90 8.3 The measured diffraction efficiencyη as a function of the probe delay ∆t in undoped. circles), iron-doped (triangles) and manganese-doped (squares) LiNbO3 samples. 96 8.6 The summary of measured values of ηpeak (in squares) and ηpl. (in triangles) for different ..pump intensities in the 70-µm LiNbO3 sample.
Volume holographic gratings and their filtering properties
Plane wave components that satisfy the Bragg condition will be diffracted on the grating, and those that do not will pass through instead. Therefore, a holographic grating can act as a filter that discriminates between different components of plane waves based on the Bragg condition.
Spatial-domain perspective
The deflection efficiency will therefore be reduced; in this case, η is 0 and no beam merging will be observed. In general, the longer the interaction length between the grating and the light field, the more selective the VHG will be.
Temporal-domain perspective
1.2(b), there is a corresponding change (tilt) of k2 as a result, giving rise to a non-zero phase mismatch indicated by the horizontal double-headed arrow. Various spatial domain properties and applications of holograms are attributed to their angular selectivity; for example, angle multiplexing[3], shift multiplexing[4], holographic storage[5], image processing[6] and pattern recognition, to name just a few.
Recording and readout of holographic gratings with polychromatic light sources
4.1), n(T0) is the refractive index of the material atλB at temperature T0, and Λ(T0) is the lattice period of the index atT0. The data of ηk(η⊥) are obtained when the polarization of the probe pulses, adjusted with the help of the Berek compensator and a polarizer in front of the photodetector, is parallel (perpendicular) to that of the pump pulses.
TE case
We therefore reach the conclusion that the matrix D−1i,T EDi+1,TE takes care of the interface effect between layer i and i+1. It is noticeable that the matrix Pi takes care of the propagation effect inside layer i with thickness hi.
TM case
Diffraction efficiency of a multilayer medium
Matrix formulation for sinusoidal gratings
This phenomenon can also be explained by the coupled mode analysis: within the bandgap of the filter,κ >|∆β|. The band gaps grow with increasing ∆n, but the frequencies of the out-of-band oscillation are the same for all three gratings.
Experimental results
- Recording holographic WDM filters in reflection geometry in LiNbO 3
- Measured filter response
The derivation of the appropriate transfer function H(ki;kd) of a VHG has been addressed by an abundance of literature[1, 2, 9]. Combination of probe absorption due to coexistence of pump pulse and excited carriers.
Theoretical consideration
Another interesting parameter is the group delay, which determines the dispersive properties of VHG. At Σo, the removed plane wave component can be written as A(ω)ejΦ(ω), where A(ω) and Φ(ω) are both real quantities and can be uniquely determined from the diffraction field representation; ω is the angular frequency of the radiation.
Numerical simulations and experimental results
- Reflection geometry
- Experimental setup
- Transmission geometry
- Experimental setup
- Wavelength selectivity
- Angular selectivity
- Diffracted beam profiles
The relationships between them can be broken down if we consider the spatial harmonic components of the incident beam. Briefly stated, the product of the angular selectivity curve of a VHG and the angular spectrum of the incident beam determines the diffraction efficiency and filter shape of the VHG. This behavior occurs because part of the diffracted beam is coupled back into the transmitted beam and η is reduced.
Conclusion
The experimental setup used to investigate the time evolution of the diffracted beam profile is shown in Fig. The time delay of the probe pulse with respect to the pump pulse can be adjusted with a variable degree of delay. The pump pulses are polarized along the c-axis and ηk(η⊥) is measured when the probe pulses are polarized parallel (perpendicular) to the pump pulses.
Theory
Temperature changes affect holographic filters primarily through two mechanisms: (Other possible effects will be neglected here, e.g. the thermal dependence of the piezoelectric tensor will become apparent when a voltage is applied.). When the temperature changes to T0+ ∆T, the Bragg wavelength of the filter will shift accordingly and shift to λB + ∆λ. The device exploits the TEC discrepancy between two properly chosen materials (in our case, aluminum and silicon) and deflects with temperature changes[9].
Experiment and results
The ζ and ξ axes are parallel and perpendicular to the direction of propagation of the diffracted probe pulse. The rest of the pulse is passed through a 1 mm thick BBO (β−BaB2O4) crystal to generate a pulse at λp= 388 nm, which is split into two identical pump pulses and then focused down to one fifth of their original diameter inside the 1 mm thick lithium niobate (LiNbO3) sample. The data obtained from pumping and probing the thin sample are shown as circles in Fig. the polarizations of both the pump and probe pulses are parallel to the c-axis of the LiNbO3 crystal.
Conclusions
Beam propagation in the 90 degree geometry holograms
The bulk index of the medium n0 and the index modulation ∆ are constants whose values are determined by the readout wavelength λ00. We therefore realize that for the Bragg-matched case, the incident and diffracted beam profiles within the uniform grating region can be analytically described by the zeroth and first-order Bessel functions of the first kind. As a result, the diffraction efficiency is defined as the total energy ratio of the incident and diffracted beams [6].
Wavelength selectivity
Numerical simulations
Obviously, the smaller the cell size, the more detail of the beam profiles we can understand. 5.6), the profile of the diffraction beam at the exit face can be described by the Bessel function of the first kind. The solid curve is plotted according to the analytical solution of the Bragg-matched case, Eq. 5.6), which is in excellent agreement with our numerical prediction.
