Direct-backward third harmonic generation (DBTHG) has been regarded as negligible or even inexistent due to the large value of wave-vector mismatch. In the past, BTHG signals were often interpreted as back-reflected or back-scattered forward-THG (FTHG). In this paper, we theoretically and experimentally demonstrate that backward third harmonic waves can be directly generated, and that their magnitude can be comparable with FTHG in nanostructures. Experimental data of DBTHG from ZnO thin films, CdSe quantum dots and Fe3O4 nanoparticles agree well with simulation results based on the Green’s function. An integral equation was also derived for fast computation of DBTHG in nano films. Our investigation suggests that DBTHG can be a potentially powerful tool in nano-science research, especially when combined with FTHG measurements.
© 2010 OSA
Third-harmonic generation (THG) is an electric-dipole-allowed process and is therefore non-vanishing in all materials, regardless of being centrosymmetric or not. Because of the Gouy phase shift, THG signals from a focused Gaussian beam can be observed only in the vicinity of interfaces, where the first- or the third-order susceptibilities are not homogeneous. This interface-sensitive feature makes THG a versatile tool in biological applications [1–12] and material science studies [13–17]. Due to momentum conservation, most of the third-harmonic waves are generated in the forward direction, and previous experimental setups were largely limited to the transmission-type geometry [5–7,18–22]. In many applications, however, collection of backward THG (BTHG) would be the only option, such as microscopy on optically-thick or highly-scattering samples. In other cases where a transmission-type configuration is possible, detection of BTHG would provide additional information when combined with signals from forward THG (FTHG) . An understanding on the behaviors of BTHG is thus essential and deserves special attention.
Applications of BTHG have been successfully demonstrated in previous years, such as microscopy ex vivo or in vivo [23–28]. The origin of such signals, however, remains controversial. Many people assert that there should be no directly-generated BTHG (DBTHG) because of the large value of wave-vector mismatch. Instead, they believe these signals should come from back-reflected or back-scattered FTHG. In this paper, we demonstrate that DBTHG can be of comparable signal strength as FTHG in nanostructures. Using the Green’s function for theory and simulation, we show that the ratio of DBTHG/FTHG is a strong function of material dimension in thin films and nanoparticles, both of which are important structures in nanosciences. An integral equation was also derived from wave equations for fast computation of BTHG from thin film structures. Experiments were performed on ZnO thin films, CdSe quantum dots and Fe3O4 nanoparticles, and the data agree well with the simulation results. Our research indicates that DBTHG plays an important role in nanostructures and can potentially reveal more information than FTHG.
2. Theory and Simulation
2.1 Calculation of THG with the Green’s function
The dyadic Green’s function was used to calculate the THG strength both in the direct-backward- and the forward-directions :
Here, R and r are coordinates of observation and source points, respectively, is the idemfactor, ∇ is the dyadic del operator, k3 is the wave-vector amplitude at 3ω in the matrix, and P (3ω)(r) is the laser-induced third-order polarization inside the sample structure. To calculate P (3ω)(r), the excitation field was modeled as a focused Gaussian beam. The refractive indices of sample materials were calculated with associated Sellmeier equations, both at 3ω to calculate and at ω to determine the material-dependent optical distance z inside the sample structure.
2.2 Integral equation for thin film structures
While the generality of the Green’s function enables it to deal with arbitrary sample shapes and various descriptions of focused laser beams, this feature also means heavy consumption of computation resources and the absence of an insightful prediction of calculation results. An easy and powerful method is thus required to effectively interpret DBTHG data in further details. In particular, thin films are commonly encountered in experiments, for example cell membranes in tissues and epitaxial layers in semiconductors. Such an approach for investigating thin films would then be valuable in the context of both qualitative and quantitative measurements. Here we propose a simple yet effective method to calculate DBTHG in layered structures. Using the paraxial wave equation, the approach is straightforward, physically intuitive and computationally economical.
