Super-transmission : the delivery of superoscillations through the absorbing resonance of a dielectric medium

The delivery of a super-oscillatory optical signal through a medium with an absorbing resonance at the super-oscillation frequency is considered theoretically and through simulations. While a regular signal oscillating at the absorption resonance frequency would be completely absorbed after a few absorption lengths, it is found that the superoscillation undergoes quasi-periodic revivals over optically thick distances. In particular revivals of extreme UV local oscillations propagating through Silica Glass over distances which are three orders of magnitude longer than the associated absorbing length are numerically demonstrated. © 2014 Optical Society of America OCIS codes: (070.7345) Wave propagation; (260.2030) Dispersion; (260.3160) Interference; (260.7190) Ultraviolet. References and links 1. Y. Aharonov, Z. D. Albert and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-half particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988). 2. Y. Aharonov, S. Popescu and J. Tollaksen, “A time-symmetric formulation of quantum mechanics,” Phys. Today 63, 27–32 (2010). 3. M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A: Math. Gen. 39, 6965 (2006). 4. M. V. Berry, “Faster than fourier,” World Scientific 5565, (1994). 5. Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, “Some mathematical properties of superoscillations,” J. Phys. A: Math. Theor. 44, 365–304 (2011). 6. F. M. Huang, Y. Chen, F. J. Garcia de Abajo and N. I. Zheludev, “Optical super-resolution through superoscillations,” J. Opt. A: Pure Appl. Opt. 9, S285 (2007). 7. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9, 1249-1254 (2009). 8. E. Greenfield, R. Schley, J. Nemirovsky, G. K. Makris and M. Segev, “Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams,” Opt. Express 21, 13425–13435, (2013). 9. T. F. R. Rogers, J. Lindberg, T. Roy, S. Savo, E. J. Chad, M. R. Dennis and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432-435 (2012). 10. K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beam,” Opt. Lett. 36, 4335-4337 (2011). 11. D. Gabor, “Theory of communication,” J. Inst. Elec. Eng. 93, 429-457 (1946). 12. L. Cohen, “Time-frequency distributions,” Proc. IEEE 77, 941–981 (1989). 13. B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal,” Proc. IEEE, 80, 520–538 (1992) 14. A. Kempf, “Black holes, bandwidths and beethoven,” J. Math. Phys. 41, 2360–2374 (2000). 15. R. Kitamura, L. Pilon and J. Miroslaw, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Optics 46, 8118–8133 (2007). 16. M. Khashan and A. Nassif, “Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2-3 m,” Opt. Commun. 188, 129–139 (2001). 17. M. R. Dennis, A. C. Hamilton and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. 33, 2976–2978 (2008). #223056 $15.00 USD Received 15 Sep 2014; revised 12 Nov 2014; accepted 14 Nov 2014; published 10 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031212 | OPTICS EXPRESS 31212 18. M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys. A: Math. Theor. 42, 022003 (2009). 19. M. S. Calder and A. Kempf, “Analysis of superoscillatory wave functions,” J. Math. Phys. 46, 012101 (2004). 20. E. T. F. Rogers, J. Lindberg, T. Roy, M. R. Dennis and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. 102, 031108 (2013). 21. E. Katzav and M. Schwartz, “Yield-Optimized Superoscillations,” IEEE Trans. Sig. Proc. 61, 12 (2013).


