A Forced Harmonic Oscillator, Interpreted as Diffraction of Light
aa r X i v : . [ phy s i c s . c l a ss - ph ] A p r A Forced Harmonic Oscillator, Interpreted as Diffraction of Light
Toshihiko Hiraiwa, ∗ Kouichi Soutome,
1, 2 and Hitoshi Tanaka RIKEN SPring-8 Center (RSC), Sayo, Hyogo, 679-5148, Japan Japan Synchrotron Radiation Research Insutitute (JASRI), Sayo, Hyogo, 679-5198, Japan (Dated: April 28, 2020)We investigate a simple forced harmonic oscillator with a natural frequency varying with time. Itis shown that the time evolution of such a system can be written in a simplified form with Fresnelintegrals, as long as the variation of the natural frequency is sufficiently slow compared to the timeperiod of oscillation. Thanks to such a simple formulation, we found, for the first time, that a forcedharmonic oscillator with a slowly-varying natural frequency is essentially equivalent to diffractionof light.
Resonance phenomena, in conjunction with forced har-monic oscillators (FHOs), are observed in a lot of dynam-ical systems, and are discussed as a fundamental prob-lem in standard textbooks on classical mechanics [1, 2].The concept of resonances is present in many branchesof science, and therefore has a wide variety of applica-tions. About three hundred years after the discoveryof a resonance-like phenomenon, theoretical models forFHOs with characteristic resonances have been well es-tablished (see Refs. [3, 4] for a recent historical reviewof FHOs). In many cases, FHOs have been discussed inthe context of resonance phenomena. Here, we presenta simple formulation of a FHO with a natural frequencyvarying with time using Fresnel integrals [5]. Thanks tosuch a simple formulation, we found, for the first time,that a FHO with a time-varying natural frequency is es-sentially equivalent to diffraction of light from a singleslit, i.e., so-called Fraunhofer or Fresnel diffraction [6–9].In this letter, we investigate a simple FHO with a time-varying natural frequency. We suppose that the driv-ing force is activated at t = 0 and is then deactivatedat t = ∆ ( > ω f ≡ πν f ) is kept constant while the natural fre-quency of the oscillator ( ω ≡ πν ) varies with time as ω ( t = 0) < ω f < ω ( t = ∆). In addition, it is assumedthat ω ( t ) varies very slowly compared to the time periodof oscillation, namely: | ˙ ω | ≪ ω , | ¨ ω | ≪ ω , (1)where ˙ ω and ¨ ω represent the first and second derivativesof ω , respectively.The basic equation of motion for the above system iswritten in the form:¨ x + ω ( t ) x = F ( t ) , (2)with the driving force: F ( t ) = t < F cos ( ω f t + φ ) (0 ≤ t ≤ ∆)0 ( t > ∆) , (3)where x denotes displacement from the equilibrium posi-tion as a function of t , F is the maximum driving force, and φ is a constant phase. Here, we neglect a dampingterm for simplicity [10].Now, the frequency ω (= 2 πν ) of the oscillator is afunction of t , and can be expanded in a Taylor’s series: ω ( t ) = ω + ω t + ω t + · · · = 2 π (cid:0) ν + ν t + ν t + · · · (cid:1) . (4)Note that we hereafter omit the second and higher orderterms of Eq. (4), which are less significant than the linearterms because of the assumption (1).Equation (2) can be approximately solved with the aidof the well-known Green’s Function method. Under theassumption (1), the Green’s function of Eq. (2) is givenby [11]: G ( t, t ′ ) = − i p ω ( t ) ω ( t ′ ) exp (cid:20) i Z tt ′ ω ( τ ) dτ (cid:21) + c.c. . (5)Using the Green’s function of Eq. (5) together withthe assumption (1), one can easily obtain a particularsolution of Eq. (2) for t > ∆ [12]: x ( t ) = Z ∆0 G ( t, t ′ ) F cos( ω f t ′ + φ ) dt ′ ≃ iF ω e − i Z t ω ( τ ) dτ + φ × h ( r ) + c.c. , (6)with