1. INTRODUCTION

Passive optical cavities are commonly used to enhance laser power for laser frequency conversion such as second-harmonic generation [1] and for sub-Doppler saturation spectroscopy [2]. If there is a nonlinear device in an optical cavity, it is well known that the optical power through the cavity can exhibit bistability [3]. Applications of the optical bistability include optical switching [4,5] and optical computation [6]. Such bistability in passive optical cavities has been observed in conventional macroscopic cavities with intracavity nonlinear materials such as saturable absorbers [7,8], and it is also observed in microcavities made of nonlinear materials such as semiconductors [5,9–11]. The bistability in microcavities can be observed for low input laser power because of high finesse and high nonlinearity. An example is nonlinearity induced by polaritons [5,10,11]. The bistability in macroscopic optical cavities is anticipated only if extremely high nonlinearity exists. For example, in the first observation of the optical bistability [7], the nonlinear device was sodium vapor, which is easily saturated by the optical beam and therefore highly nonlinear.

In this study, an optical bistability with a macroscopic passive ring cavity is observed in the near-infrared region (952 nm). In this cavity, there are no designed nonlinear materials, and hence the bistability observed has been unexpected. This bistability is undesirable when the cavity should be stabilized at resonance for purposes of second-harmonic generation and saturation spectroscopy. We consider that the saturated absorption of water vapor is responsible for the bistability, and, based on this idea, a theoretical model is constructed. The absorption by water vapor observed in this experiment seems to be very small compared with that of sodium in [7] because the absorption cross section by water is much smaller than that by sodium. In fact, the linear optical absorption in one round trip of the cavity is 3% in this study, whereas that of the sodium in [7] is over 90%. The laser frequency is far-detuned from the resonance (4 GHz), which is larger than the spectral linewidth of water in atmosphere ($\sim 3\mathrm{GHz}$ for pressure broadening, and $\sim 300\mathrm{MHz}$ for Doppler broadening). Hence the bistability described in this paper seems to be dispersive rather than absorptive [12]. Although dispersive bistability appears easily compared to absorptive one [7], it seems that the effect of water vapor dispersion is very small.

To find out what causes the optical bistability, a theoretical model is constructed. As a result of the model calculation, it is found that the bistability is induced by power-dependent refractive index of water indeed, but the mechanism is not straightforward. Gaussian power distribution of the optical beam in the cavity induces spatial distribution of the refractive index of atmosphere with water vapor. This is considered as an effective graded-index (GRIN) lens, which gives rise to the self-(de)focusing effect. Then the Rayleigh range of the beam is modified, and as a result the Gouy phase is shifted. This Gouy phase shift causes the dispersive optical bistability.

The model calculation agrees with the experimental results qualitatively. Discussions of the optical bistability with the Gaussian beam have been given [13,14], but as far as we know, there are no discussions with the Gouy phase shift. The observed optical bistability is a particular case in which the effect of the Gouy phase shift is drastic. In some cases, the Gouy phase shift can give some effect that is not so drastic as optical bistability. For example, the Gouy phase shift may distort spectral line profiles of molecular absorption observed in cavity-enhanced saturation spectroscopy. Another example is that free-spectral range of a cavity becomes dependent on the optical power because of the Gouy phase shift. These two examples suggest that the Gouy phase shift is especially important to noise-immune cavity-enhanced optical heterodyne molecular spectroscopy (NICE-OHMS) [15,16].

2. EXPERIMENT

The experimental setup is schematically shown in Fig. 1. The optical cavity is of a bow-tie ring configuration, and one of the mirrors is attached on a piezo-electric transducer (PZT, 60 nm/V) for modulation of the cavity length. This cavity is originally for enhancement of second-harmonic generation efficiency (for simplicity, the nonlinear crystal is taken out). The laser beam is coupled into the cavity through an input coupler of a partial-reflection mirror (transmittance is 2.5%). A fused silica plate at the Brewster angle for astigmatism compensation is inserted inside the cavity [17].

 

Fig. 1. Experimental setup. ECLD, external cavity laser diode; TA, tapered amplifier; IC, input coupler; M, high-reflective flat mirror; CCM, concave mirror; BP, Brewster plate; PZT, piezoelectric transducer; PD, photodiode.

