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A Forced Harmonic Oscillator, Interpreted as Diffraction of Light

We investigate a simple forced harmonic oscillator with a natural frequency varying with time. It is shown that the time evolution of such a system can be written in a simplified form with Fresnel integrals, as long as the variation of the natural frequency is sufficiently slow compared to the time period of oscillation. Thanks to such a simple formulation, we found, for the first time, that a forced harmonic oscillator with a slowly-varying natural frequency is essentially equivalent to diffraction of light.

We investigate a simple forced harmonic oscillator with a natural frequency varying with time. It is shown that the time evolution of such a system can be written in a simplified form with Fresnel integrals, as long as the variation of the natural frequency is sufficiently slow compared to the time period of oscillation. Thanks to such a simple formulation, we found, for the first time, that a forced harmonic oscillator with a slowly-varying natural frequency is essentially equivalent to diffraction of light.
Resonance phenomena, in conjunction with forced harmonic oscillators (FHOs), are observed in a lot of dynamical systems, and are discussed as a fundamental problem in standard textbooks on classical mechanics [1,2]. The concept of resonances is present in many branches of science, and therefore has a wide variety of applications. About three hundred years after the discovery of a resonance-like phenomenon, theoretical models for FHOs with characteristic resonances have been well established (see Refs. [3,4] for a recent historical review of FHOs). In many cases, FHOs have been discussed in the context of resonance phenomena. Here, we present a simple formulation of a FHO with a natural frequency varying with time using Fresnel integrals [5]. Thanks to such a simple formulation, we found, for the first time, that a FHO with a time-varying natural frequency is essentially equivalent to diffraction of light from a single slit, i.e., so-called Fraunhofer or Fresnel diffraction [6][7][8][9].
In this letter, we investigate a simple FHO with a timevarying natural frequency. We suppose that the driving force is activated at t = 0 and is then deactivated at t = ∆ (> 0), and that the frequency of the driving force (ω f ≡ 2πν f ) is kept constant while the natural frequency of the oscillator (ω ≡ 2πν) varies with time as ω(t = 0) < ω f < ω(t = ∆). In addition, it is assumed that ω(t) varies very slowly compared to the time period of oscillation, namely: whereω andω represent the first and second derivatives of ω, respectively. The basic equation of motion for the above system is written in the form: with the driving force: where x denotes displacement from the equilibrium position as a function of t, F 0 is the maximum driving force, and φ 0 is a constant phase. Here, we neglect a damping term for simplicity [10]. Now, the frequency ω (= 2πν) of the oscillator is a function of t, and can be expanded in a Taylor's series: Note that we hereafter omit the second and higher order terms of Eq. (4), which are less significant than the linear terms because of the assumption (1). Equation (2) can be approximately solved with the aid of the well-known Green's Function method. Under the assumption (1), the Green's function of Eq. (2) is given by [11]: Using the Green's function of Eq. (5) together with the assumption (1), one can easily obtain a particular solution of Eq. (2) for t > ∆ [12]: with a response function: Here, we define r ≡ ω f /ω 0 (= ν f /ν 0 ), u 1 and u 2 are given by: and two functions C(u) and S(u) are so-called Fresnel integrals, defined as: Note that, in deriving Eqs. (6) and (7), we neglect rapidly-oscillating terms in the integrand.
As we see from Eq. (6), the particular solution obtained here is of a characteristic form: that is, the first part of the r.h.s. of Eq. (6) represents a propagating wave with frequency modulation, whereas the last one represents a response of the oscillation amplitude to the frequency ω f of the driving force. Furthermore, as we see in the response function of Eq. (7), the imaginary argument of the exponent in the integrand is a quadratic function of a variable t ′ (t), thus yielding Fresnel integrals.
Our formulation can be also extended to the other case where the frequency ω of the oscillator is kept constant while the frequency ω f of the driving force varies slowly with time as ω f (t = 0) < ω < ω f (t = ∆), as discussed in Refs. [13,14]. In this case, a response function is a bit modified, but is written in almost the same form as that in Eq. (7):  which yields Fresnel integrals as well. Here, we write the frequency ω f as: f , and neglect rapidly-oscillating terms in the integrand. Note that, strictly speaking, the assumption of the "slow change" of ω f is not necessary for the 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 from a single slit (Fig. 1) based on the so-called Fresnel-Kirchhoff diffraction integral with the Fresnel approximation (see, e.g., Ref. [9]). Fresnel's formulation of singleslit diffraction approximates the imaginary argument of the exponent in the integrand, which represents a phase difference between secondary spherical waves from the wavefront at the aperture, to be a quadratic phase variation, which results in a characteristic form of Fresnel integrals multiplied by a constant phase factor [cf. Eqs. (6) and (7)].
By comparing Eq. (7) with Fresnel's formula, one can obtain correspondence relations between the FHO and the single-slit diffraction, as listed in Table I. It is obvious from Eq. (7) that the FHO with slowly-varying frequencies can be viewed as diffraction of waves in the frequency domain with time t to be an independent variable, whereas the single-slit diffraction here is discussed in the space domain. Thus, the frequency ω, moving in the frequency domain during a time window ∆, is interpreted, in the case of single-slit diffraction, as the incremental space coordinate on the slit from −a to +a, and the constant frequency ω f as an observation point, i.e., the space coordinate on the screen.