Experimental results
- Beam profile experiment
- Filtering properties of the 90 degree geometry holograms
It is obvious that as ∆nin increases, the relative intensity of the diffracted beam near the input surface will also increase. To explain the increasing trend of the total diffraction efficiencyη, we recall that the analytical solution to the diffraction efficiency (from Eq. of the crystal (and thus the width of the incident beam) or the modulation of the index ∆n.
Conclusion
The peak intensity of each of the pump pulses in the sample is approximately 180 GW/cm2. The broadening factor as a function of the half-angle between pump pulses δ(θp) is plotted in Figure 6.5: the dotted line is calculated numerically according to theory, and the solid curve is plotted using Eq. 6.12) under the assumption of paraxial approximation. Om Trp. to calculate, it is necessary to take into account the case when the pump pulse has already passed, that is, the probe pulse does not overlap with the pump pulse.
Theory
- Coupled mode equations for pulse holography
- Solution of the diffracted pulse: undepleted incident probe
The diffraction broadening is negligible because the Rayleigh range of the pulse is much longer than the sample thickness. The main difference between the coupled mode equations Eq. 6.4a) and (6.4b) for pulse holography and that for continuous wave (cw) holography is the presence of the time derivative to account for the short duration of the pulse. The peak value of the diffraction efficiency ηpeak = η(∆t = 0) and the dimensionless broadening factorδ are given by.
Experimental results and discussion
In our experiment, this factor turns out to be 8.2±0.4; the discrepancy can be explained by the deviation from Kleinman symmetry [14]. As can be seen from the expression, Dc is independent of the spatial dimension Dp of the pump pulses and determined exclusively by the time duration of the pulse τp. We can now consider the effect of the composite pump on the probe pulse as in the two-beam case.
Conclusion
If we increase the angle 2θp between the pump pulses, the angle of incidence θ3 of the probe pulse must also increase to satisfy the Bragg condition. The reduction in the transverse width of the composite pump continuously counteracts the effect of an increased probe angle of incidence and leaves the temporal resolution in this configuration almost unchanged. The influence of the excitation geometry on the temporal resolution in femtosecond pump-probe experiments.
Introduction
Collinear pump-and-probe experimental setup
The spatial FWHM of the pulses (after passing through the collecting lens) on the input side is approximately 0.6 mm. A photodiode is used to measure the output energy of a pump or probe pulse by selecting an appropriate filter to block one of them.
Two-photon absorption process for a single femtosecond pulse
- Motivation
- Theory of two-photon absorption
- Experimental results
The solution to the differential equation Eq. 7.1) subject to the boundary condition at the input level = 0. where a Gaussian intensity profile of the pulse is assumed; τp and Dp specify the temporal and spatial extent of the pulse, respectively. If we ride on the top of the pulse as it propagates within the non-linear medium, i.e. setting-z/vp= 0, the intensity profile will appear to us as depicted in Fig. We see that the preferential absorption near the center of the pulse leads to the flattening of the Gaussian intensity profile.
Collinear pump-and-probe experiment and modeling
- Experimental results
- Modeling of collinear pump-and-probe experiment
- Modeling the dip of T r (∆t)
- Modeling the plateau of T r (∆t)
- Comparison between theory and experiments: the determination of parameters
On the other hand, the plateau portion in the dependence Tr(∆t) is attributed to the absorption of the probe pulse by charge carriers (electrons and/or holes) excited by the intense pump pulse; the concentration of the excited free carriers due to pump and probe photons is negligible in comparison. The polarization dependence of the plateau value is explained by the anisotropy inherent in the LiNbO3 crystal. In these expressions, ∆t is the delay of the probe pulse, and δ0 = (np− nr)d/ctpis is the parameter explaining the difference between pump and probe speeds.
Conclusion
The experiment is performed with the c-axis of an undoped LiNbO3 crystal oriented parallel to the polarization of the pump pulses. The dependence of the peak and plateau values of η on the intensity of the pump pulses is shown in Fig. plotted. Since the coupling constantκ of the mixed lattice is proportional to the carrier concentration, it is natural that the value of ηpl.
Experimental observation
- Experimental setup
- Polarization dependence
- Dependence on dopants
Four percent of the pulse energy is tapped and serves as a probe pulse shifted by a variable delay stage. The maximum intensity of each of the pump pulses within the sample is about 165 GW/cm2; the maximum pump pulse fluence is. Each data point corresponds to the measured diffraction efficiency value averaged over 100 single-pulse experiments: after each pump and probe measurement, uniform illumination is applied to erase the residual photorefractive grating.
Theoretical justification and comparison with experimental data
- Decoupling of pump pulses when q p = 2β p I p0 d ≤ 1
- Curve-fitting and the extracted Kerr coefficient of LiNbO 3
- Mixed grating due to the excited carriers
- Intensity dependence of η peak and η pl
The index lattice arising from the Kerr effect accounts for almost 95% of the peak diffraction efficiency η (∆t = 0), which agrees well with the polarization dependence shown in Figure. As in the previous section, we take advantage of the condition qp ≤ 1; neglecting the DC termα, Eq. To get an estimate of the value of ηpl., we use the approximation: θ1=θ2=θ3=θ4= 0, which is reasonable near the paraxial region.
Conclusion
However, the distribution of data points (in squares) indicates a "reversal" of the intensity dependence;. In the low-intensity region, the fitted exponent is ~ 2.3, which can be explained by a similar reasoning for ηpl. In the high-intensity range, the dependence is almost linear (b≈1.25), which can be attributed to the inevitable strong absorption of pump pulses in the sample: a decreasing fraction of the pump pulse energy is used to establish the mixed lattice responsible for ηpeak.