Download Full Size | PDF

The grating-feedback external-cavity laser diode (ECLD) at 952 nm is amplified by a tapered amplifier. The laser beam is spatially filtered by a pinhole for transversal mode cleaning. A fraction of the beam is introduced into a wavelength meter. The maximum input power is 180 mW in this setup.

The laser wavelength $\lambda$ is set at 952.18 nm or 952.07 nm (in vacuum). The laser wavelength is tuned by adjusting the diffraction grating angle of the ECLD. Finesse of the cavity is measured as follows. A triangle wave voltage (10 Hz, $10{\mathrm{V}}_{\mathrm{p}-\mathrm{p}}$) is applied to the PZT, and transmission through one of the cavity mirrors is observed by a photodiode. Resonance curves are recorded with the input beam power of less than 1 mW to estimate the finesse. As a result, the value of finesse for $\lambda =952.07\mathrm{nm}$ and $\lambda =952.18\mathrm{nm}$ are 250 and 120, respectively. In the case of $\lambda =952.07\mathrm{nm}$, the finesse is determined mainly by the transmission loss at the partial-reflective mirror of 2.5%. In the case of $\lambda =952.18\mathrm{nm}$, it is determined by the water molecule absorption in addition to the partial mirror transmission loss. The observed values of finesse imply that the absorption loss at 952.18 nm is estimated to be $\sim 3\%$. There is a ro-vibrational absorption line of the transition between $v_1=v_2=v_3=0$, $J=4$, $K_a=1$, $K_c=3$ and $v_1=2$, $v_2=0$, $v_3=1$, $J=3$, $K_a=1$, $K_c=2$ in the electronic ground state of water molecules at $10502.0601{\mathrm{cm}}^{-1}$ (transition A) [18], which is close to $\lambda =952.18\mathrm{nm}$ ($10502.2{\mathrm{cm}}^{-1}$). This corresponds to the detuning of $\sim 4\mathrm{GHz}$. The detuning of the laser at 952.07 nm is $-20\mathrm{GHz}$ from the transition at $10504.17{\mathrm{cm}}^{-1}$ (transition B) and $+40\mathrm{GHz}$ from the transition A [18] (the transition A is approximately five times stronger than the transition B).

There may be other factors to cause the optical loss, such as reflection at the Brewster plate and transmission at the other mirrors in the cavity. These factors are negligible because the transmission at the input coupler and the absorption by the water vapor account for the estimated finesse quantitatively.

Normally, optical bistability is observed by increasing or decreasing the input power [7]. In this study, however, the input laser power is kept constant, and the optical cavity length is swept. We think that this setup is practical when the optical cavity is to be stabilized at resonance. Even with sweeping the cavity length, optical bistability has been observed [14]. The optical power into the cavity is set between 30 and 180 mW by tuning the driving current to the tapered amplifier. Then the transmission out of the cavity is recorded as a function of the voltage applied to the PZT (the cavity length displacement) to observe the resonance curve.

The result is shown in Fig. 2. On the upper horizontal axis, the corresponding cavity length displacement is also given. When $\lambda$ is 952.18 nm and the input power is high (180 mW), the hysteresis is clearly observed, implying the optical bistability [14]. For convenience, the state of the higher intracavity optical power is denoted as the HP (high-power) state, and the other state is as the LP (low-power) state. In the experiment, the HP state is observed when the cavity length is swept from shorter to longer (from right to left in the figure). In the bistable range, it depends on the sweeping direction which of the HP or LP state is realized. Meanwhile, when $\lambda =952.07\mathrm{nm}$ the hysteresis is hardly observed. In the case of the input laser power of 180 mW, the bistability within a narrow range is observed.

 

Fig. 2. Experimental results. The transmission through the optical cavity is observed as the cavity length is swept by means of applying the ramp voltage to PZT. Direction of the sweep is indicated by the arrow in the figure. The lower horizontal axis is for the applied voltage to PZT, and the upper one is for the corresponding value of the cavity length displacement. Input laser power and laser wavelength are given in the figure. The relative humidity (RH) is 63% for (a)–(d) and 46% for (e).

