In sTPM, a cylindrical lens is used to focus the collimated beam to a line beam allowing single directional scan in a transverse plane. At the Fourier plane after cylindrical lens, one axis is a focused beam while the other axis is a plane wave. To simplify nomenclature, the axis along the focused beam direction is named a Fourier axis and the other is named an Imaging axis. In this configuration, the reconstruction problem is also reduced to 2D instead of 3D since the Imaging axis can be treated independently.
To utilize the synthetic aperture algorithm, both phase and amplitude of the transmitted E-field have to be recorded. Interferometry is widely used to measure the E-field [14
]; however a coherent plane wave is typically used as a illumination to generate uniform contrast of interference across the field of view. For a focused beam, limited dynamic range of detector impedes the proper recording of the large intensity variation. To reduce spatial intensity variation of the focused beam, a cylindrical lens is installed after the image plane to carry out optical Fourier transform along the Fourier axis. The optical transformation of the high spatial variation of the focused beam gives the uniform intensity distribution, which is ideally for recording. By numerical process with inverse Fourier transform to the recorded image, the E-field image of the line focus beam at the image plane can be obtained from the numerical Fourier transform of the recorded image along the Fourier axis.
Another modification is made to sTPM solely to be more applicable to flow. Instead of scanning the illumination beam along the Fourier axis as introduced in theory, the sample is translated across the line focused beam. The relative position of the sample with respect to the illumination beam is then used for numerical processing.
The detailed experimental setup is shown in . The E-field measurement is based on a heterodyne Mach-Zehnder interferometer [15
]. The output of a HeNe laser (20 mw, λ = 632.8 nm from Melles Griot, USA) is split by an optical beam splitter (not shown). One beam (the object beam, shown in red) travels through the object located on the sample stage of an inverted microscope (Axiovert S100, Zeiss Germany). The other beam (reference beam, shown in blue) travels outside of the microscope, with its frequency shifted by a pair of acousto-optic modulators (not shown) to modulate the relative phase between the object and the reference in time (see [10
]). The focused beam at the sample is produced by a 4F configuration of a cylindrical lens (C1) and a high NA condenser (L1) lens (1.4 NA oil immersion, Nikon Japan). With the use of the cylindrical lens, the optical system is presented in two orthogonal axes: Imaging and Fourier axes. Planar axis () has a plane wave illuminating beam at the sample and Fourier axis () has a focused illuminating beam. The transmitted beam, which now is shaped as a line focus beam is magnified by an objective (L2; 100x oil immersion, Zeiss Germany,) and tube lens (L3), and relayed to the image plane (IP) at microscope’s output port. A beam splitter (B1) is placed right after IP. The reflected beam containing the image is relayed to the video camera (M1; TC-87, Sony Japan) via a lens (L4). is a typical bright field image of a 10 µm polystyrene bead measured at M1. Note that a broadband LED (not shown) is used as a source and C1 is temporarily taken out. shows the line-focused beam illuminating the center of the same bead in (c), with the C1 in position.
Fig 1 Experimental setup. Two views depend on the axes of cylindrical lenses are shown: (a) Imaging axis in which the illumination beam is a plane wave, (b) Fourier axis in which the illumination is a focused beam. C1-4 are cylindrical lenses with focal lengths (more ...)
The transmitted beam from B1 and the reference beam interfere to give the phase and amplitude of the field at the image plane. Note that the variation of intensity in line focus beam cannot simply be digitized by a CCD detector due to limited dynamic range. The focused beam is expanded in the Fourier axis by Fourier transforming with a cylindrical lens (C2). Simultaneously two cylindrical lenses C3 and C4 relay the image on the Imaging axis. This makes the Fourier axis in spatial frequency coordinate (kx
) and the Imaging axis in spatial coordinate (y
) at the camera plane. The reference beam, a planar beam whose frequency is shifted by 1.25kHz using two acousto-optic modulators as mentioned previously [5
] is brought to interfere with the expanded focused beam at a fast CMOS camera (M2; Fastcam 1024PCI, Photron Japan). The camera records four interferometric images at 5kHz frame rate. Typical interferogram images are shown in , when line focus beam is positioned about the center of a 10 µm bead. By applying a four-frame phase shifting interferometry (4f-PSI) algorithm [16
], the phase image ϕ(kx, y)
is obtained (). To generate a scanning point source for this synthetic aperture tomography, the sample is translated across the line focus beam, along the Fourier axis by a precision micro-position translation stage (PI M-216, Physik Instrumente Germany) with a step size of 0.1 µm.