Consider a focused Gaussian beam propagating in the + z direction; the electric field of the fundamental wave can be expressed as 30], the FTHG equation can be derived as
The nonlinear wave equation for a backward-propagating wave can be written, in SI units, as 
Now that THG is only significant around the beam waist, and that we are considering nano thin films that only extend to a small fraction of a wavelength in the z direction, we imposed an approximation of quasi-planar wave front and neglected the curvature of wave fronts in the equation. In Eqs. (5) and 6, in the exponential terms (radial field profiles) were discarded, but in the longitudinal terms were kept since they represent the important information of Gouy phase shift across the beam waist. The equation can then be solved as
In reality, a nanometer-scale thin film rarely appears isolated in air or vacuum, and when it is attached to or sandwiched between other media, the equations for THG have to be modified accordingly. Following similar approaches of Schins et al. , we consider the following two conditions when an interface is crossed. For one thing, the q-parameter, or more specifically the medium-dependent optical distance to the focal plane, is changed because the laser beam enters a region with a different refractive index. For the other, the phase difference is maintained constant between the generated third-harmonic wave and the material third-order polarization, and a phase correction term is introduced at each interface. Equation (7) is then modified as
2.3 Simulation results of THG from nano thin films
First we consider an isolated thin film in air, as depicted in Fig. 1(a) . The laser beam was supposed to focus on the surface of the thin film, which is modeled as ZnO for further comparison with the results in the experimental section. The refractive indices were calculated to be n1 = 1.934 at 1230 nm and n3 = 2.229 at 410 nm, according to the Sellmeier equation . The observation cone angle was defined as θ = 157° to 180° and 0° to 33° for BTHG and FTHG, respectively, corresponding to the numerical aperture (NA) 0.55 aspheric lens used in the experiment. All calculated values are normalized to the maximum of FTHG strength. Figures 1(b) and 1(c) show the calculated results of FTHG and BTHG, respectively, as a function of film thickness. While the FTHG amplitude is well known to be monotonically increasing, the DBTHG trace is heavily oscillatory due to the large phase mismatch Δk, which mainly determines the period between peaks. The ratio of BTHG/FTHG is then plotted in Fig. 1(d). As the film grows thicker, the BTHG intensity is much weaker than FTHG because the large phase mismatch introduces significant destructive interference and impedes the buildup of signal strength. This is the traditional regime of bulk optics or micro-photonics, where effects of constructive and destructive interference make FTHG much stronger than DBTHG. In this regime, DBTHG is typically ignored or confused with back-scattered FTHG. It should be noted, however, that this ratio increases rapidly when the film thickness becomes thinner; eventually the ratio reaches unity as the thickness approaches zero. This is intuitively correct because in this extreme case, the excitation polarization becomes a sheet of element dipoles, whose far-field radiation patterns are supposed to be symmetric in both the epi- and forward directions. It should be also noted that the simulations results from the Green’s function and from the integral equation are very similar to each other, which validates the assumptions made during the derivation of the simplified equation.
2.4 Simulation results of THG from nanospheres
Next we consider an isolated nanosphere placed at the center of the beam waist. Again, for future comparison we model the nanosphere as a CdSe quantum dot in aqueous solution. Because the χ(3) value of CdSe is much larger than that of water , it can be regarded as an isolated sphere and the volume of integration V was limited to the particle region. The refractive index of CdSe at 1230 nm was calculated to be 2.523 . For , n3 = 1.344 is the refractive index of water in this case . Because the particle size was much smaller than the wavelength, diffraction at the particle-solvent interface was ignored in our calculation. The observation cone angle was defined as θ = 115° to 180° and 0° to 65° for BTHG and FTHG calculations, respectively, corresponding to the NA 1.2 objective used in the experiment. The value was calculated by α = sin−1 (NA/n1) = 65°, where n1 is the refractive index at ω in the matrix  and is chosen to be 1.325 for water here . Again the calculated values are normalized to the maximum of FTHG strength. The values of FTHG, BTHG and BTHG/FTHG ratio are then plotted in Figs. 2(a) , 2(b) and 2(c), respectively.
