Photoacoustic Brillouin spectroscopy of gas-filled anti-resonant hollow-core optical fibers: supplement

YAN ZHAO,1,2 YUN QI,1,2 HOI LUT HO,1,2 SHOUFEI GAO,1 YINGYING WANG,3 AND WEI JIN1,2,∗ 1Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, China 2Photonics Research Center, The Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen, China 3Institute of Photonics Technology, Jinan University, Guangzhou, China ∗Corresponding author: wei.jin@polyu.edu.hk

Photoacoustic Brillouin spectroscopy of gas-filled anti-resonant hollow-core optical fibers: supplement 1

. GEOMETRY OF AR-HCF MODEL BUILT FROM SEM
The geometry model of anti-resonant hollow-core fiber (AR-HCF) used for simulation is precisely built from the scanning electron microscopy (SEM) of the fiber. SEM image of fiber A and its geometry used for simulation are shown in Figs. S1(a) and (b), respectively. The simulated electric field ψ i is shown in Fig. S1(c). The diameter of the whole hollow region is 60.5 um and its center is set at (0, 0). The inner diameter, thickness and center of capillaries estimated from SEM are tabulated in Table. S1.

A. Absorption coefficient of acetylene
We focus on the P(13) line in the ν 1 + ν 3 band of acetylene (C 2 H 2 ) at 1532.83 nm. Foreign gas broadening affects the absorption coefficient (α) of C 2 H 2 . For simplicity, we assume nitrogen has the same foreign broadening coefficient with that of air. The absorption coefficients of C 2 H 2 versus gas pressure are shown in Fig. S2. They are calculated based on the HITRAN database [1]. In the Doppler regime (i.e. pressure less than 1 kPa), the wall collisional broadening in AR-HCF estimated with hard sphere model is about 7 MHz [2,3], which is much smaller than the Doppler linewidth and is ignored in the simulation.

Fig. S2.
The absorption coefficient at 1532.83 nm at different gas pressures for pure C 2 H 2 and trace C 2 H 2 in nitrogen. The absorption coefficient of nitrogen balanced C 2 H 2 is normalized by its concentration. The increase of absorption coefficient for pressure higher than 1 bar is due to the neighbor absorption lines.

B. Thermal relaxation time of acetylene
Thermal relaxation time (τ) characterizes the speed of heat generation after optical absorption. It determines the efficiency of generating a harmonic heat source in the MHz regime. For C 2 H 2 , it is determined by the V-R, T relaxation process and is inversely proportional to gas pressure p [4]. We measure the thermal relaxation processes in AR-HCF with a 10-ns optical pulse and the result is shown in Fig. S3. The result yields a pressure-dependent thermal relaxation time of: pτ = 74 ns · atm(N 2 balanced C 2 H 2 ) (S1) Fig. S3. The transient response of thermal relaxation and thermal conduction in AR-HCF. The fiber is filled with 106 ppm C 2 H 2 balanced with nitrogen.

C. Pump light and heat source
AR-HCF mainly supports two sets of LP-like optical modes as shown in Fig. S1(c). The intensity of the pump is modulated by an acoustic optical modulator. The optical intensity of pump is 11 , and ψ i are the power normalized electric field of LP 01 and LP 11 -like modes, η the fractional power in LP 01 , P P the pump power, z the direction of light propagation, k P the angular wavevector of pump light, n r = n 01 − n 11 the differential mode index between LP 01 and LP 11 and Ω = 2π f is the angular frequency of pump modulation. For AR-HCFs, more than 99.9% power of light is in the gas (n gas ≈ 1) and hence it is a good approximation to normalize that dSψ 2 i = 1 over the fiber cross-section. For modulation frequency of a few MHz and fiber length of meters or shorter, the phase term of optical intensity qL ≈ 2π f L/c π (where L is the length of fiber) and could be ignored (the effective mode indices are all very close to unity). For simplicity, the heat source is estimated with a two-level model and is expressed by where Q 0 is the DC component of the heat source,Q the harmonic heat source, C the concentration of C 2 H 2 and φ = tan −1 (Ωτ) is the phase angle.

THERMO-VISCOUS ACOUSTICS IN AR-HCF
For simulation of acoustic problems in small dimension, viscous and thermal dissipation at the boundary play an important role and a thermo-viscous acoustic model is often used [5]. Meanwhile, we need to consider the coupling between the thermo-viscous acoustics and the silica capillaries [6]. The thermo-viscous model and the solid mechanics model are used for simulation in COMSOL Multiphysics. The parameters used in the simulation is listed in Table. S2 [7][8][9][10]. The density of gas is assumed to be linear dependent on gas pressure. The thermo-viscous acoustic model is coupled with the solid mechanics model with an isothermal boundary condition. For low concentration of C 2 H 2 in nitrogen, parameters of N 2 is used for simplicity. The steady state is calculated by assuming Q = Q 0 . For simplicity, we ignore the deformation of the fiber geometry at steady state and only consider the steady state temperature field T s and pressure field p s . Then we calculate the harmonic perturbation in the frequency domain. The harmonic perturbation is accurate only when the harmonic perturbations of pressure, temperature and density are much smaller than their steady state values. Since the variation of heat source along z-axis is much smaller than in the fibre cross-section, hence the simulation is performed for simplicity with a 2-D model. We have where ρ 1 and w 1 are the harmonic solutions when only the first term ofĨ on the right hand side (RHS) of Eq. S4 is applied, ρ 2 and w 2 are the harmonic solution when only the last term ofĨ on the RHS of Eq. S4 is applied. As demonstrated in Fig. S4, the modal interference gives a periodic optical intensity distribution I P = I 0 P +Ĩ P e −jΩt (and hence the heat source distribution Q = Q 0 P +Qe −jΩt as shown in Fig.  S4(a) along fiber with a period of d = λ P /n r ≈ 1.6mm. The periodic heat source results in a periodic distribution of acoustic field, which is described by the density change ρ = ρ 0 +ρe −jΩt , and the displacement of capillary w = w 0 +we −jΩt . In our experiment, L/d 1 and hence the total phase modulation is equivalently induced by the equivalent acoustic fieldsρ = ρ 1 and w = w 1 . The cosine terms in Eq. S8 and Eq. S9 are averaged out. The acoustic fieldsρ andw are induced by an equivalent harmonic pumpQ. The equivalent fields (ρ,w and etc.) are equal to the fields at the quarter period as demonstrated in Fig. S4.

