In this example, we investigate the effect of photothermal heating in plasmonic nanostructures based on the published work in the referenced article [1]. Using Lumerical's FDTD for optical simulation and Heat Transport Solver for thermal simulation, we look at two types of thermoplasmonic antennae; dipole and diabolo. In the first part of this example we consider arrays of both these antennae under varying optical intensity and compare their performance. In the second part of this example we look at a single diabolo antenna and demonstrate how its performance can be greatly enhanced by employing a plasmonic lens.
Part 1: Dipole vs. Diabolo Arrays
Simulation setup: Optical (FDTD)
Based on the experimental setup of Ref. [1], in this part of the example we consider two arrays of antennae, one using diabolo antennae and the other one using dipole antennae, both made out of gold. Open the diabolo_array.fsp project file in FDTD. The array of diabolo antennae is sitting on top of a thick Sapphire (Al2O3) substrate. The dimensions of each antenna are taken from Ref. [1]. The array has a pitch of 340 nm in both X and Y axes [1]. The FDTD simulation region covers one unit cell centered on one diabolo antenna and has a width equal the period of the array in both X and Y direction. The length (along Z) of the simulation region is kept at 1 micron. A plane wave source is used to illuminate the antennae array at an wavelength of 1064 nm. The symmetric nature of the antenna structure is used to reduce the simulation region to 1/4 th of its volume by using symmetric and anti-symmetric boundary conditions at the appropriate boundaries.
An (advanced) "Power absorbed" analysis group is used to measure and save the absorbed optical power in the antenna. The analysis group is modified so that it can calculate the absorbed power for a certain input power defined by the "input_power" parameter and can save the data in a .mat file whose name is defined by the "filename" parameter.
Optical simulation setup in FDTD
The dipole_array.fsp project file uses a similar setup using dipole antennae where each antenna is 215 nm long, 50 nm wide, and 50 nm thick [1].
Simulation setup: Thermal (HEAT)
Open the diabolo_array.ldev project file. The thermal simulation is performed for one unit cell centered on one gold diabolo antenna and so the structure in the project file contains a single antenna on top of a Sapphire (Al2O3) substrate. The dimensions of the antenna is identical to the setup in FDTD. The thickness of the substrate is set to 250 micron which is a typical value for commercially available sapphire wafers. An Import heat source is used to import the optical absorption data saved by FDTD to be used as heat input to the simulation. Two temperature monitors are placed surrounding the antenna to record the temperature profile. A sweep (Pin) is set up in the project file to apply a scaling factor on the heat source to perform multiple simulations under varying input powers.
The fixed temperature thermal boundary condition is applied at the bottom of the simulation region to set it at room temperature (300 K). The top of the simulation region is covered by air which has a convective boundary condition defined at its interface with gold and sapphire to model heat loss at the top surface of the structure through convection. The model for convection is chosen to be constant with a convective heat transfer coefficient h = 10 W/m2K from the Material Interfaces.
Convective boundary condition at the material interfaces with air
Results and discussion: Optical (FDTD)
Optical Resonance
Open the diabolo_array.fsp project file in FDTD. Open the property editor for the plane wave source (incident) and change the frequency range to start from 0.9 micron and stop at 1.2 micron. Run the simulation and visualize the Transmission from the "transmission" DFT monitor. Use scalar operation "-Re" to make the sign of the real part of transmission positive. The plot will show that the transmission has a minima around 1064 nm which is consistent with the findings of Ref. [1]. Alternatively, right click on the analysis group and plot Pabs_total to visualize the absorbed power which also shows a maxima around 1064 nm indicating that the diabolo antenna has a resonance at a wavelength of 1064 nm. The same simulation can be done for the dipole antenna using the dipole_array.fsp project file.
Transmission past the diabolo antenna as a function of wavelength
Optical absorption
Next, edit the property of the plane wave source again (in the diabolo_array.fsp project file) and change it back to a single wavelength of 1064 nm by setting both the start and stop wavelengths at 1.064 micron. Run the simulation again and once the simulation is run, right click on the analysis group and click "Run analysis". The analysis group will calculate the absorbed power in the antenna and save it in a .mat file whose name can be defined under from the Analysis tab. By default the name of the .mat file is set to be Pabs_diabolo_array_1mW.mat. The input power is scaled in the analysis group to an input power of 1mW for the entire array. Consistent with the experimental setup of Ref. [1], we assume that the incident beam has a FWHM of 19.5 micron covering 2583 antennae. Thus, in the analysis group, we use an input power of 1/2583 mW or 3.87147e-07 W for the unit cell to scale the absorbed optical power. After the analysis is done, we can right click on the analysis group to visualize the absorbed power (Pabs).
Absorbed power at a certain depth (Z value) inside the diabolo antenna
Following the same approach, simulate and save the absorbed power for the dipole antenna array using the dipole_array.fsp project file.
Results and discussion: Thermal (HEAT)
Open the diabolo_array.ldev project file. The heat source (heat) already has the optical absorption data loaded but you can also load the Pabs_diabolo_array_1mW.mat file onto the heat source and select the Pabs attribute to import the data saved by FDTD simulation. Set the scaling factor in the heat source to 55 corresponding to an optical input of 55 mW and run the simulation. Once the simulation is run, the results will be loaded in the HEAT solver region and in the temperature monitors. Right click on the first monitor (monitor_1) and visualize temperature to see the temperature profile.
