Phase contrast microscope

In many specimens such as living cells there is only a small difference in transparency between the structure being imaged and the surrounding medium. In these cases, conventional bright field microscopy often fails to create an image with sufficient contrast. Phase contrast microscopy can be used to create a image with a strong contrast ratio by coherently re-interfering a reference, or surround beam (S), with a diffracted beam (D) from the specimen. The index contrast between the specimen and the surrounding medium creates a phase delay between the two beams, which can be augmented by a phase plate in the S beam. The resulting image, where the phase difference is translated into an amplitude variation at the image plane by interference, can have a high contrast ratio particularly if both beams have the same amplitude at the image plane.

A standard phase contrast microscope design is shown below. The S beam is shown in yellow, and the D beam is shown in blue.

FDTD uses the finite difference time domain technique to rigorously solve for the object fields at the specimen plane, correctly accounting for all the phase delay and diffractive effects even for wavelength scale structures. All diffraction, refraction, interference, absorption and polarization effects are calculated in the near field of the specimen without approximation. By post-processing the FDTD simulation data, the equivalent bright field microscope image and the phase contrast image can be reconstructed at the image plane. In the case of the phase contrast microscope, the image can be calculated for an arbitrary phase delay due to the phase plate without having to re-simulate the fields at the specimen.

The example on the following pages will show how to use FDTD to calculate the image of a complex, low index, structure with a phase contrast microscope. We will compare the results to the image obtained by bright field microscope of the same specimen.

In this example, FDTD only simulates the blue region (from Specimen to Image). The simulation and analysis will proceed with the assumptions listed below. Most of the above assumptions can be modified easily.

1. We will model the specimen illuminated by plane waves at a polar angle of 30 degrees. This angle depends on the dimensions and properties of the condenser annulus and condenser lens and is determined using other ray tracing programs.
2. We will assume that the specimen is periodically replicated, but we will only image one period at the image plane.
3. We assume that the plane waves at different azimuthal angles are incoherent. Ideally, we would simulate a large number of azimuthal angles, but in this example we will consider only 4 azimuthal angles.
4. We assume that the plane waves are unpolarized.
5. We assume that the collection optics has a numerical aperture of 0.8.
6. We assume that the magnification factor is 4.

Simulation

For the FDTD simulations, we consider a complex shape as shown in the screenshot below. The specimen is a glass rectangle with index of n=1.5 and dimension of 4μm x 4μm x 0.25μm. The specimen is etched with half toroid hole, inside of which is a glass ellipsoid with index n=1.4 and dimensions of 1μm x 1μm x 0.25μm in the x, y and z directions, respectively.

The light source is a plane wave of polar angle 30 degrees and wavelength 500 nm.

The fsp file phase_contrast1.fsp contains the template of the structure and source. The script file phase_contrast_batch_run.lsf  will run simulations for 4 azimuthal angles. For each angle, it will calculate 2 orthogonal polarization states for a total of 8 simulations. It will then repeat the simulations without the structure in order to calculate the precise phases of the surround (S) beam. Each simulation will be saved as an fsp file with a different name for later analysis.

The simulations with the specimen will re-use the mesh from the first simulation, thereby saving computation time.

Analysis

After running the script file phase_contrast_batch_run.lsf, the script file phase_constrast_analysis.lsf can be used to calculate the fields at the image plane.  This script will do the following analysis:  

1. The near field image is decomposed into grating order (gratingpolar command). This is similar to the near to far field projection, but in a periodic situation, which calculates the angular distribution of the fields. The end result is E_r(u_x,u_y), E_θ(u_x,u_y) and E_φ(u_x,u_y) where r, θ, φ refer to a spherical coordinate system and u_x and u_y are the direction cosines that describe the angle of the light. The in-plane wave vectors for each plane wave are given by k_x = k * u_x and k_y = k * u_y, where k = 2*pi/λ.  Now that we have E(kx,ky) we can do Fourier optics!  
2. We assumed periodic (or Bloch) boundaries in this example. This is not strictly necessary (please see the Imaging section for an example with a finite sized beam). However, since we used a plane wave source with Bloch boundaries, we know that light can only diffract at specific angles that is given by the Bragg condition. We use the internal function gratingpolar to calculate the strength of each order in spherical coordinates.  
3. The magnification factor is applied. A lens system that provides magnification is only modifying the angle of the light. So it is as simple as multiplying the direction cosines, u_x and u_y by a desired factor. Normally, this would lead to a lot of complications because of the vectorial nature of the E field. However, the beauty of working in spherical coordinates (E_r, E_θ, E_φ) is that the vectorial components do not changes when u_x and u_y are modified because they are a local coordinate system that is tied to the value of u_x and u_y.
4. We can also apply a numerical aperture which clips any light that has a very steep angle, (i.e. would not be collected by the lens system). This means that all beams with u_x² + u_y² > NA² are clipped.
5. We return to Cartesian coordinates because this is what we want at the image plane. This is a straightforward coordinate change.
6. We calculate the inverse of the original far field projection.  This is easier with plane waves than the original calculation because we only need to sum all the plane waves multiplied by the correct phase factor:$$E_{-} \text {image }=\sum E\left(k_{x}, k_{x}\right) e^{i k_{x} x+i k_{y} y+i k_{z} z}$$

The focal plane is where z = 0, so we simply have a Fourier transform to calculate. This is achieved with the chirped z-transform (czt), which is just a more convenient form of fft.

1. The bright field images E2_image and E2_ref_image are calculated by incoherently summing the contribution from each azimuthal angle and polarization.  
2. The phase contrast image E2_pc_image can be calculated by coherently summing the scattered and reference beams, with some desired phase offset.  The script calculates the image for 5 different phase ofsets.  The contribution from each azimuthal angle and polarization are then added incoherently.  

Results

The script phase_constrast_analysis.lsf will generate the following figures.

The Near field (object field) at one polar angle, azimuthal angle and polarization:

Image plane of the phase contrast microscope for various phase delays of the S beam.  We see that a significantly better image resolution can be achieved with the phase contrast microscope for the appropriate phase delay of the S beam.

For comparison purposes, the script also calculates an image of the specimen as produced by a bright field microscope.  It is clear that the bright field image does not resolve the structure as well as the phase contrast microscope.