Download Full Size | PDF

We consider that the absorption by water vapor in atmosphere and the associated dispersion account for the optical bistability. For confirmation, the result with the identical setup with Fig. 2(a) except the humidity is shown in Fig. 2(e), in which the hysteresis is not so significant as in Fig. 2(a). The relative humidity for Figs. 2(a)–2(d) is 63%, and that for Fig. 2(e) is 46%. In Fig. 2(e), the finesse of the cavity is estimated as 160. In the next section, a detailed theoretical discussion is presented.

3. DISCUSSION

A. Optical Field in the Cavity

Here we calculate the intracavity optical power taking the absorption and the dispersion of water vapor in atmosphere (1 atm, 300 K, relative humidity is 63%) into account. Because the PZT voltage is swept slowly, we assume that the intracavity optical power is always in stationary state.

In this study, the optical beam is supposed to be Gaussian. Namely, the intracavity optical electric field propagating along $z$ axis (optical axis) is

In general, $|E|$ depends on $z$ because of the absorption, loss at the mirrors, and $z$ dependence of $w$. Therefore, to evaluate the single-round-trip attenuation, integration of the absorption coefficient in $z$ from $z=0$ to $L$ (the cavity length, 670 mm in the experiment) is necessary. In this model, however, we consider that $|E|$ does not depend on $z$ for simplicity (note that $w$ is considered as an independent parameter of $z$ as well). With this approximation, the single-round-trip attenuation coefficient is $r_c{e}^{-\alpha L}$, where $r_c$ is the reflection coefficient of the partial reflection mirror (${|r_c|}^2=0.975$), and $\alpha$ is the nonlinear absorption coefficient of water vapor. This approximation may be valid for weak absorption and not-so-tight focusing of the beam.

The intracavity optical electric field amplitude satisfies

B. Water Absorption and Dispersion

We emphasize that $\alpha$ and $n$ are functions of the intracavity optical electric field amplitude $E$ because of the nonlinear absorption and the associated dispersion by water vapor. Because the Doppler broadening ($\sim 300\mathrm{MHz}$) may not be ignorable compared with the pressure broadening ($\sim 3\mathrm{GHz}$), we suppose the Voigt profile for the absorption lineshape. $\alpha$ and $n$ for the laser at the angular frequency detuning of $\delta$ from the absorption line center are expressed as, respectively,

Here, transition saturation of water molecules is supposed to be a nonlinear process that induces the observed bistability, so that $\chi$ is the function of $E$. For the evaluation of $\chi$, some assumptions are made; that is, the closed transition between two energy levels is considered, and there are two relaxation processes of water molecules, namely, population (longitudinal) relaxation and phase (transversal) relaxation.

Then the complex susceptibility is [19]

The value of $a$ is evaluated so that the absorption loss in a single round trip of the cavity is 3% for 952.18 nm in the limit of $E\to 0$ with $\delta /2\pi =4\mathrm{GHz}$. Because $\chi$ is the function of $E$ or $I$, $\alpha$ and $n$ in the right-hand side of Eq. (3) also depend on $E$. Then the resonant curve can be simulated by solving Eq. (3) for $E$ as a function of the cavity length displacement $\mathrm{\Delta }$. In the numerical calculation, because $|\mathrm{\Delta }|\ll L_0$, we approximate the exponential factor in Eq. (3) as $e^{-k{\chi }^{\prime \prime }{L}_0/2}{e}^{ik({\chi }^{\prime }{L}_0/2+\mathrm{\Delta })}$.