a response function: h ( r ) = ω ω Z ∆0 exp h i n ω (1 − r ) t ′ + ω t ′ oi dt ′ = √ ν √ ν exp (cid:20) − i πν (1 − r ) ν (cid:21) × Z √ ν ∆0 exp i π ( ˜ t + √ ν (1 − r ) √ ν ) d ˜ t = √ ν √ ν exp (cid:20) − i πν (1 − r ) ν (cid:21) × [ { C ( u ) − C ( u ) } + i { S ( u ) − S ( u ) } ] . (7)Here, we define r ≡ ω f /ω (= ν f /ν ), u and u are givenby: u = √ ν (1 − r ) √ ν u = u + √ ν ∆ , (8)and two functions C ( u ) and S ( u ) are so-called Fresnelintegrals, defined as: C ( u ) = Z u cos (cid:16) i π v (cid:17) dvS ( u ) = Z u sin (cid:16) i π v (cid:17) dv . (9)Note that, in deriving Eqs. (6) and (7), we neglectrapidly-oscillating terms in the integrand.As we see from Eq. (6), the particular solution obtainedhere is of a characteristic form: that is, the first part ofthe r.h.s. of Eq. (6) represents a propagating wave withfrequency modulation, whereas the last one representsa response of the oscillation amplitude to the frequency ω f of the driving force. Furthermore, as we see in theresponse function of Eq. (7), the imaginary argument ofthe exponent in the integrand is a quadratic function ofa variable t ′ (˜ t ), thus yielding Fresnel integrals.Our formulation can be also extended to the other casewhere the frequency ω of the oscillator is kept constantwhile the frequency ω f of the driving force varies slowlywith time as ω f ( t = 0) < ω < ω f ( t = ∆), as discussed inRefs. [13, 14]. In this case, a response function is a bitmodified, but is written in almost the same form as thatin Eq. (7): h (˜ r ) = ω (1) f ω Z ∆0 exp " − i ( ω (0) f (1 − ˜ r ) t ′ + ω (1) f t ′ ) dt ′ , (10) ScreenSlit r D = 2a x
Wavefront
FIG. 1. Diffraction of light from a single slit. Each solidline represents an undiffracted wavefront, while a solid curverepresents a diffracted wavefront determined from Huygens’wavelets at the aperture (dashed-line circles). Here, we definethe wavelength λ , the aperture size D = 2 a , and the distance r between the slit and the screen. TABLE I. Correspondence relations between the FHO andthe single-slit diffraction. The definition of each symbol forthe single-slit diffraction is the same as in Fig 1. For Fresnelnumbers, see the text.FHO Single-slit Diffraction ω f (Driving force) Position on screen ω (Oscillator) Position on slitVariation of ω in ∆, 2 πν ∆ Aperture size D = 2 a Phase slippage betweenoscillator and force Variation of opticalpath lengthA quantity ˜ N F ≡ ν ∆ / N F = a / ( λr ) which yields Fresnel integrals as well. Here, we write thefrequency ω f as: ω f ( t ) = ω (0) f + ω (1) f t + ω (2) f t + · · · , (11)define ˜ r ≡ ω/ω (0) f , and neglect rapidly-oscillating termsin the integrand. Note that, strictly speaking, the as-sumption of the ”slow change” of ω f is not necessary forthe derivation of Eq. (10).One may encounter a quite similar form as in Eqs. (6),(7) and (10) in a description of diffraction of light froma single slit (Fig. 1) based on the so-called Fresnel-Kirchhoff diffraction integral with the Fresnel approxima-tion (see, e.g., Ref. [9]). Fresnel’s formulation of single-slit diffraction approximates the imaginary argument ofthe exponent in the integrand, which represents a phasedifference between secondary spherical waves from thewavefront at the aperture, to be a quadratic phase varia-tion, which results in a characteristic form of Fresnel in-tegrals multiplied by a constant phase factor [cf. Eqs. (6)and (7)].By comparing Eq. (7) with Fresnel’s formula, one canobtain correspondence relations between the FHO andthe single-slit diffraction, as listed in Table I. It is ob-vious from Eq. (7) that the FHO with slowly-varyingfrequencies can be viewed as diffraction of waves in the