MULTIWAVELENGTH METASURFACES
Appendix 3.3: Additional information and discussion for TPM
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Figure 3.A2: Measurement and simulation results for the lenses with a lower NA. (a)and(b)Measured intensity in the focal plane of a double wavelength lens (1000-µm focal length, 300-µm diameter) at 915 nm a, and 1550 nmb. At 915 nm the lens actually focuses the light 980µm away from its surface, so the focal spot shown here is imaged at ≈980 µm away from the surface. The error in focal distance is probably due to the approximation made in the mode diameter of the fiber (see Fig. 3.A1S1), which affects the focusing distance of a low-NA lens more than that of a high-NA lens. (c)Intensity measured in the axial planes of the lens for 915 nm. (d)The same axial plots for 1550 nm. (e) and(f)Simulated focal plane intensity of a lens with the same NA as the one shown in (a–d) but with a diameter of 75 µm at wavelengths of 915 nm (e), and 1550 nm(f). (g)and(h)Simulated intensity profiles in the axial planes at 915 nm and 1550 nm, respectively, calculated for the same lens described ine.
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Figure 3.A3:Double wavelength blazed gratings based on the meta-molecule design. (a)Schematic of the simulated grating. The 5 degree grating is 322 meta- molecules long, while the 20-degree one is 146 meta-molecules long. Periodic boundary conditions were used in side boundaries and perfectly matched layers were used to terminate the simulation domain in top and bottom directions. (b) and(c)Real part of the electric field a few wavelengths after the 5-degreeb, and 20-degreecgratings. (d)and(e)Distribution of transmitted and reflected power in different angles for the 5-degreed, and 20-degreeegratings.
generated with EBL in a positive electron-beam resist. An Al2O3layer was deposited on the generated pattern after development, and was used to reverse the pattern through lift-off. The patterned Al2O3layer was then used as a hard mask in the dry etching process of the p-Si layer. Finally, a∼2-µm-thick layer of SU-8 was spin coated on the metasurface, and a gold aperture was deposited around the lens to reduce the background light.
Measurement procedure. The setup in Fig. 3.A6 was used for the DW-ML characterization measurements. A collimated beam was used to illuminate the DW-ML and the intensity distribution was imaged at multiple planes around the focal plane and parallel to it. These images were compiled to form the axial intensity distribution profiles of Figs.3.19band3.19c. The two-photon image in Fig.3.20bis captured by replacing the objective lens in a TPM by the DW-ML. The DW-ML was oriented in such a way that the excitation axis of the metasurface (corresponding to the 820-nm wavelength) overlapped with the polarization direction of the excitation laser. The details of the TPM are shown in Fig.3.A4. Both the DW-ML and the refractive objective are used to image a polyethylene microsphere about 90µm in diameter and coated with a fluorophore (UVPMS-BR-0.995 10-90mµm, Cospheric).
Effects of chromatic dispersion on efficiency. In this section we first study the effect of the finite bandwidth of the light used to characterize the DW-ML at 605 nm.
Second, we estimate the effect of the bandwidth of the pulsed laser on the excitation efficiency of the two-photon imaging. In both cases, we model the lens as a transmission mask with constant amplitude and phase over the bandwidth. This model results in an upper-bound for the efficiency, as it only takes the diffractive chromatic dispersion into account, and overlooks the wavelength dependence of the nano-posts over the bandwidth of the pulse.
To estimate the apparent reduction in measured efficiency due to the finite bandwidth of the 600-nm source, we simulated the focusing of the DW-ML at 41 wavelengths (580 nm to 620 nm, at 1-nm separations). This bandwidth was chosen to completely cover the pass-band of the used filter. The simulation was performed through modeling the DW-ML as a complex transmission mask, and propagating the fields after the DW-ML using a PWE code. To get the total intensity in the focal plane, the weighted intensities (using the transmission values of the bandpass filter) were added for all the wavelengths. The integral of this total intensity in a disk with a 5-µm diameter was divided to the total power before the DW-ML to achieve a
∼38% focusing efficiency. In addition, we calculated the efficiency at the center
wavelength of 600 nm to be∼63%. Therefore, we estimated the experimental single wavelength focusing efficiency to be∼45%. As expected from our previous work, the experimental focusing efficiency at the shorter wavelength is more sensitive to fabrication errors [137].
The excitation efficiency in the two-photon process is proportional to the intensity squared. To estimate the ratio of the peak intensity squared for the DW-ML and the conventional objective lens, we simulated both cases using the method explained above. We modeled the DW-ML by its transmission mask at 820 nm, and modeled the objective as a perfect aspheric phase mask. The pulsed laser has a/120 fs width, and assuming a bandwidth-limited Gaussian pulse, we find that it has a'9 nm bandwidth.
We calculated the electric field distribution in the focal plane for both the DW-ML and the conventional objective for all the wavelengths. Since the exact waveform of the pulse after passing through the setup is not known, we calculated the peak intensity ratio in two extreme cases. First, we assumed that all different wavelengths add up in phase in the focal plane (corresponding to a case with the shortest possible pulse width and the largest peak-to-average power ratio). Second, we assumed that the pulse is completely broadened (with a peak-to-average power ratio of one). To model this, we added the simulated intensities of different wavelengths in the focal plane. With equal input powers for the DW-ML and the conventional objective, the peak intensity ratios in the two cases were∼12.7 and∼4.9, respectively. In reality, the ratio must be between these two values because the pulse reaching the focal point is broadened due to dispersion, yet it’s not completely incoherent.
Using these pieces of data, we can estimate the contribution of different factors to the lower excitation-collection efficiency of the DW-ML compared to the conventional objective. As observed in Figs. 3.20b and 3.20c, the collected power with the DW-ML is about 0.06 of the collected power with the objective. In addition, the excitation laser power with the DW-ML is about 4.7 times larger than the objective.
The collected power is proportional to the peak excitation intensity squared, and the collection efficiency. Therefore we can write
POLCol
PDW−MLCol = ηColOL
ηDW−MLCol ( POLExc PDWExc−ML
IOLp
IDW−MLp )2, (3.1) where the subscripts determine the utilized lens (OL denoting the conventional objective lens), Col and Exc denote collection and excitation, and Ip is the peak intensity for the lenses calculated with equal excitation powers. Using this equation
and the 22.5% collection efficiency of the DW-ML, we can estimate the IOL
IDW−MLp ratio to be∼9.1. This ratio falls well within the 4.9-12.7 range that was calculated.