In this model, transition saturation is the nonlinear effect; therefore, the estimation of the saturation broadening ${\gamma }_2\sqrt{I/I_S}$ is important. The value of ${\gamma }_2$ is $1.9\times {10}^{10}{\mathrm{s}}^{-1}$ for the transition A [18]. For $\mu$, we can estimate as $\mu =5\times {10}^{-33}\mathrm{C}·\mathrm{m}$ from the spontaneous emission rate ($\sim 0.8{\mathrm{s}}^{-1}$) [18]. For the estimation of ${\gamma }_1$, it should be discussed what mainly causes the population relaxation of water molecules in atmosphere. In the present case, we consider that the diffusion of water molecules out of the optical beam (transit time across the optical beam) is responsible for the population relaxation. Because the mean free path of water molecules in standard atmospheric pressure is in the order of a few tens nanometers, typical time for which the water molecule stays in the optical beam (radius is $w$) can be estimated by the Fick’s law of diffusion as $w^2/(4D)$, where $D$ is the diffusion coefficient of water vapor in air ($\sim 2\times {10}^{-5}{\mathrm{m}}^2·{\mathrm{s}}^{-1}$) [20]. Then ${\gamma }_1$ can be estimated roughly as $4D/w^2$, resulting ${\gamma }_1\sim 3\times {10}^4{\mathrm{s}}^{-1}$. This value of ${\gamma }_1$ might be underestimated because the population relaxation also can be induced by other reasons such as air flow. The population relaxation due to the spontaneous emission is insignificant because its rate is known to be very small as previously stated. Regardless of inaccuracy of the estimated value of ${\gamma }_1$, we think that this estimation is reasonable enough to give qualitative discussions.

C. Numerical Calculations

In the numerical calculation, Eq. (3) is solved self-consistently. For a given $\mathrm{\Delta }$, a trial value of $E$ ($E_r$) is substituted into the right-hand side of Eq. (3), and the obtained $E$ ($E_l$) in the left-hand side is again substituted into the right-hand side as a new trial value. This process is repeated until $|E_l-E_r|/|E_r|$ is less than ${10}^{-3}$. The same calculation is done for the next value of $\mathrm{\Delta }$ to obtain the resonance curve.

This procedure simulates the actual process occurring inside the cavity when $\mathrm{\Delta }$ is swept. Then we expect that the hysteresis (optical power dependence on the sweeping direction) can be reproduced. The sweeping direction dependence is taken into account by choosing the initial trial value of $E$ in the above-mentioned procedure. In this model, the final value of $E$ in each step is the initial trial value of $E$ for the next step.

In the experiment, the saturation occurs when $I/I_S$ is higher than $1+{\delta }^2/{\gamma }_2^2$. This condition is satisfied when the optical power is over $\sim 9.8\mathrm{W}$. The power in the cavity on resonance is $\sim 60\mathrm{W}$; therefore, it seems likely that the transition saturation causes the bistability. Figure 3(a) shows the result of the numerical calculation. As shown in the figure, no results of the bistability are obtained. By artificially tuning the parameters, the bistability is somehow reproduced, as shown in Fig. 3(b), but the parameter values are impractical [$a$ is 10 times larger than that for Fig. 3(a) and ${\gamma }_1$ is $1/10$ of that for Fig. 3(a)]. In addition, the sweeping direction with which the HP state appears does not agree with the experimental result (Fig. 2). These calculations imply that the nonlinear absorption and the associated dispersion of water vapor are insufficient to cause the bistability observed in the experiment.

 

Fig. 3. Calculation results with taking the power-dependent nonlinear absorption and the refractive index into account. Red dashed lines and black solid lines correspond to the sweeps of right and left directions, respectively, as indicated by the arrows in the figure. In (a), the parameter values are set as estimated from the experimental setup as stated in the text. In (b), the value of ${\chi }^{\prime \prime }$ is 10 times larger than that set in (a), and the value of ${\gamma }_1$ is $1/10$ of that set in (a), in order to artificially obtain the result of the optical bistability. The scale of the horizontal axis is magnified by a factor of 10 compared with that in Fig. 2.

Download Full Size | PDF

D. Gouy Phase Shift

As in Eq. (7), the refractive index depends on the optical power. The power distribution of the Gaussian beam gives transversal spatial dependence of refractive index, resulting in a self-focusing or self-defocusing effect on the beam. Self-focusing is known to induce optical bistability [21]. This effect is taken into account in the improved model as follows. In the ring cavity, there are two beam waists, as shown in Fig. 4 (at BW1 and BW2). The optical path from the beam waist BW1 to the other beam waist BW2 is considered (half of the round trip). For the other path, the same estimation explained here is applicable. The self-(de)focusing effect is taken into account by inserting an effective GRIN lens of the length of $L/2$ and the transversal refractive index distribution of

 

Fig. 4. Gaussian beam inside the ring cavity. $w_1$ and $w_2$ are beam radii at the beam waists denoted by BW1 and BW2, respectively.