Introduction
Superoscillations were introduced in the context of weak quantum measurements [1,2].A weak quantum measurement can yield values much larger than the largest eigenvalue of the related observable.Yet, upon ensemble averaging the mean result is identical to the mean of a regular measurement.Superoscillation [3][4][5] is an interference phenomenon between the Fourier components of a band-limited signal giving rise to local rapid (super-) oscillations which do not appear in the signal's spectrum.Thus the superoscillation is not directly susceptible to operations manifested on spectral components.This fact was used in optics to demonstrate a phenomenon known as super-resolution without evanescent waves [6][7][8][9][10].There, sub-diffraction superoscillatory features of a light beam were carried from the near to the far field by propagating modes.In contrast, a sub-diffraction spatial Fourier component is evanescent.
In this work we consider the propagation of temporal optical signals having a superoscillation at an absorbing resonance of the dielectric medium.Because the absorption only acts on Fourier components, the superoscillation is not absorbed.Still, one can expect that dispersion acting on the same components would eventually destruct the interference leading to the superoscillation.However, we show that as the signal propagates through the medium, the superoscillation revives periodically or quasi-periodically.Thus, a superoscillatory signal could be used to deliver fast oscillations to a target within or after a dielectric medium in highly absorbed frequency bands.The superoscillation is thus super-transmitted through an opaque medium.This in particular is relevant to extreme UV (EUV, ranging between 124nm to 10nm) where typical absorption lengths are about 100um in air and 10nm to 100nm in solid dielectrics.Such large absorption requires vacuum conditions and precludes the use of transmission optical elements.We assert that EUV superoscillations can be super-transmitted through a solid dielectric thicker by orders of magnitude than the corresponding absorbing length.
In the following we first show how to construct a superoscillatory optical signal made of any desired number of Fourier components.Then, a measure for deducing the degree to which a signal is superoscillatory is devised.The crux of this work is the analysis of the propagation of superoscillatory signals through a dispersive medium, predicting super-transmission by virtue of ordered revivals of the superoscillation.Simulations follows in which an optical signal having a superoscillation in the extreme UV, is propagated through Silica glass (SiO 2 ).Ordered revivals of the superoscillation are demonstrated over a length scale of 100μm.

Constructing a temporal optical superoscillatory signal
Berry and Popescu [3] devised a family of complex valued superoscillatory functions: Although the fastest Fourier component oscillates at frequency of Nω 0 , around t = 2πm/ω 0 (for integer m) it super-oscillates at a faster frequency of aNω 0 .For example, around t = 0: The real part of Eq. ( 1) can serve as a physical superoscillatory signal.It can be represented with the following binomial expansion: This expression allows us to construct a superoscillatory signal comprised from any desired number M of Fourier modes.M is N/2 for even N or (N + 1)/2 for odd N.For propagation in a dielectric medium it is convenient to construct an analytic signal: where * stands for convolution and ω m = mω 0 .The real valued C m coefficients are derived from Eq. ( 5) Thus an optical superoscillatory signal can be constructed by superposing together several narrow band frequency sources.The propagation of the signal in a dielectric medium is given by: Where α(ω m ) are the absorption coefficients, k m = ω m n(ω m )/c are the modes' wave vectors, n(ω m ) is the refractive index and c is the speed of light in vacuum.At any specific coordinate z = z 0 the signal can be expressed in a form containing an amplitude and a phase:

The degree to which a signal is superoscillatory
Treating ω(t) as the local (instantaneous) frequency is justified only when A(t) is slowly varying [12,13].This is not always the case and so we need to characterize the local frequency in a different manner.For this purpose we develop an adaptive version of the well known Gabor transform [11] to calculate a time-frequency distribution of the signal: Our adaptive version uses a variable width σ (t) for the Gaussian window G (t, τ).Such a dynamic window width allows the calculation of the local frequency for a superoscillatory signal, where the dynamic range of the local frequency can become very large.The width Using the distribution F(t, ω), the local frequency is derived as a mean conditional frequency (In our numerics, if for some t 0 , F(t = t 0 , ω) is a multi-component signal [12], the integration limits in Eq. (11) and the following Eq. 12 are modified to isolate the relevant component.See further discussion in Appendix B): Ω(t) would be identical to ω(t) of Eq. ( 8) when A(t) is a constant of time [12].
Examples illustrating the usefulness of the above procedure to determine the local frequency of different signals are presented in Appendix C. Another useful quantity is a distributionderived local amplitude: Now, at each location in the dispersive medium, we quantify the propagating signal as a better superoscillatory function if the following characteristics are increased: the ratio between the instants in which the function contains superoscillations to the temporal extent of the whole function; the superoscillations local frequencies and local amplitudes.In general, there is a trade off between these characteristics [14].Still, we construct a measure that takes all of these into account (For the measure, we use a scaled version of σ (t), see further discussion in Appendix A): Here ω max is the highest spectral (Fourier) component of the function f (t).For a function which contains no superoscillations M = 0, while for an idealized (non realistic) function which is always superoscillatory M = 1.M > 0 when there are local frequency components that exceed ω max .