When Figs. 2(b) and 1(c) are compared, it can be observed that size-dependent BTHG amplitudes from nanoparticles and nano films exhibit rather different trends, which can be interpreted with a relatively simple and straightforward picture. Since a focused Gaussian beam was used for simulation, the beam profile would be symmetric to the z-axis, the direction of beam propagation. Referring to Fig. 2(d), we can consider the laser beam as a collection of concentric, cylindrical layers infinite in number. Each cylindrical layer would change its amplitude and phase along the z-direction, and each layer can be further decomposed as a set of infinite rays. We now can follow a similar procedure as in  to predict the behaviors of BTHG. The strength of BTHG can then be expressed asFig. 1(c), in which the values local maxima and minima remain approximately the same over the simulated range of thickness.
3. Experimental Results and Discussion
To corroborate the aforementioned theory and simulation, we conducted experiments of FTHG and BTHG on ZnO thin films, CdSe quantum dots and Fe3O4 nanoparticles. Since isolated nano films are difficult for synthesis and manipulation, we designed the sample structure to be ZnO thin films deposited on sapphire. Because the χ(3) value of nanofilm ZnO (~ esu , ) is much larger than that of sapphire (esu , ), we expected the THG response of such samples should be close to isolated films. These thin films were grown on c-plane sapphire substrates by either atomic layer deposition (ALD)  or pulsed-laser deposition (PLD) . Five groups of samples were used in this experiment and the thicknesses were determined by scanning electron microscopy (SEM) to be 45, 73, 113, 163 and 200 nm, respectively. The excitation source was an optical parametric oscillator (Mira-OPO, Coherent, U.S.A.) synchronously pumped by a Ti:sapphire laser (Mira Optima 900-F, Coherent, U.S.A.), and the wavelength was locked at 1230 nm. Figure 3 shows the schematics of the experimental setup; Fig. 3(a) is the configuration for BTHG and FTHG measurements, and Fig. 3(b) is for calibration procedures. In our setup, two identical aspheric lenses (C230TM-C, NA = 0.55, Thorlabs, U.S.A.) were used for illumination and collection so that the relative strength of BTHG and FTHG can be quantitatively calibrated. First, a piece of sapphire substrate was z-scanned as in Fig. 3(a) to obtain the position-dependent trace of FTHG. The experimental setup was then switched to Fig. 3(b), and the half-wave plate was rotated accordingly so that the laser power before the illumination lens remained the same. The FTHG z-scan was performed again in this configuration. Let s1 and s2 be the measured FTHG strength when the laser beam is focused on the interface closer to the illumination lens in Figs. 3(a) and 3(b), respectively. Since the illumination-collection set was symmetric, these two values represented the same signal level and their difference accounts for other factors of dissymmetry, such as different PMT sensitivities and different degrees of misalignment between the collection lens and the PMT in these two arms. Suppose for the configuration 3(a) the raw signals of FTHG and BTHG were measured to be f0 and b0, respectively. The BTHG/FTHG ratio can then be calculated as BTHG: FTHG = (b0 / s2): (f0 / s1).
