images of the scene. The image resolution of SPMI is 128×128-pixels. (a) DM SPMI experiments and illustration. (b) Grating-free SPMI experiments and illustration.

Fig. 4. Experimental results for dual modulation and grating-free method. Objective, 4× (NA 0.1) and 20× (NA 0.4). PD sampling rate, 1 MHz. DLP projecting rate, 22.7 kHz.

Fig. 5. Comparison of (a) the contrast maximization method and (b) this method.

Fig. 6. Experimental setups of SPI autofocus methods. (a) Dual modulation SPMI and (b) grating-free SPMI.

Fig. 7. The out-of-focus and in-focus spectra. (a) The spectra of cervical polyp slices, which are acquired under 4× (NA=0.1) out-of-focus SPMI experiments. (b) The spectra of cervical polyp slices, which are acquired under 4× (NA=0.1) in-focus SPMI experiments.

Fig. 8. The out-of-focus and in-focus spectra. (a) The spectra of esophageal cancer slices, which are acquired under 4× (NA=0.1) out-of-focus SPMI experiments. (b) The spectra of esophageal cancer slices, which are acquired under 4× (NA=0.1) in-focus SPMI experiments.