The effects of propagation on the superoscillatory nature of the signal
When the propagating Fourier modes are far enough from any absorbing resonance, the major concern would be the effect of dispersion on the phases of the modes (we treat absorption later).Now, an important feature of the superoscillatory signal is as follows: if the same constant phase φ is added to all of the modes, it is easy to show that close to t = 0 the real signal would be cos(aNt − φ ).Hence the signal is still superoscillatory as its highest local frequency is maintained.Our simulations indicate that the measure M{ f (t)} is modulated by no more than 15% and Ω(t) by no more than 10% when φ is varied (see detailed results in Appendix D).
During propagation each mode is shifted by k m z.With no dispersion the phase shifts would amount to a common time delay and the signal would stay superoscillatory.With dispersion the superoscillation would be destroyed.However, the phases could realign to revive the superoscillation when either they are all equal or when they constitute a time delay of the signal up to an added common phase.We consider such a case as a revival of the superoscillation.Formally the condition for superoscillation revival at some location z is that the phases k m z would be ordered on a straight line: where mod stands for the modulo operator.Here A(z)m constitutes a common time delay while B(z) adds a common phase shift.For two modes this condition is trivially satisfied for all z.A superoscillatory signal made of two modes would thus stay superoscillatory, regardless of dispersion (while neglecting absorption).For three ordered modes m = p, q, l with p < q < l it can be shown using a straightforward mathematical argument (see detailed derivation in Appendix E) that the three modes would be ordered on a straight line at a frequency of In other words, every distance interval of Λ = 2π/Δk the signal would be superoscillatory (for the case of no dispersion: Δk = 0 and the signal is always superoscillatory).This is a major result of this work.Despite dispersion and the delicate nature of a superoscillatory signal, if it is carried by three modes -the superoscillation would revive in an exact periodic manner.
Another important question is -how long the superoscillation would persist at each revival.Superoscillation is a sensitive interference effect because the relevant time scale concerning phase shifts is the period of the local superoscillation.As a worst case estimate, the superoscillation would be destroyed when the deviation from condition Eq. ( 14) is translated to a relative time shift which is half the period of the superoscillation.For a superoscillation made of two modes this would never happen.For three modes it would happen after propagating a distance of (see Appendix F for derivation): where Δk is given by Eq. ( 15), m is the highest mode number and a and N were defined in Eq. ( 1).In essence l c is the coherence length of the superoscillation.Note that a revival would persist for twice this value (accounting for first building the superoscillation and then for destroying it).
For a superoscillatory signal constructed by M > 3 modes -the modes can be ordered m 1 < m 2 < ... < m M and M − 2 groups of three modes can be constructed: m 1 , m 2 , m l with m l ∈ {m 3 , m 4 , ..m M }.For these groups, M − 2 periodicities Λ m can be calculated (with Eq.15).Then a perfect revival would occur when a set of M − 2 integers p l are found to satisfy: where z is a coordinate function.Generally the periodicities Λ l would be incommensurate and so this last condition can only be approximated (although locations can be found where the approximation can be made with an arbitrarily small error).Still, as long as the inaccuracy in this condition is within the coherence length of the superoscillation -the superoscillation would revive (while the coherence length is now taken as the minimum of M − 2 coherence lengths of the above mentioned groups of ordered three modes).
To complete the picture we need to consider when the absorption in the medium would destroy the superoscillation.Here, considering the different amplitudes of the Fourier modes it is important to keep their ratio.We also note that at z = 0 the amplitude of the superoscillation is 1 while the amplitudes of the other modes are C n (scaling by a common amplitude would not change the following result).If all the amplitudes were decaying at the same rate than the superoscillation would have survived while decaying at the same rate.For different decay rates, we develop a rough estimate for the survival of the superoscillation.We require that the decay of each mode with its own decay rate relative to its decaying with another mode's decay rate would be smaller than the superoscillation decay: In this case, if all the modes satisfy α n z 1 then the superoscillation would persist as long