frequency domain with time t to be an independent vari-able, whereas the single-slit diffraction here is discussedin the space domain. Thus, the frequency ω , moving inthe frequency domain during a time window ∆, is inter-preted, in the case of single-slit diffraction, as the incre-mental space coordinate on the slit from − a to + a , andthe constant frequency ω f as an observation point, i.e.,the space coordinate on the screen.A key feature common to both phenomena is aquadratic term in the phase of integrated waves, whichyields Fresnel integrals, as in Eq. (7). In the FHO case,such a phase term comes from the difference of phase ad-vance, i.e., the phase slippage between the oscillator andthe driving force. In the single-slit diffraction case, onthe other hand, such a phase term comes from Fresnel’sapproximation of the optical path lengths of accumulated − − − − x [mm] I n t en s i t y ( a . u . ) (b)-(i) = 0.3 F N − − − − x [mm] (b)-(ii) = 1 F N − − − − x [mm] (b)-(iii) = 10 F N ω / f ω r = − × | h (r) | (a)-(i) = 0.3 F N~ ω / f ω r = − × (a)-(ii) = 1 F N~ ω / f ω r = − × (a)-(iii) = 10 F N~ FIG. 2. (a) Frequency responses h ( r ) for (i) ˜ N F = 0 . N F = 1 and (iii) ˜ N F = 10, with ν = 10 [Hz] and ν = 0 . − ]. The ranges of resonant frequencies eval-uated by Eq. (13) are marked by dashed lines. (b) Diffractionpatterns from a single slit for (i) N F = 0 .
3, (ii) N F = 1 and(iii) N F = 10 with λ = 1 [ µ m] and r = 1 [m]. Dashed linesrepresent the positions of x = ± δ ¯ x . For the definition of δ ¯ x ,see the text. spherical waves.In the theory of single-slit diffraction, a Fresnel number N F [= a / ( λr )] is often defined to characterize diffrac-tion patterns with different configurations: for N F ≪ Fraunhoferdiffraction ). On the other hand, for N F &
1, Fresnel’sformula is called a Fresnel transformation, and a result-ing diffraction pattern is a perfect shadow of the aper-ture (i.e.,
Fresnel diffraction ). In the FHO case, a corre-sponding quantity is given by:˜ N F ≡ ν ∆ . (12)As is the case of the single-slit diffraction, systems withthe same value of ˜ N F will have a response function h ( r )of equivalent properties.Figures 2 (a) show the frequency responses h ( r ) withdifferent values of ˜ N F (i.e., different values of ∆). Forreference, the intensity patterns of single-slit diffractionwith the same values of N F (i.e., corresponding values of a ) are plotted in Figs. 2 (b). For both the phenomena, adramatic change of the frequency responses h ( r ) (or thediffraction patterns) takes place around ˜ N F ( N F ) ≈ h ( r ) on ˜ N F is in excellentagreement with that of the diffraction patterns on N F .For quantitative discussion, we evaluate the range ofresonant frequencies for the driving force, 2 δ ¯ ω f , using theanalogies between the FHO and the single-slit diffrac-tion. To clarify the situation, we start with the lightdiffraction case: for the Fraunhofer regime ( N F ≪ δ ¯ x of a principal peakis obtained from the slit-screen distance r and the an-gle θ , which defines a destructive phase relation betweenthe wavelets from the both edges of the aperture, andis given by 2 δ ¯ x ≈ r θ ≈ λr / (2 a ). For the Fresnelregime ( N F & δ ¯ x ≈ a .Now, the derivation of δ ¯ ω f is straightforward: with thecorrespondence relations in Table I, we obtain:2 δ ¯ ω f = ( π/ ∆ (for the Fraunhofer regime)2 πν ∆ (for the Fresnel regime) . (13)The evaluated ranges for different values of ˜ N F are in-dicated by dashed lines in Figs. 2 (a). As we see fromthe figures, our evaluation is valid both for the Fraun-hofer and Fresnel regimes. We notice that, for theFHO case, the centre of resonant frequencies is given by¯ ω f = 