We compared the ∆

R responses of the Co and Co

_{50}B

_{50} transducers (

Figure 3a). In the case of the Co transducer, ∆

R showed a typical temperature response (∆

R arising from ∆

T) without any acoustic oscillations (∆

R arising from ∆

ε): a rapid increase in ∆

T at zero time delay by the pump pulse, followed by a gradual decrease because of slow heat dissipation to the substrate (

Figure 3b). In contrast, the Co

_{50}B

_{50} transducer showed a strong acoustic oscillation in addition to the slow temperature response (

Figure 3c). The magnitude of acoustic oscillation was similar to that of the temperature signal.

By subtracting the temperature signal from the raw data, we were able to extract acoustic oscillations. Then, we compared the acoustic oscillations obtained with different Co

_{50}B

_{50} thicknesses (

Figure 4). The oscillations were fitted with a damped cosine function,

$A\mathrm{cos}\left(2\pi ft\right)\mathrm{exp}\left(-t/\tau \right),$ where

A is the oscillation amplitude,

f is the oscillation frequency,

t is the time delay between the pump and probe, and

τ is the exponential decay time. A smaller Co

_{50}B

_{50} thickness resulted in a larger

f and smaller

τ. Such a strong oscillation of acoustic waves indicated that acoustic waves were reflected from the sap/Co

_{50}B

_{50} interface with a high reflection coefficient (

r). The reflection of acoustic waves was determined by acoustic impedance,

$Z=\rho {v}_{\mathrm{s}}$, where

ρ is the density of the material. When two materials with

Z of

Z_{1} and

Z_{2} have infinite thicknesses,

r is expressed as [

34,

35]

We extracted the thickness of the soft intermediate layer by analyzing acoustic oscillations. The frequency dependence on the transducer thickness is well understood by the speed of sound of the metal transducer (

Figure 5a). Using the relationship

$f=\frac{{v}_{\mathrm{s}}}{2{d}_{\mathrm{tot}}}$, where

d_{tot} is the thickness of Co

_{50}B

_{50}+Pt and

v_{s} is the effective speed of sound of Co

_{50}B

_{50}+Pt, we determined

v_{s} to be 4900 m s

^{−1}. Then, by combining

f and

τ, we obtained a damping parameter of

$\alpha =1/2\pi f\tau $ (

Figure 5b). α describes the amplitude attenuation during one period of oscillation; it ranges from zero for no attenuation to

$1/2\pi $ for full attenuation after one period. The attenuation of acoustic waves in a medium is caused by the wave transmission at interfaces and bulk attenuation inside the medium. Considering the energy conservation during the wave propagation at interfaces, the wave transmission is related to the wave reflection. Then,

α can be expressed as

where

r_{int1} is the reflection coefficient at the bottom interface between the substrate and transducer,

r_{int2} is the reflection coefficient at the top interface between the transducer and air, and

β_{bulk} is the bulk attenuation factor in the transducer layer (we note that

r can be negative when

Z_{1} <

Z_{2} in Equation (2), so that the reflected wave can change its sign. Since

α describes the attenuation of the absolute magnitude of the acoustic wave,

α should be related to |

r| but not

r).

r_{int2} was found to be close to one, considering the large mismatch in the

Z values of the transducer and air. The

β_{bulk} value of metals depends on the frequency [

36,

37], and there are no experimental reports regarding ferromagnetic metals at a high frequency of 200 GHz. When

β_{bulk} is negligible and

α is dominated by

r_{int1}, an

α of 0.034 ± 0.011 leads to an |

r_{int1}| of 0.79 ± 0.07 at the Co

_{50}B

_{50}/sap interface. When

β_{bulk} is as high as ≈10

^{7} m

^{−1} [

38], the bulk attenuation can explain most of

α, and |

r_{int1}| will be close to one. Such a large |

r_{int1}|of >0.79 cannot be explained by the

Z of the transducer and substrate alone; thus, a soft layer should exist at the interface. Then,

r_{int1} is expressed with the acoustic properties of the soft layer [

1]:

where

$\varphi =2\pi f{d}_{3}/{v}_{3}$; here,

d_{3} is the thickness of the soft layer and

v_{3} is the speed of sound of the soft layer. Assuming the same

Z of 4 × 10

^{7} kg m

^{−2} s

^{−1} for CoB and sapphire, and a typical

Z_{3} of 1.7 ± 0.5 × 10

^{6} kg m

^{−2} s

^{−1} and

v_{3} of 1.5 ± 0.2 km s

^{−1} for soft materials [

34,

35], an |

r_{int1}| of >0.79 ± 0.07 leads to a

d_{3} of >2 ± 1 Å at an

f of 200 GHz.