Propagation simulation through Silica glass of an optical signal superoscillating at the extreme UV
For simulations it is important to consider the required accuracy at which we need to know the indices of refraction of the Fourier modes.Here a conservative estimation would require that the phase shift acquired by the error in the index of refraction n e over the whole propagation length L, for all of the modes, needs to be smaller than π/aN.In this case we require that n e < λ 0 2aNL where λ 0 = 2πc/ω 0 .For the purpose of the simulation we chose the fundamental frequency to be ω 0 = 1.5849 × 10 15 rad s (corresponding to a wavelength of λ 0 = 1189.32nm).The parameters a and N of the superoscillatory signal were chosen to be a = 2 and N = 5.According to Eq. ( 5), this means there are three Fourier components oscillating at frequencies: ω 0 , 3ω 0 , 5ω 0 (corresponding to wavelengths of λ 0 = 1189.32nm,λ 0 /3 = 396.44nmand λ 0 /5 = 237.86nmrespectively) with three corresponding amplitudes: C 1 = 90/16, C 3 = −195/16 and C 5 = 121/16.The signal's highest local frequency is the superoscillation's frequency -10ω 0 corresponding to an extreme UV wavelength of λ 0 /10 = 118.93nmwhich is located at the absorption resonance of Silica glass (SiO 2 ) where the absorbing length is only α −1 (λ 0 /10) = 47nm [15,16].In the simulation, the spectral width of each mode was chosen to be a practical value of 1GHz, which is roughly equivalent to a pulse width of 1ns.The spectral modes were numerically propagated in the medium using interpolated values for the index of refraction and for the absorption coefficients taken from tabulated data [15,16].Here, the whole signal was propagated over a distance of L = 120μm for which the error in the refraction index needs to be smaller than ∼ 5 × 10 −4 , so we require to have four significant digits after the decimal point for the indices of refraction.The refraction and absorption coefficients for the Fourier components are: n 1 = 1.4482, n 2 = 1.4713, n 3 = 1.5155 and α 1 = 0.636 1 m , α 2 = 2.862 1 m , α 3 = 975.3 1 m respectively [15,16].In addition, according to Eq. (18) the amplitudes decay would destroy the superoscillation completely after 676μm (our simulations indicate a shorter distance of ∼ 500μm, see Appendix G).So for propagating up to 120um, the effect of absorption is very small.
The results of the simulation are shown in Fig. 1 where the evolution of the local frequency Ω(t) and the measure M{ f (t)} are calculated.The signal is plotted at z = 0, at a location where the superoscillation is destroyed and at the location of the final revival.The periodic pattern of the revivals of the superoscillation is evident from the evolution of the measure indicating super-transmission of the superoscillation.Equation ( 15) predicts a revival periodicity of 6um in agreement with the simulation.The width of each revival matches well our prediction of 2l c = 1.2μm made with Eq. ( 16).This value by itself is an order of magnitude longer than the absorbing length for the superoscillation frequency.It is important to note that a high value for the maximum local frequency does not guarantee by itself that the signal is a good superoscilltory signal, that is -have a high value of the measure.This can be seen at the vicinity of each revival location through the rather erratic evolution of the highest local frequency.This is reminiscent of superoscillations found in random signals [17] compared to specifically designed superoscillatory signals.(e) Evolution of the highest value of the local frequency Ω(t).The red dashed lines denote the periodic locations of the superoscillation revivals.The green dashed line denotes a location where the signal is not superoscillatory.