A key feature common to both phenomena is a quadratic term in the phase of integrated waves, which yields Fresnel integrals, as in Eq. (7). In the FHO case, such a phase term comes from the difference of phase advance, i.e., the phase slippage between the oscillator and the driving force. In the single-slit diffraction case, on the other hand, such a phase term comes from Fresnel's approximation of the optical path lengths of accumulated spherical waves.
In the theory of single-slit diffraction, a Fresnel number N F [= a 2 /(λr 0 )] is often defined to characterize diffraction patterns with different configurations: for N F ≪ 1, where the screen is far from the slit, or where the slit aperture is narrow, a quadratic term in the phase is negligible so that Fresnel's formula is reduced to a Fourier transform of the shape of the aperture (i.e., Fraunhofer diffraction). On the other hand, for N F 1, Fresnel's formula is called a Fresnel transformation, and a resulting diffraction pattern is a perfect shadow of the aperture (i.e., Fresnel diffraction). In the FHO case, a corresponding quantity is given by: As is the case of the single-slit diffraction, systems with the same value ofÑ F will have a response function h(r) of equivalent properties. Figures 2 (a) show the frequency responses h(r) with different values ofÑ F (i.e., different values of ∆). For reference, the intensity patterns of single-slit diffraction with the same values of N F (i.e., corresponding values of a) are plotted in Figs. 2 (b). For both the phenomena, a dramatic change of the frequency responses h(r) (or the diffraction patterns) takes place aroundÑ F (N F ) ≈ 1. Furthermore, the behavior of h(r) onÑ F is in excellent agreement with that of the diffraction patterns on N F .
For quantitative discussion, we evaluate the range of resonant frequencies for the driving force, 2δω f , using the analogies between the FHO and the single-slit diffraction. To clarify the situation, we start with the light diffraction case: for the Fraunhofer regime (N F ≪ 1), it is well known that the width 2δx of a principal peak is obtained from the slit-screen distance r 0 and the angle θ, which defines a destructive phase relation between the wavelets from the both edges of the aperture, and is given by 2δx ≈ 2r 0 θ ≈ λr 0 /(2a). For the Fresnel regime (N F 1), the width of a rectangular pattern is almost the same as that of the aperture, namely, 2δx ≈ 2a. Now, the derivation of δω f is straightforward: with the correspondence relations in Table I, we obtain: 2δω f = 2π/∆ (for the Fraunhofer regime) 2πν 1 ∆ (for the Fresnel regime).
The evaluated ranges for different values ofÑ F are indicated by dashed lines in Figs. 2 (a). As we see from the figures, our evaluation is valid both for the Fraunhofer and Fresnel regimes. We notice that, for the FHO case, the centre of resonant frequencies is given bȳ ω f = 2πν 0 + πν 1 ∆.
As another example of the analogies between FHOs with time-varying frequencies and light diffraction, let us consider a HO with a time-varying natural frequency exposed continuously to a sinusoidal force with a constant frequency. Here, we suppose that the frequency ω of the oscillator coincides with the frequency ω f of the driving force at t = 0. In this case, a particular solution is obtained just by setting r = 1 and replacing ∆ with t in the response function of Eq. (7), namely: which is in turn a function of t, and thus describes the time evolution of the oscillation amplitude. Equation (14) is a quite similar expression to a diffraction formula for so-called knife-edge diffraction. Figure 3 (a) illustrates the time evolution of the squared oscillation amplitude (or, equivalently, the energy of the oscillator), together with an intensity pattern for light diffraction from a knife-edge obstacle [ Fig. 3 (b)]. We see that the time evolution of the oscillation energy behaves like a knife-edge diffraction pattern: that is, the energy increases monotonically until t 60 [sec] and then exhibits small beating (in other word, we could say that the oscillator is in a quasi-stationary state). Asymptotically, it approaches to [see Eq. (14)]: The time duration δt in which the driving force efficiently supplies kinetic energy to the oscillator is estimated by using the analogies: in knife-edge diffraction, a "good measure" of the fringe width of diffraction patterns, δx, is given by the condition that the corresponding Fresnel number, N F = δx 2 /(λr 0 ), becomes unity [see Fig. 3 (b)]. Similarly, we have the time duration δt: with ν 1 = 0.0003 [sec −2 ]. To summarize, we investigated a simple FHO with slowly-varying frequencies. We demonstrated that the time evolution of such a system can be written in a simplified form using Fresnel integrals. As a result, we found that FHOs with slowly-varying frequencies can be viewed as diffraction of waves in the frequency domain, and therefore are equivalent to diffraction of light. Also we showed two examples to see the similarities between the two phenomena, and derived simple formulae for the quantities which characterize the systems. We expect that our formulation as well as such simple formulae is applied to, e.g., accelerator physics and provides a sim-ple and intuitive approach to the phenomenon of "resonance crossing", which is a central issue in a ring-type particle accelerator design [15,16]. As a matter of fact, we applied our formulation to the design of an abortedbeam-handling system for a new synchrotron light source accelerator [17]. Our findings will shed light on another aspects of fundamental physics and stimulate related research fields.