Download Full Size | PDF

The value of ${\alpha }_G$ depends on the longitudinal position $z$ because the beam radius $w$ is a function of $z$. Here let us examine the dependence of ${\alpha }_G$ on $w$. Except the explicit dependence ($w^{-2}$) in Eq. (10), ${\alpha }_G$ depends on $w$ implicitly because ${|E|}^2$ ($I$) is proportional to $w^{-2}$. When $I$ is much less than $I_S(1+{\delta }^2/{\gamma }_2^2)$, the transition saturation does not occur, and Eq. (8) implies that $n$ is independent of $I$, so that ${\alpha }_G=0$. In the case of $I\gg I_S(1+{\delta }^2/{\gamma }_2^2)$, the refractive index deviation [${\chi }^{\prime }/2$ in Eq. (7)] is proportional to $I^{-1}$ [see again Eq. (8)]. Then the derivative factor in Eq. (10) is proportional to $I^{-2}$. As a result, ${\alpha }_G$ is independent of $w$ in the case of saturation.

Hence ${\alpha }_G$ depends on $w$ or $z$ only when $I\sim I_S(1+{\delta }^2/{\gamma }_2^2)$; in this model, we do not consider the dependence of ${\alpha }_G$ on $z$. With this simplification, the GRIN lens can be approximated as a propagation of $L/2$ in a uniform medium plus a thin lens of a focal length of $f_G=-1/({\alpha }_{G}L)$ [22]. Note that $\frac{dn(I)}{dI}$ in Eq. (10) is negative for positive $\delta$, resulting that $f_G$ is also negative. Then the effective GRIN lens is a concave lens.

The thin lens location is assumed at the concave mirror, whose curvature $R$ is then modified in effect as

The approximations applied here may be appropriate as far as $R\ll |f_G|$, which turns out to be the case later.

Because of the modified mirror curvature by the effective GRIN lens, the beam radii at the beam waists are also modified. The values of $w_{\mathrm{1,2}}$ are obtained by the use of the ABCD law for Gaussian beams [23]. The change in the beam radii causes not only the shift of the coupling efficiency $\eta$ but also the change in the Rayleigh ranges $z_{\mathrm{1,2}}=\frac{1}{2}k{w}_{\mathrm{1,2}}^2$. The Gouy phase in one round trip along the optical axis in the cavity is

The intracavity optical beam undergoes not only the phase shift by the spatial propagation [$e^{inkL}$ in Eq. (3)] but also the Gouy phase shift ($e^{-i\mathrm{\Psi }}$). Then Eq. (3) is now replaced with

Here, ${\mathrm{\Psi }}_0$ is the Gouy phase in one round trip with the infinitesimal beam power. In addition, now $\eta$ is not supposed to be 1; instead, $w_2$ is calculated by the above-mentioned model to let $\eta$ be $2w^{\prime }{w}_2/(w^{\prime 2}+w_2^2)$, where $w^{\prime }=50\mathrm{\mu m}$.

E. Numerical Calculations with the Gouy Phase Shift

The numerical calculation results with the improved theoretical model are summarized in Fig. 5. As seen in Fig. 5(a), the optical bistability is reproduced qualitatively in agreement with the sweeping direction dependence and the range of the cavity length displacement in which the bistability appears.

 

Fig. 5. Calculation with the self-(de)focusing effect of the Gaussian beam in the cavity. Red dashed lines and black solid lines correspond to the sweeps of right and left directions, respectively, as indicated by the arrows in the figure. (a) Optical power of the transmission. (b) Focal length of the effective GRIN lens. It is evaluated as a negative value, implying concave lenses, but, for the logarithmic plot, its inverted value is plotted. (c) Gouy phase shift induced by the effective GRIN lens. (d) Beam radius $w_2$.

