We used surface profilometry to characterize the phase mask quality and conducted optical tests to establish the optical efficiency and PSF performance.
3.1. Profilometric measurement
A white light interferometer (Zygo New View) was used to verify that the fabricated mask follows the required phase design of the DH-PSF mask. shows the design height relief for 514 nm wavelength and the mask design from Ref. [25
], while (b) shows the surface profile of the fabricated mask as measured by the profilometer.
The fabricated mask matches the design relief in terms of position of phase vortices, phase gradient and location of the extrema. It is important to note that phase vortices and their locations are critical for generating the desired DH-PSF. The mask was fabricated and tested for λ = 514 nm. The 2π phase delay in wavefront corresponds to a thickness of 816 nm for this photoresist. The imperfections shown in contribute to some scattering, but as we show below, they lead to minor deviations from the ideal DH-PSF design.
3.2. Efficiency measurement
The transmission efficiency of the fabricated mask and its ability to localize energy into the characteristic pattern was measured to determine the deviation from ideal performance. The testing setup is shown in
. The 514 nm wavelength line of an Ar + laser (Coherent) is spatial filtered and collimated with a lens of 100 mm focal length (L1). The resulting collimated beam is apertured right after the lens to a 2.7 mm diameter. A second lens (L2) of focal length f = 100 mm focuses the beam onto a CCD camera (Point Grey Research, Chameleon CMLN-13S2M-CS). The fabricated phase mask is placed at the front focal plane of L2 to generate the DH-PSF, and mounted on an x-y-z translation. Alignment is determined via observation of the PSF as the stage is adjusted. Images of the standard PSF are acquired by removing the fabricated mask. The camera is adjusted to maximize dynamic range using the standard PSF and these parameters are stored for later testing of the DH-PSF with the fabricated phase mask in place.
Fig. 2 (a) Setup for efficiency measurement of the fabricated DH mask. The 514 nm Ar + laser line is spatially filtered. L1 and L2 are achromatic lenses of focal length f = 100 mm. The phase mask is positioned at the back focal plane of L1 (front focal plane (more ...)
show the experimental standard PSF and DH-PSF images, respectively. show the PSFs simulated numerically [10
] with the same system parameters including noise. The dark regions in the experimental PSF images, shown in , were characterized and found to have single pixel 0/1 noise with density 0.3. This noise was added to the simulated PSF images so as to make theoretical efficiency calculations consistent with the experimental conditions.
One method of measuring the transmission efficiency, η, through the mask is to observe the reading of an optical power meter placed behind the Fourier plane with and without the fabricated mask in the beam path, resulting in η = 87%
Alternatively, the CCD measurements shown in were used to determine the transmission efficiency. The relative energy in the DH-PSF image with respect to the energy in the standard PSF image was found to be 83.4% and 95.8%, for the experimental and theoretical cases, respectively. The transmission efficiency of the phase mask is given by the ratio of these numbers leading to η = 87%. The theoretical relative energy was used to compensate the experimental relative energy for the effects of a finite detector. The transmission efficiency found by the two methods described above turned out to be the same.
The remaining 13% of the energy that is not transmitted is lost in reflection, absorption, and scattering. Specular and diffused reflection account for about 11% and can be easily reduced using anti-reflection coatings. Absorption and scattering can be reduced by improving the optical material quality, for instance, by transferring the mask to glass or quartz.
Another measure of the functional efficiency for the fabricated mask is the ratio of the experimental peak intensities for the standard and DH-PSF, [ISTD/IDH]EXP, as compared to the theoretical value [ISTD/IDH]THEO. These ratios were found to be 8.7 and 7.2 respectively giving a functional efficiency, [ISTD/IDH]THEO/[ISTD/IDH]EXP x100 = 83%. If this efficiency is compensated to take into account the transmission loss, it results in a functional efficiency of 83/.87 = 95%.