A. Phase modulation of probe beam
The probe beam in the AR-HCF experiences phase modulation (or modulation of mode index n i ) due to shifting boundary (SB), gas density change (GD) and photoelasticity (PE): where x is an infinitesimal perturbation (i.e. a normalized equivalent harmonic heat sourcē Q/αCP P ) and subscript m refers to the label of optical modes (m = 01, 11a, 11b). Since we only care about the harmonic phase modulation near the steady state, we only consider the phase modulation due to harmonic perturbations. Considering the pump depletion due to optical absorption and ignoring the fiber loss, the phase modulation is calculated by where L is the length of AR-HCF and dn m /dx is independent of z when considering a normalized equivalent heat sourceQ/αCP P . Hence from Eq. S11 and S12 we have, where L e f f is the effective absorption length and it approximates the length of fiber L when αCL 1.The modulation of refractive index is related with modulation of eigenfrequencies of optical modes by [11] dn m dx = − n g m ω L dω dx (S15) where n i g = n i + dn i /dω is the group mode index and ω L is the angular frequency of the probe. The shifting boundary induced perturbation is calculated as [12] wheren is the normal vector on the interface from silica to gas, E i the electric field, D m = εE m the electric displacement field, ∆ε 12 = ε 1 − ε 2 and ∆(ε −1 12 ) = ε −1 1 − ε −1 2 are the differential permittivity between silica (ε 1 ) and gas (ε 2 ). The density change and photoelasticity induced perturbations are calculated by the standard perturbation theory [13,14] where the electrostrictive constant γ e = 2(n gas − 1) = Aρ 0 for dilute gas and A is a constant [13]. S ij is the strain tensor and for fused silica, p 11 = 0.121, p 12 = 0.27 and p 44 = (p 11 − p 12 )/2 [14]. The phase modulation of modal interferometer (MI) is For example, Fig. S5 shows the simulation result with 106 ppm C 2 H 2 balanced with nitrogen at a pressure of 1 bar. As shown in Fig. S5(a), for optical modes in the gas, the contribution of phase modulation from photoelastic effect could be ignored. The contributions from gas density change and shifting boundary are comparable. Different optical modes also experience different phase modulation as we shown in Fig. S5(b). However, the two polarizations of an optical mode shows little difference as we found in the simulation.

B. Calibration of the phase modulation
The light from AR-HCF is coupled into a single-mode-fiber (SMF) and forms a modal interferometer. The phase modulation of probe is calibrated with the interferogram measured by detuning the wavelength of probe (Fig. S6). The lock-in amplifier measures the root-mean-square (RMS) voltage of the small harmonic signal (V RMS ). Hence, the experimental phase modulation could be calibrated as

C. Raman gain of SMF pigtails
The harmonic pump modulates the Raman gain on the probe. The harmonic Raman gain is detected by the lock-in amplifier and then gives an equivalent background 'phase modulation'. The Raman gain with a frequency difference ∆ν = ν P − ν L = 72 cm −1 is about 0.14 of its peak value [15] g R (∆ν) = 0.14g R,peak ≈ 7.8 × 10 −15 m/W (S21) In the experiment setup, the harmonic pump power is about P P = 162 mW. The length of SMF pigtails is about L SMF = 25 cm and the effective cross-section of SMF is about A e f f = 76 µm 2 .
The Raman gain on probe is about The Raman gain induced equivalent background 'phase modulation' is Since the Raman induced background ∆φ R is incoherent with ∆φ MI , the experimental phase modulation |∆φ expr | is compared with the total phase modulation |∆φ tot | = |∆φ MI | + |∆φ R | (S24)

PRESSURE DEPENDENCE OF PABS OF AR-HCF
As shown in Fig. S7, we measure the PABS spectrum of two different AR-HCFs with a gas pressure from 1 bar to 11 bar. The fiber is filled with 106 ppm C 2 H 2 balanced with nitrogen. The phase modulation due to the radial air mode is approximately proportional to the gas pressure. The shift of the resonances is due to the coupling between the capillary mode and the air mode.
Because of the different LP-mode of probe beam, the PABS spectrum at a pressure of 1 bar is different from Fig. 4(a) in the main text.