Download Full Size | PDF

From the result in Fig. 5, it turns out that the optical bistability experimentally observed in the passive cavity is indeed induced by the power-dependent refractive index of water vapor. The spatial distribution of the refractive index makes the GRIN lens in effect, and the change in the Rayleigh ranges shifts the Gouy phase. Therefore, the Gouy phase shift depends on the optical power, and this nonlinearity induces the optical bistability. In order to confirm this interpretation, the focal length of the induced GRIN lens and the Gouy phase shift as functions of the spatial cavity length displacement are given in Figs. 5(b) and 5(c), respectively [in Fig. 5(b), the sign of the negative focal length is inverted to show in a logarithmic vertical scale]. It should be noted that $|f_G|$ is always over a few meters, which is much longer than the concave mirror curvature (75 mm). Then the effect of the GRIN lens seems not so remarkable, but the resultant shift in the Gouy phase is $\sim 0.8\mathrm{rad}$. Corresponding spatial displacement is $\sim 0.8\mathrm{rad}/k=0.1\mathrm{\mu m}$, which is comparable with the linewidth of the resonance curve in Fig. 2.

In the improved model, the coupling efficiency $\eta$ is also considered as a function of the optical power. The result in Fig. 5(d), in which the beam radius is plotted, suggests that this effect is partly responsible to the optical bistability because the variation of $w_2$ in Fig. 5(d) is $0.5<\eta <0.9$. However, as explained as follows, it is found that the nonlinear coupling efficiency is not so important to cause the optical bistability. When artificially the effect of the Gouy phase shift is turned off ($\mathrm{\Psi }-{\mathrm{\Psi }}_0$ is always zero), and only the effect of $\eta$ dependence on the input laser power is considered, the model calculation result does not show the optical bistability. Conversely, when artificially $\eta$ is always set to 1, the optical bistability is reproduced with non-zero $\mathrm{\Psi }-{\mathrm{\Psi }}_0$. Then it can be concluded that the Gouy phase shift is important to interpret the mechanism of the optical bistability.

In comparison between the experimental results and the model calculation, there is one qualitative disagreement, which is that the left-direction sweep in Fig. 2(a) (black curve) shows a relatively sharp resonance as high as the peak of the right-direction sweep (red curve). This resonance is not reproduced in the calculation [Fig. 5(a)]. This disagreement can be explained as follows. In Fig. 6, the overall phase shift in one round trip is shown as a function of $\delta$. This phase shift is the sum of the spatial displacement of the cavity length, the nonlinear refractive index deviation, and the Gouy phase. When the overall phase shift is zero, the cavity is on resonance. In the case of the left-direction sweep (red), the Gouy phase shift compensate the spatial phase shift. Then the resonance condition is nearly kept even when the cavity displacement is swept. Conversely, in the case of the right-direction sweep (black), the overall phase jumps when it approaches to zero (indicated by a blue arrow). When the phase jump occurs, the overall phase is across zero. Practically, the phase jump must occur within a finite time, and then, during the jump, the resonance condition is temporarily satisfied. This temporal resonance is considered to be observed in Fig. 2(a). Because the present theoretical model is static, this resonance cannot be reproduced.

 

Fig. 6. Overall phase shift in one round trip. Red dashed line and black solid line correspond to the sweeps of the right and the left directions, as indicated by the arrows in the figure. The area inside the rectangle is magnified in the inset. Blue arrow indicates the phase jump for the right-direction sweep.

Download Full Size | PDF

According to this experiment and the model calculation, it is found that the optical bistability can be observed even when the input laser is far-detuned (4 GHz) and not so intense (180 mW). By the calculation, the bistability can be observed when the detuning is 10 GHz, as shown in Fig. 7(a). In Fig. 7(b), the parameter values are set to reproduce the result in Fig. 2(c) ($\delta /2\pi =-20\mathrm{GHz}$, and the values of ${\gamma }_2$, $a$, and $\mu$ are for the transition B [18]). In Fig. 2(b), the bistability is observed, whereas in the model calculation it is not. This disagreement may be due to approximations applied in the model. In addition, although the transition B (detuning of $-20\mathrm{GHz}$) is the closest to the laser wavelength in Fig. 2(b) (952.07 nm), the transition A (detuning of 40 GHz) can give some effect. Noting that transition A is five times more intense than transition B, transition A is more influential on the refractive index than transition B because ${\chi }^{\prime }$ is proportional to ${\delta }^{-1}$ in a far-detuned case.