3.3. PSF characterization and calibration
The phase mask is placed in a custom table top fluorescence microscope depicted in
. The setup consists of a 488 nm Argon line for the excitation beam and a 405 nm laser diode for the activation beam (for PALM experiments). Both beams are coupled into the microscope objective (Plan-Neofluar Zeiss, NA = 1.3) using lens Lc of focal length f = 100 mm. A dichroic D1 (Omega QMAX 410) is used to combine the 488 nm and 405 nm beams, while dichroic D2 (Semrock Di01-R405/488/561/635) separates the excitation and activation beams from the emitted signal (at 514 nm). An achromatic lens, TL with f = 150 mm, together with the objective provides a magnification of 91X at the intermediate image plane (dashed line). The fabricated phase mask is placed in the Fourier plane of a 4F relay system such that the back focal plane of the microscope objective (which is inaccessible) is imaged onto the phase mask. The first lens L1 of the relay system is of focal length f = 100 mm. A second lens L2, f = 150 mm focuses the field onto the camera (Andor iXon EMCCD) for a total system magnification of ~137X. An iris is placed close to the phase mask to stop the beam down to the diameter of the fabricated mask, 2.7 mm. This limiting aperture reduces the NA of the system to 1.23. The fabricated mask is mounted on an x-y-z translation stage for precise alignment. A band pass emission filter (Omega QMAX 510-560) is placed in the beam path. The camera is cooled to −80°C for all experiments. The samples are mounted on a 3D, nanometer resolution piezo translation stage (Physike Instrumente PZ 164E).
Fig. 3 The setup for DH-PSF characterization and PM-DH-PALM experiment. The objective, Obj is 1.3NA Zeiss Plan-NeoFluar, Lc is f = 100 mm achromat. TL, L1 and L2 are achromatic lenses with focal lengths, fTL = 150 mm, fL1 = 100 mm and fL2 = 150 mm, respectively. (more ...)
The calibration of the DH-PSF is accomplished by imaging a slide containing quantum dots (QD-525, Invitrogen) which are mounted in a 90% glycerol solution. Quantum dot blinking was not observed in this medium at the imaging rates used in all the reported experiments.
The quantum dots were chosen so that the emission spectrum closely resembles that of PA-GFP (photo-activatable green fluorescent protein) for future use in PM-DH-PALM.
shows the image of a quantum dot cluster generating the shape of two lobes rotating with defocus to form a double-helix. The axial range of this system, corresponding to 180° rotation of the two lobes of double-helix, is 4 µm. The sample is translated through the full range in 100 nm intervals for calibration purposes. 60 images with 100 ms exposure are acquired at each axial position.
Fig. 4 The transverse (x-y) images for different z positions are shown for (a) the experimental PSF obtained with the fabricated DH phase mask, (b) the simulated PSF using the measured surface profile of the DH mask, and (c) the simulated PSF using the original (more ...)
For comparison, the PSFs generated numerically using the measured mask (from profilometer data), as well as the ideal design, are shown in respectively. The experimental PSF in is blurred relative to the PSFs in because the imaged particle was a quantum dot aggregate, which contributes to an increase in the lobe size relative to an ideal point source.
The relation between axial position and the rotation angle of the two lobes, generates a calibration curve, which is calculated with an angle estimator (AE) [12
] and is shown in . The AE calculates the weighted centroid of each lobe to give the rotation angle of the PSF at each z
position. The transverse position is estimated by the midpoint of the weighted centroids. The calibration curve matches the theoretical prediction very well in the axial region from z
= 1 µm to 3.9 µm. Due to aberrations in the intensity profile for the region z
< 1 µm, we did not consider it further in the analysis.
Interestingly, in the calibration curve for the simulated PSF, the full range of the rotation angle established by the AE is ± 70°, even though the two peaks of the PSF are seen to rotate by ± 90°. This difference is because the AE used here searches for two lobes by adaptively thresholding and calculating the weighted centroid of each lobe. The resulting thresholded lobes are not restricted to a particular shape around the peak. As a result, the weighted centroid is shifted with respect to the intensity peak for each lobe leading to a smaller rotation angle value.
The calibration data in is also used to estimate the precision of the 3D position estimator along the axial range of operation [12
]. The AE provides the x
position and rotation angle from the image. With the known rotation angle, the z
position is then determined from the calibration curve. The precision of the estimations with varying z
position are shown in
. For the 3 µm axial range the average precision is σ(X
) = (3.8 nm, 2.4 nm, 3.7 nm). For this quantum dot cluster, the number of photons detected in the two lobes at focus were 56000 ± 9400. If necessary, this precision can be improved by using the pattern matching maximum-likelihood estimator (MLE) [29
The precisions of the estimated x, y and z positions using the Angle Estimator for the quantum dot cluster used in the calibration.