The ratio of BTHG/FTHG from experimental and simulation results are plotted in Fig. 4(a) . To simulate the generation of third-harmonic waves in sapphire, n1 = 1.752 and n3 = 1.785 were calculated from its Sellmeier equation . The final BTHG/FTHG ratios were calculated by incorporating surface reflection at the sapphire/air and the ZnO/air interfaces at 410 nm. Although a large number of factors were not considered in our simulation (etalon effect in thin film, possible deviation from normal incidence, inhomogeneity of ZnO χ(3), etc.), the experimental data agreed well with the simulation trace from the Green’s function or from the integral equation. Most importantly, when the film thickness is thin enough the ratio can significantly exceed the reflectivity value of the sapphire-air interface, which is 7.94% at 410 nm. Our results indicate that the collected BTHG signals were directly-generated backward THG, not back-reflected or back-scattered FTHG. As the film thickness becomes larger, the FTHG strength far exceeds the BTHG and the ratio would decrease greatly, as shown in the figure. The results also justified our choice of sample design, in which the large contrast of film and substrate χ(3) values can simulate an isolated thin film. To illustrate on this point, Fig. 4(b) shows the simulation results in which the χ(3) value of ZnO was set to 100X even larger. In Fig. 4(a), when the film thickness approaches zero the BTHG/FTHG ratio would drop rapidly, because in this case the sample resembles a piece of sapphire and the contribution from the substrate dominates. In Fig. 4(b), the THG response from such a sample would be closer to an isolated thin film like in Fig. 1(d), as expected.
To investigate the BTHG behavior of nanospheres, CdSe lipid-enclosed quantum dots (LEQDs) and Fe3O4 LEQDs in aqueous solution were imaged by our epi-THG microscope. The excitation source was a home-built femtosecond Cr:forsterite laser centered at 1230 nm, with a 130 fs pulsewidth, a 110 MHz repetition rate and a 20 nm spectral full width at half maximum [23,25]. After being shaped by a telescope, the laser beam was directed into a modified beam-scanning system (Olympus FV300) and an upright microscope (Olympus BX51). A water-immersion objective (Olympus UplanApo/IR 60X/ NA 1.2/WD 0.2 mm) was used for illumination and collection. Full details of sample preparation, experimental configurations and data analysis can be found in our previous work [41,42]. The refractive index of Fe3O4 was calculated to be 2.059 at 1230 nm ; its χ(3) value is also much larger than that of water , so the model of an isolated nanosphere still holds. The experimental and simulation results are then plotted in Fig. 5 . The polydispersity of particles were also considered by further weighting the calculated curves with a Gaussian-shaped size distribution [41,42]:
The standard deviation is calculated by , where the value of σref is obtained from transmission electron microscopy (TEM) statistics. As can be seen in the figure, the experimental data fit well to the simulation on direct-epi-THG with polydispersity (solid lines) but deviate obviously from the trace of the FTHG (dashed lines), indicating that the signals we observed were indeed DBTHG, not FTHG back-scattered after generation.
It can be observed in both cases that due to the large phase mismatch between the excitation polarization and the generated wave, BTHG is much more sensitive to nano-scale variations of sample dimensions than FTHG. This feature may be potentially useful for nanophotonic investigation. If FTHG measurements are also feasible in an experiment, simultaneous retrieval of BTHG and FTHG signals can be used to calculate the BTHG/FTHG ratio, which can be a potentially powerful tool for quantitative analysis in the nano regime. With proper calibration of a microscopy system, for example, the BTHG image can be divided by the FTHG image pixel by pixel to generate a “ratio image,” which can be juxtaposed with BTHG and FTHG images for further interpretation. Previously unavailable information, such as dimensions of cellular membranes or organelles in medical imaging, may be obtained in this approach.
In summary, we demonstrated that backward THG can be directly generated and that its strength can be on the same order of FTHG in nanostructures. Using the Green’s function as the theory, we performed numerical simulations on BTHG and FTHG in thin films and nanoparticles. We also derived an integral equation for BTHG calculation in nano films, which was shown to yield similar results to those from the Green’s function. We experimentally investigated size-dependent BTHG from ZnO thin films on sapphire as well as CdSe quantum dots and Fe3O4 nanoparticles in aqueous solution, and the data agreed very well with the simulation results. Our research suggests that BTHG signals can potentially provide more information than FTHG in nanophotonic measurements, particularly when combined with FTHG signals to calculate the BTHG/FTHG ratios.
The authors gratefully acknowledge the financial support from the National Health Research Institute (NHRI-EX99-9936EI) and the National Science Council (NSC98-2120-M-002-001), Taiwan.
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