Conclusions and Discussion
To summarize, using an optical superoscillatory signal, rapid local oscillations, even if located spectrally at an absorption resonance of a given medium, can be super-transmitted through that medium over macroscopic length scales, much longer than the absorption length.This is made possible due to ordered revivals of the superoscillation along the propagation coordinate.We numerically demonstrated delivery of extreme UV oscillations at local wavelength of ∼ 120nm through 120μm of Silica glass which is three orders of magnitude longer than the associated absorption length for a spectral component at this wavelength (∼ 50nm).To actively determine the locations at which the superoscillation is manifested the phases and amplitudes of the Fourier modes of the signal injected to the medium need to be controlled.On the other hand, this suggests that the sensitivity of a superoscillatory signal to the phases of its Fourier components can be used as a probe for the dispersion properties of a given material.As the phenomenon we present here is not intuitive -one can ask if there is some fault in describing the interaction of the medium with the light field by its spectral response.As long as the description is consistent (e.g.not violating any physical principle like causality) we must accept its validity.Further, one might ask how is it possible that locally the signal can oscillate very fast yet the medium does not respond to it and absorbs these superoscillations.The answer lies with the fact that response of the medium depends not only on the instantaneous value of the electric field but also on its previous values, which is a direct consequence of dispersion.This explains why the response can be different for a regular signal that would be absorbed compared with a superoscillation.Furthermore, if the superoscillation would have been absorbed, while the Fourier modes of which the signal is made of were not absorbed, energy conservation would have been violated.An important question is how well can the superoscillation be utilized for measurements and for interaction with a detector.For this, it is important to realize when considering a superoscillation that even though there is no energy in the associated Fourier component, still power and energy can be delivered at the time of the superoscillation as the associated Poynting vector is not zero at these times.This is in accordance with the spatial case which enabled the use of spatial superoscillations in microscopy [9].The force exerted by the fields at the superoscillation is naturally much smaller than the force exerted at times around the superoscillation, but this should be compared with the alternative of generating high frequency oscillations through multi-photon frequency conversion which is also characterized by a small ratio between the amplitudes of the generated and generating fields.It is important to note that the superoscillation can still be "projected" into an actual Fourier component when a nonlinear interaction is involved.This happens when the interval at which the superoscillation exists is cut from the rest of the signal.For a spatial superoscillation [18] this can be done by projecting the function through a slit [19] or a mask [20].For a temporal superoscillation this can be manifested with frequency mixing with a gate pulse.Such a nonlinear projection would be useful for measuring or utilizing the rapid oscillation and is especially suited for superoscillatory signals having several consecutive fast oscillations.For this we note that the numerical example we gave used a signal with superoscillations that are comprised of a single fast oscillation at each period of the signal while superoscillatory signals can be constructed in ways to optimize their yield [21] creating superoscillations with several consecutive fast oscillations.Our results are general and should apply to any such signal.

Appendix A: Constructing the local width function σ (t) for the adaptive Gabor transform
For our adaptive Gabor transform operating on the function f (t) we use the following procedure to construct σ (t): we define half an oscillation of the function f (t) as a section between two consecutive extremum points.The set of all extremum points {(t n , f (t n ))} is found.Finally, σ (t) is constructed by a spline interpolation of the set of points {(x n , y n )} = {((t n + t n+1 ) /2,t n+1 − t n )} and scaled with a width factor σ 0 .This procedure creates a local width function that matches the local half-period of the signal.σ (t) is then used in the window function: 2σ 2 (t) .After the width function is determined according to the signal, the scaling factor fine tunes the window's width around the superoscillatory feature.In our propagation simulations, fine tuning the window's width changes the visibility of the measure M{ f (t)} and of the local frequency Ω(t) independently.With a scaling of σ 0 = 1 we achieve optimal measure visibility, while with a scaling of σ 0 = 0.5 we achieve optimal local frequency visibility as well as an optimal matching between Ω(t) and the analytic expression ω(t) for signals with a slowly varying envelope.As such, unless otherwise stated, σ 0 = 1 is used when calculating the measure and σ 0 = 0.5 is used when calculating Ω(t) independently.
Figure 2 demonstrates in stages how the dynamic width function is calculated for a superoscillatory signal.The set of all extremum points {(t n , f (t n ))} is found for the superoscillatory function f so (t). (b) σ (t) is constructed by a spline interpolation of the set of points The window function G (t, τ) is shown for a few representative delay values τ.

