Coherent anti-Stokes Raman Fourier ptychography

Abstract

We present a theoretical and numerical study of coherent anti-Stokes Raman scattering Fourier ptychography microscopy (CARS-FPM), a scheme that has not been considered so far in the previously reported CARS wide-field imaging schemes. In this approach, the distribution of the Raman scatterer density of the sample is reconstructed numerically from CARS images obtained under various angles of incidences of the pump or Stokes beam. Our inversion procedure is based on an accurate vectorial model linking the CARS image to the sample and yields both the real and imaginary parts of the susceptibility, the latter giving access to the Raman information, with an improved resolution.

Fig. 1

Phase-matching and lens transfer: a) Linear scattering: the incident plane wave of wave vector ${\mathbf{k}}_1^{\prime }$ is scattered into direction k₁ if the object contains the spatial frequency vector K following ${\mathbf{k}}_1^{\prime }+\mathbf{K}={\mathbf{k}}_1$. The scattered wave is collected through the pupil of the microscope if its transverse component, ${\mathbf{k}}_{\parallel ,1}$ satisfies, $k_{\parallel ,1}\le k_1\text{NA}$, i.e. if k₁ ends on the cap of the sphere outlined in dark blue. b) Nonlinear four-wave-mixing: the effective incident wave vector in direction ${\mathbf{k}}_{aS}^{\prime }={\mathbf{k}}_p+{\mathbf{k}}_p-{\mathbf{k}}_S$ for an homogeneous infinite non-linear medium is scattered in direction ${\mathbf{k}}_{aS}={\mathbf{k}}_{aS}^{\prime }+\mathbf{K}$ if the sample contains the spatial frequency vector K. The scattered anti-Stokes wave is collected by the microscope objective if $k_{\parallel ,aS}\le k_{aS}\text{NA}$. c) Classical phase-matching condition for a homogenous sample, i.e. K = 0 under plane-wave excitation. d) Conventional representation of the transfer function displaying the k_z-projection of the Ewald’s spheres of sub-Figs. a) and b).

Fig. 2

Declaration of variables: r coordinates of the object; R coordinates of the far-field; $\mathbf{R}\prime$ coordinates of the image plane; k wavevector in the object space; k’ wavevector in the image space; ${\mathbf{p}}^{(\prime )}$, ${\mathbf{s}}^{(\prime )}$ unit vectors of plane wave polarization; θ_S polar angle of the Stokes beam; θ_p polar angle of the pump beam; ϕ_S azimuth angle of the Stokes beam; ϕ_p azimuth angle of the pump beam; θ_l polar angle alteration after both lenses; f₁ focal length of lens ${\mathrm{L}}_1$; f₂ focal length of lens ${\mathrm{L}}_2$; z₀, z₁, z₂ axial sample coordinates; Z₀ axial image coordinates.

Fig. 3

Simulated 25 wide-field CARS images for different illumination angles of the Stokes beam and normal pump beam incidence. The images were calculated using Eq. (22). Parameters - Raman shift: 2850 cm⁻¹, λ_S = 1030 nm, λ_p = 796 nm λ_aS = 649 nm, magnification: 10x, defocus: z₀ = 20 μm, detection NA: 0.2. While normal incidence of the pump beam is assumed the illumination angles of the Stokes beam are outlined within each image (θ_S polar angle, ϕ_S azimuthal angle). The imaginary and real part of the ground truth (GT) of the 2D sample is displayed in Fig. 4 a and e. Poisson noise was added assuming that the brightest pixel on the camera collects a mean value of 1000 photons. The scale bar equals 200 μm. The blue circles on top of the Fourier images of the anti-Stokes polarization is intended as a guide to the eye to estimate the overlap of neighboring illuminations.

Fig. 4

Reconstruction results: The real and imaginary part of the sample’s ground truth (GT) are displayed in the sub-Figs. a and e, respectively. The images b and f were reconstructed from the center 9 wide-field CARS images presented in Fig. 3. Sub-Figs. c and g display the corresponding reconstruction results for 49 wide-field CARS images. Scanning the pump beam the images d and h were obtained. Note that though more images were included in the reconstruction the maximum excitation angle θ_p equals the maximum excitation angle in Stokes scanned configuration in sub-Figs. c and g. The added Poisson noise assumes 1000 photons for the brightest image pixel, i.e. a standard deviation of 3.2 %. Sub-Figs. i and j display the calculated in-focus amplitude and phase images resulting from the reconstructed complex sample structure of sub-Figs. c and g. The natural logarithm of the residue of the cost functional F as defined in Eq. (24) is plotted in the lower center of the Fig. as a function of the iteration number for 49 Stokes scanned wide-field CARS images. The continuous, dotted and dash-dotted lines account for a standard deviation of 3.2 %, 10 % and 31.6 % of added Poisson noise, respectively. The blue and white scale bar equal 10 μm and 100 μm, respectively.

Fig. 5

Proposed experimental implementation:1 High power pump laser at 1029-1065 nm (FWHM <2 nm) with a repetition rate 10 kHz - 1 MHz and a pulselength 1 ps, 2 Optical parametric amplifier (OPA) tunable within 730 - 1010 nm (200 - 4000 cm⁻¹), 3 laser-scanning mirrors, 4 4f-imaging system, 5 linear polarizer, 6 dichroic beam combiner, 7 sample and linear polarizer as well as dielectric filters, 8 low NA 4f-system, 9 low noise (cooled) CCD camera.

Fig. 6

Declaration of variables: r coordinate at the object plane; r₁ coordinates of the intersection of the object plane and the center ray (L₁) with propagation direction k; Rʹ coordinate of the image plane; R₁ coordinates of the intersection of the image plane and the center ray (L₂) with propagation direction k’; m coordinates of the intersection of the Fourier plane and the center ray (L₁) with propagation direction k.