Theory of cavity ring-up spectroscopy

Cavity ring-up spectroscopy (CRUS) provides an advanced technique to sense ultrafast phenomena, but there is no thorough discussion on its theory. Here we give a detailed theoretical analysis of CRUS with and without modal coupling, and present exact analytical expressions for the normalized transmission, which are very simple under certain reasonable conditions. Our results provide a solid theoretical basis for the applications of CRUS.


I. INTRODUCTION
Whispering-gallery-mode (WGM) microresonators confine light through internal total reflection and can support optical modes with high quality factors and small mode volumes [1]. These features make them an important platform for studying the interaction between light and matter [2][3][4][5][6]. When a continuous laser is utilized to sweep the optical modes of a WGM microresonator via a fiber taper at very slow speed, the steady-state transmission spectrum of the system can be obtained [7]. By monitoring the change of the steady-state spectrum, such as the mode shift [8], broadening [9] and splitting [10], highly sensitive sensing can be achieved [11,12]. However, these methods can only sense the environmental change on a timescale of milliseconds due to the slow sweeping speed of the laser.
Recently cavity ring-up spectroscopy (CRUS) is proposed [13]. Experiments with WGM microresonators show that CRUS can be measured within tens of nanoseconds and thus it provides a way to achieve ultrafast sensing [13]. CRUS can be understood through an abrupt turn on of a far-detuned monochromatic laser that is connected to the input of the fiber taper. Although the laser frequency is far detuned from the mode frequency of the microresonator, some light of the mode frequency can still be coupled into the microresonator due to the broadening of the laser frequency, which is caused by the sharp rise of the laser intensity. The light into the microresonator will leak back to the fiber taper and then interfere with the directly transmitted light, which can lead to an oscillation of the light intensity in the output of the fiber taper. As a demonstration of its ability in sensing ultrafast phenomena, CRUS has shown its application in monitoring the optomechanical vibrations of the WGM microresonator [13]. It is believed that CRUS can be used in the study of ultrafast phenomena such as protein folding, light harvesting and cavity quantum electrodynamics [13]. However, there is no detailed theory of CRUS that may limit its future application [14]. It is the purpose of this paper to provide a theory of CRUS.

II. THEORY OF CRUS
The idea of the CRUS is illustrated in Fig. 1, where a fiber taper is used to couple light into and out of a WGM microresonator. The input field in the fiber taper is denoted by E in , which will excite the clockwise (CW) mode in the microresonator denoted by E cw . The evolution of E cw satisfies the equation [1] where w c is the angular frequency of the CW mode, κ is the total loss rate, and κ e (κ e < κ) is the loss rate associated with the fiber taper. In CRUS, a monochromatic laser with a special amplitude modulation is coupled into the fiber taper and the input field can be written as where s is the amplitude of the monochromatic laser, w l is the angular frequency of the monochromatic laser and f (t) is a step-like modulation function as shown in Fig. 1. The value of the function f (t) is 0 when t ≤ 0, changes from 0 to 1 in a short rise time t r , and keeps the value 1 when t ≥ t r . The input in Eq. (2) describes the abrupt turn on of a monochromatic laser at t = 0.
It is natural to assume that E cw (0) = 0. Based on the method of variation of constants, an integral expression for E cw (t) can be obtained, The correctness of Eq. (3) can be verified by the facts that it satisfies the initial condition E cw (0) = 0 and the differential where δ = w l − w c is the detuning of the angular frequency of the laser relative to the angular frequency of the CW mode. Using the method of integration by parts and f (0) = 0, we can obtain The interesting time interval in CRUS is t ≥ t r where there is f (t) = 1 and df (t)/dt = 0. Define There is which has two terms, one with the angular frequency w l of the laser and the other with the angular frequency w c of the CW mode. The parameter α describes the effect of f (t) on the field E cw (t). A detailed discussion on the parameter α will be given in section 4, where we will show that there is |α| < 1, and α ≈ 1 can be achieved if the rise time t r is short enough.
The output field E out in the fiber taper is related to the input field and the field in the microresonator through the expression Substitute Eq. (8) into the above, we can get where the complex numbers Ae jθA and Be jθB are written in the polar form, and there are The normalized transmission T (t) = |E out (t)/E in (t)| 2 will be which describes the content of CRUS. The third term in the above shows that T (t) will oscillate with the period T p = 2π/|δ|, the oscillation lifetime τ = 1/κ and the initial oscillation amplitude A p = |2AB|. Within the lifetime τ , the number of the oscillation will be N p = τ /T p = |δ|/(2πκ), which indicates that |δ| ≫ κ is necessary even when we only want to observe a small number of oscillations within the lifetime. Suppose the condition |δ| ≫ κ is satisfied, then Ae jθA ≈ −1, Be jθB ≈ 2κ e αj/δ and which will lead to where θ α is the phase of α and the term proportional to (κ e /δ) 2 is omitted. Since the initial oscillation amplitude of T (t) is A p = 4κ e |α|/|δ|, it is better to experimentally observe CRUS in the over coupling condition with an optical mode of ultrahigh intrinsic quality factor so that κ e can be large enough to get a relative large A p .

III. THEORY OF CRUS WITH MODAL COUPLING
A WGM microresonator can also support a counterclockwise (CCW) mode that is degenerate to the CW mode. Now we consider the case where there is a coupling between the degenerate CW and CCW modes [15]. The CCW mode is denoted by E ccw . Instead of the Eq. (1), the dynamical equations are [15] where β is the coupling strength between the CW and CCW modes. Define It can be obtained that which are decoupled equations similar to Eq. (1). Therefore similar to Eq. (8) there are where δ + = δ + β, δ − = δ − β and Note that E cw (t) = {E + (t) + E − (t)}/2. According to Eq. (9), the exact expression for the normalized transmission T (t) = |E out (t)/E in (t)| 2 is where In the following, with some approximations we will present a simple expression of T (t).
It can be seen that the approximate T (t) agrees well with the exact one.

IV. INFLUENCE OF THE MODULATION FUNCTION
The modulation function f (t) affects the normalized transmission T (t) through the parameter α. We first note that there is |α| < 1, because from its definition in Eq. (6), where df (t)/dt ≥ 0 is assumed, which is reasonable in CRUS. If the rise time t r is so short that |δ|t r ≪ 1 and κt r ≪ 1, then α ≈ 1 because from Eq.
where only the first term of the Taylor series of the exponential function in Eq. (6) is kept. The result means that CRUS will not depend on the detailed shape of the modulation function f (t) if the rise time t r is short enough. A correction to α ≈ 1 can be obtained if the first two terms of the Taylor series of the exponential function in Eq. (6) are kept, i.e., The parameter γ is defined as where the method of the integration by parts is used. It can be seen that there is 0 < γ < 1.
In the case f (t) = sin( πt 2tr ) (0 ≤ t ≤ t r ), there is where δ ′ = δ − π/(2t r ) and δ ′′ = δ + π/(2t r ). These two exact expressions demonstrate that α ≈ 1 if the rise time t r is short enough. We finally note that the discussions in the above two sections will not change when f (t) (0 ≤ t ≤ t r ) is a complex function with f (t r ) = 1 (this requirement will not make f (t) loss of generality). And we will still have α ≈ 1 if the rise time t r is short enough.