Goos-Hänchen Effect: An Intriguing Phenomenon from Optics to Acoustics

Throughout human history, people have been fascinated by sound and light, by what they saw and heard. Understanding the relationship between optical and acoustic phenomena has been an ongoing scientific endeavor. Analogies between optical and acoustic theories have been mutually beneficial, yielding theoretical advances in both fields. Studies of the analogy between sound and light involved many aspects, including optics, music, mathematics, and physics, and historical discussions have been summarized in the literature [1]. Industrial applications of the acoustic analogs to optics have also been fruitful in recent years, e.g., the acoustic dispersive prism [2], acoustic analogies of high-index optical waveguide devices [3], etc.


Introduction
Throughout human history, people have been fascinated by sound and light, by what they saw and heard. Understanding the relationship between optical and acoustic phenomena has been an ongoing scientific endeavor. Analogies between optical and acoustic theories have been mutually beneficial, yielding theoretical advances in both fields. Studies of the analogy between sound and light involved many aspects, including optics, music, mathematics, and physics, and historical discussions have been summarized in the literature [1]. Industrial applications of the acoustic analogs to optics have also been fruitful in recent years, e.g., the acoustic dispersive prism [2], acoustic analogies of high-index optical waveguide devices [3], etc.

Physical Models
Consider an interface between fluid and solid, where a harmonic P-wave strikes at the interface, which generates the reflected and refracted waves [42], as shown in Figure 1.
To examine the Goos-Hänchen effect in acoustics, the two proposed lateral-displacement models are as follows: (i) a virtual lateral-displacement model (VLDM), where the effective speed is equivalent to the propagation speed of the acoustic wave; and (ii) a real lateral-displacement model (RLDM), where the effective speed is the real speed of the acoustic signal, as shown in Figure 2. The effective propagation speed (either an equivalent propagation speed or the real propagation speed) of the reflected P-wave are dependent not only on the media of the two sides of the interface but also the incident-angle of the P-wave.
In the reported studies, the interface between fluid water and a solid Perspex has been used as a prototype testing system for these models [35].

Transition Time
The transition time of the acoustic signal reflected from the interface can be used as an alternative measurement for lateral displacement.
To  shows that there is indeed a virtual lateral displacement at this interface, which is again a demonstration of the acoustic Goos-Hänchen effect. Along the RLDM path, the calculated results are similar to but slightly different from that of Figure   5, which provides the same conclusion as that of the VLDM path.

Additional Questions
In the two lateral-displacement models at the water-Perspex interface, there is clearly an acoustic Goos-Hänchen effect.
While this result is exciting, we are still puzzled by the insufficient physical interpretation and unanswered questions. Foremost, the Goos-Hänchen displacement in optics is a coherent effect of the total reflection of a finite-sized optical beam. The transition of an acoustic signal, as discussed here, is incoherent and is a non-total reflection of different frequency components. It is somewhat unexpected to find the acoustic analog of the Goos-Hänchen effect, where the acoustic wave has such different physical properties from optics. Clearly, an insightful physical explanation is needed to enhance our understanding.
Secondly, the transition time calculated from the VLDM model is different from that of the RLDM model. Even though both models have provided the same conclusion, it is not clear which model is more physically meaningful. Are they both meaningful but in different physical domains?
Finally, the discussion is based on a specific interface system between fluid and solid, i.e., the interface between water medium and Perspex. We are still waiting to see if the discussed result is generic and can be applied universally to all systems.
Overall, the Goos-Hänchen effect is an intriguing topic from optics to acoustics that warrants further investigation. There remain many critical questions, that need to be answered. Until these questions are properly answered, we are unable to provide a tangible conclusion.