Temperature profile at the antenna and the top surface of the substrate
Next, from the optimizations and sweep window, run the "Pin" sweep to vary the input optical power from 5 mW to 125 mW in 7 steps and run the simulations. The sweep saves the "boundary" dataset available in the HEAT solver region as a result which records the power flow at the boundaries. However, the result that we are interested in is the temperature rise in the simulation structure under different optical intensities. The script file get_Pin_sweep_result.lsf script file can be used to read the temperature profile from each of the simulation files and calculate the average increase in temperature so that it can be plotted against input optical power. Make sure that the "project_name" variable in the script is set to the name of the .ldev file and run the script.
The script will load the temperature data in monitor_2 of each simulation file to get the surface temperature for each incident power. We then assume that the 2583 antennae in the center of the array that gets illuminated by the incident light are at an uniform temperature given by the peak temperature in monitor_2 and the surrounding region where the antennae are not illuminated is at a constant temperature equal to the room temperature (300 K). the temperature distribution is thus assumed to have the top-hat profile shown below (left). The total surface area is considered to be 85 micron x 85 micron as per Ref. [1]. By comparing the ratio of the illuminated and unilluminated area, the increment in average surface temperature is found to be equal to 0.0413*(Tsim - 300). The script calculates this rise in average temperature and plots it as a function of input optical power. This averaging is done so that the simulation result for a single unit cell can be compared against the measured temperature of the entire experimental setup. The figure below (right) shows the temperature rise for both the diabolo and dipole arrays as a function of input power. The plot is in agreement with Fig. 4(b) of Ref. [1].
Temperature profile for the entire array of antennae
Temperature rise versus input optical power for both antenna arrays
Part 2: Single Diabolo with Plamonic Lens
Simulation setup: Optical (FDTD)
Open the diabolo_superstructure.fsp project file in FDTD. The simulated structure now contains the Sapphire substrate with a single gold antenna and a plasmonic lens surrounding the antenna (also referred to as the 'superstructure' in Ref. [1]). The material for the plasmonic lens is also gold and it has a pitch of 720 nm [1]. The setup is illuminated with a Gaussian beam source with a FWHM of 4.44 micron which translates to a beam radius of 3.774 micron at 1/e2 power which is the input parameter for the source. The span of the simulation region in X and Y dimension is much larger in this setup so that the intensity of the Gaussian source goes to zero at the simulation boundaries. PML boundary condition has been used in all three dimensions. The Pabs analysis group calculates the power absorbed in the antenna for an input power of 110 mW for comparison with Ref. [1]. The diabolo_single.fsp project file contains a similar setup but without the plasmonic lens so that we can see the effect of the lens on the amount of heating from the single antenna.
Optical simulation setup for the diabolo antenna with plasmonic lens (superstructure)
Simulation setup: Thermal (HEAT)
Open the diabolo_superstructure.ldev project file. The setup contains the single antenna on top of the Sapphire substrate. The thickness of the substrate is kept at 250 micron. However, the X and Y span of the simulation region has been made larger (4.44 micron) in this simulation to match the FWHM of the incident beam in order to account for the spreading of the heat in the surrounding area. The same two temperature monitors are used to capture the temperature and the same fixed temperature thermal boundary condition is used at the bottom surface of the substrate to keep it at 300 K. The top of the simulation region if filled with Air with convective boundary conditions defined in the Material Interfaces. The import heat source has been loaded with the result saved by FDTD simulation of the diabolo_superstructure.fsp project file. The diabolo_single.ldev project file also has an identical setup with the difference that the heat source has been loaded with result from the simulation of the diabolo_single.fsp project file.
Results and discussion: Optical (FDTD)
Run the diabolo_superstructure.fsp project file and then run the analysis. The analysis group is set up to save the power data in a .mat file named Pabs_diabolo_superstructure.mat. Similarly, open and run the diabolo_single.fsp project file and run the analysis. The power absorption data will be saved in a .mat file named Pabs_diabolo_single.mat. After the analysis in both files are done, right click on the analysis groups and visualize Pabs to compare the absorbed power between the two cases. As can be seen in the figure below, the absorbed power increased approximately by a factor of 4 in the superstructure due to the plasmonic lens.
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NOTE: Due to the large simulation volume, the run time of these simulations will be quite long as compared to the periodic array simulations. |
Pabs for the single antenna and the superstructure at Z= 0.00785 micron
Results and discussion: Thermal (HEAT)
Open and run the diabolo_superstructure.ldev project file. Once the simulation is run right click on monitor_1 and visualize temperature. Similarly, run the diabolo_single.ldev project file and visualize the temperature profile from monitor_1. The figure below shows that the peak temperature in the superstructure is considerably larger than in the single antenna. The peak rise in temperature in the superstructure (antenna with lens) is approximately 280 K where the peak rise in temperature for the single antenna is about 70 K. These results are in agreement with the simulation results from Ref. [1].
Temperature profile for the single antenna and the superstructure with the plasmonic lens
Related publications
[1] Z. J. Coppens, W. Li, D. G. Walker, and J. G. Valentine, "Probing and Controlling Photothermal Heat Generation in Plasmonic Nanostructures," Nano Letters, vol. 13, pp. 1023-28, 2013.