 

Fig. 7. Calculated transmission power as a function of the cavity length displacement. Red dashed lines and black solid lines correspond to the sweeps of right and left directions, respectively, as indicated by the arrows in the figure. Parameter values are identical to those used in Fig. 5, except (a) the detuning of 10 GHz; (b) the detuning of $-20\mathrm{GHz}$, and the values of ${\gamma }_2$, $a$, and $\mu$ are for the transition B; (c) input laser power of 18 mW; (d) input laser power of 1.8 W.

Download Full Size | PDF

The laser frequency within the range of $\sim 20\mathrm{GHz}$ about any intense spectral line can exhibit the optical bistability, depending on the optical power and the beam radius. Even if the optical power is less and the optical bistability does not occur, the Gouy phase shift can cause distortion of the resonance curve. Typically there is one intense transition line of water vapor per 80 GHz in 952 nm range [18], and therefore in the near-infrared region (900–980 nm), the optical bistability in passive cavity is not an uncommon phenomenon.

The bistability observed in this study is undesirable for the cavity stabilization at resonance for second-harmonic generation and intra-cavity absorption spectroscopy. In the case of the HP state in Fig. 5(a), the cavity length for the peak transmission is identical to that at which the phase jump occurs. Therefore, when the cavity is stabilized at the peak power in the HP state, the phase jump is induced even with the infinitesimal fluctuation of the cavity length. This phase jump makes error signals of the phase-detecting stabilization schemes deteriorate. These schemes include the Pound–Drever–Hall scheme [24] and the Hänsch–Couillaud scheme [25]. To avoid the bistability, it is straightforward to purge the water vapor. Of course, the other way to avoid the bistability is to reduce the input laser power [calculation is shown in Fig. 7(c), in which the input laser power is set at 18 mW], but it may be interesting that the stabilization can be unaffected by the bistability in the case of high input laser power. The calculation result for the input power of 1.8 W is shown in Fig. 7(d). The bistability still exists, but the peak, at which the cavity is to be stabilized, is not where the phase jump occurs (it occurs at a little longer cavity length). This means that a small fluctuation of the cavity length does not cause the phase jump, and the cavity length can be stabilized by the negative feedback servo loop to keep resonance.

The optical bistability observed in this study is an instance of which the nonlinear Gouy phase shift gives a remarkable effect. In some cases, the Gouy phase shift can give some effect that is not so drastic as optical bistability. One example is cavity-enhanced saturated absorption spectroscopy in the Fabry–Perot cavity. In such a case, the absorption gas is inside the cavity; therefore, the experimental setup is similar to that in this study. The optical power in this study is high compared with that in typical experiments of cavity-enhanced saturation spectroscopy; therefore the optical bistability may not be observed. However, some effect such as distortion in spectral line profiles of molecular absorption can occur by the Gouy phase shift. In addition, with the effect of the Gouy phase shift, the free-spectral range of the cavity may become dependent on the optical power. Then in NICE-OHMS experiments [15,16], in which the laser is to be phase-modulated at a frequency of free-spectral range of the cavity, the Gouy phase shift must be considered in discussions of molecular spectral line profiles in detail.

4. CONCLUSION

Optical bistability in a passive ring cavity with no designed nonlinear materials is experimentally observed. This optical bistability is unexpected and undesirable when the optical power is enhanced in the cavity for second-harmonic generation and saturation spectroscopy. By the model calculation, it turns out that this bistability is caused by the power-dependent refractive index of water vapor. The effective GRIN lens induced by the nonlinear refractive index causes the modification in the Rayleigh ranges and the Gouy phase shift of the Gaussian beam inside the cavity. The model calculation qualitatively agrees with the experimental result.

Because there are many intense spectral lines of water vapor in the near-infrared region, the bistability discussed in this paper is not uncommon. Purging water vapor or using a low-power laser is a straightforward scheme to avoid the bistability, but, by utilizing a high-power laser, difficulties involved in the cavity stabilization may be overcome. The Gouy phase shift can give some effect on cavity-enhanced molecular spectroscopy and NICE-OHMS, although the effect may not be so drastic as the optical bistability. For example, distortion of spectral line profiles is anticipated. Detailed discussions are for future subjects. In addition, if the cavity is applied to second-harmonic generation, nonlinearity caused by a nonlinear crystal such as the thermal-lens effect [26] and the optical Kerr effect can induce the optical bistability with the mechanism discussed in this paper.