Appendix B: Taking into account multi-component signals
When calculating the local frequency and local amplitude through the time-frequency distribution F(t, ω) at a given t = t 0 , the distribution might represent a multi-component spectrum.Such a spectrum contains several well defined spectral features.A trivial example for a multicomponent signal is the superposition of two signals oscillating at different carrier frequencies.
During the analysis of the distribution at some time instants the local oscillation reside on a pedestal.The local spectrum then exhibit the oscillation component together with a zerofrequency (DC) component.As we do not want the DC component to contribute to the calculation of the local frequency and to the associated local amplitude (at this local frequency) we filter the DC component by restricting the limits of the integrals defining Ω(t) and ρ(t) (Eq.12 and Eq.13 in the main text, respectively).
In this case we can write that The integral limits ω 1 and ω 2 which isolate the relevant spectral component are set by the extremum points to either side of the component.This is illustrated in Fig. 3 showing a function, its time-frequency distributions at a specific instant exhibiting fast oscillations and the limits for isolating the highest spectral component.

Appendix C: Local frequency calculation example
Fig. 4 shows a local frequency calculation example of four different functions: a cosine, a chirped frequency function, a superoscillatory function with a slowly varying amplitude and a superoscillatory function with a rapidly varying amplitude.The figure contains for each function its time-frequency distribution, analytically calculated local frequency ω(t) (Eq.8 in the main text), the distribution-derived local frequency Ω(t) and its spectrum.The superoscillatory functions contain local oscillations that do not appear in their spectra.It is evident that although the analytically derived local frequency fails to detect the superoscillation for the function with the fast evolving amplitude, the adaptive time-frequency distribution detects it.If there exists a distance z c where all φ m (z c ) become a common phase φ c : The super-oscillatory section will remain intact only to be phase shifted by the common phase − cos(aNt − φ c ). Fig. 5 demonstrates that our numerical method produces a roughly steady The reason for the observed oscillations in the value of the measure and of the maximum local frequency is that when φ c is varied the instantaneous frequency and amplitude remain constant only for very short time scales.In addition the superoscillation effective time duration is changing.These would change in turn any value which depends on finite time windowing.

Fig. 1 .
Fig. 1.Evolution of a superoscillatory signal propagating in SiO 2 as a function of coordinate.(a) The measure M{f(t)} as a function of propagation.(b)-(d) A few instances of the signal along the propagation direction: superoscillatory at z = 0, with no superoscilations at z = 69μm and superoscillatory at the location of the final revival z = 120μm, respectively.(e)Evolution of the highest value of the local frequency Ω(t).The red dashed lines denote the periodic locations of the superoscillation revivals.The green dashed line denotes a location where the signal is not superoscillatory.

Fig. 2 .
Fig. 2. Construction of the dynamical window function for an adaptive Gabor transform.(a) The set of all extremum points{(t n , f (t n ))} is found for the superoscillatory function f so (t). (b) σ (t) is constructed by a spline interpolation of the set of points {(x n , y n )} = {((t n + t n+1 ) /2,t n+1 − t n )}. (c)The window function G (t, τ) is shown for a few representative delay values τ.

Fig. 3 .
Fig. 3. Treating time-frequency distributions with local multi-component spectra.(a) Superoscillatory signal (continuous line) f so (t) with the window function (dashed line) centred on a superoscillation.(b) Local spectrum at the superoscillation.The limits defining the relevant spectral component are marked with ω 1 and ω 2 .

Fig. 4 .
Fig. 4. Using the adaptive Gabor transform to analyze the local frequency of four functions: (a) a cosine cos(5t) (b) a chirped function cos(t 2 ) (c) a superoscillatory function with a slowly varying amplitude created with a = 2 and N = 5 parameters (d) a superoscillatory function with rapidly varying amplitude created using the following three mode's amplitudes: A(ω 1 = 1) = 0.014, A(3ω 1 ) = −0.311,A(5ω 1 ) = −0.219.For each function f (t): F(t, ω) is the time-frequency distribution, ω(t) is the analytically derived local frequency, Ω(t) is the distribution-derived local frequency and f (ω) is its Fourier transform.For visibility, the distributions F(t, ω) are graphically normalized for each time value.The calculated Ω(t) are traced in a dashed white line over the distribution.