2 πν + πν ∆.As another example of the analogies between FHOswith time-varying frequencies and light diffraction, let usconsider a HO with a time-varying natural frequency ex-posed continuously to a sinusoidal force with a constantfrequency. Here, we suppose that the frequency ω of theoscillator coincides with the frequency ω f of the drivingforce at t = 0. In this case, a particular solution is ob-tained just by setting r = 1 and replacing ∆ with t inthe response function of Eq. (7), namely: h ( t ) = √ ν √ ν Z √ ν t exp (cid:16) i π t (cid:17) d ˜ t = √ ν √ ν (cid:2) C (cid:0) √ ν t (cid:1) + iS (cid:0) √ ν t (cid:1)(cid:3) , (14) − × | h ( t ) | = 1 F N~ (a) I n t en s i t y ( a . u . ) = 1 F N (b) FIG. 3. (a) Time evolution of the squared amplitude | h ( r ) | with ν = 10 [Hz] and ν = 0 . − ]. An arrow indicatesthe time t = δt corresponding to ˜ N F = 1. For the definitionof δt , see the text. (b) Intensity pattern for light diffractionfrom a knife-edge obstacle with λ = 1 [ µ m] and r = 1 [m].he obstacle is placed at x ≤
0. An arrow indicates the screenposition x corresponding to N F = 1. which is in turn a function of t , and thus describesthe time evolution of the oscillation amplitude. Equa-tion (14) is a quite similar expression to a diffractionformula for so-called knife-edge diffraction.Figure 3 (a) illustrates the time evolution of thesquared oscillation amplitude (or, equivalently, the en-ergy of the oscillator), together with an intensity patternfor light diffraction from a knife-edge obstacle [Fig. 3 (b)].We see that the time evolution of the oscillation energybehaves like a knife-edge diffraction pattern: that is, theenergy increases monotonically until t .
60 [sec] and thenexhibits small beating (in other word, we could say thatthe oscillator is in a quasi-stationary state). Asymptoti-cally, it approaches to [see Eq. (14)]: | h ( t ) | t → + ∞ −−−−→ ν ν (cid:2) C (+ ∞ ) + S (+ ∞ ) (cid:3) = ν ν . (15)The time duration δt in which the driving force effi-ciently supplies kinetic energy to the oscillator is esti-mated by using the analogies: in knife-edge diffraction,a ”good measure” of the fringe width of diffraction pat-terns, δx , is given by the condition that the correspond-ing Fresnel number, N F = δx / ( λr ), becomes unity [seeFig. 3 (b)]. Similarly, we have the time duration δt : δt = 1 √ ν ≈
60 [sec] , (16)with ν = 0 . − ].To summarize, we investigated a simple FHO withslowly-varying frequencies. We demonstrated that thetime evolution of such a system can be written in asimplified form using Fresnel integrals. As a result, wefound that FHOs with slowly-varying frequencies can beviewed as diffraction of waves in the frequency domain,and therefore are equivalent to diffraction of light. Alsowe showed two examples to see the similarities betweenthe two phenomena, and derived simple formulae for thequantities which characterize the systems. We expectthat our formulation as well as such simple formulae isapplied to, e.g., accelerator physics and provides a sim- ple and intuitive approach to the phenomenon of ”reso-nance crossing”, which is a central issue in a ring-typeparticle accelerator design [15, 16]. As a matter of fact,we applied our formulation to the design of an aborted-beam-handling system for a new synchrotron light sourceaccelerator [17]. Our findings will shed light on anotheraspects of fundamental physics and stimulate related re-search fields. ∗ [email protected][1] R. P. Feynman, R. B. Leighton, and M. Sands, The Feyn-man Lectures on Physics , 1st ed., Vol. 1 (Addison Wesley,1971).[2] H. Goldstein, C. P. Poole Jr., and J. L. Safko,
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