## 1 Introduction  
Ultrasonedestructive evaluation allows the detection of defects within a material or structure. This plays an important role in preventing failures and is especially important in inspecting load-bearing components used in the industry [1]. So-called micro-defects or damage, such as internal stress, closed cracking, and zero-volume delamination, are generally the forerunners of material cracking and subsequent failures. Microscopic changes due to precipitation, dislocation and embrittlement affect the overall properties of the material. However, measurements of conventional linear ultrasonic parameters, such as sound velocity, attenuation, and backscattering, are not sensitive to these inhomogeneities and do not provide quantitative information on material condition [2]. Nonlinear acoustic methods, such as second harmonic generation (SHG) and measurement of acoustical nonlinear parameters, have proven useful for detecting these types of material defects [3, 4, 5]. When finite amplitude ultrasonic waves propagate in a material, the nonlinear elastic properties of the material distort the waveform and generate harmonics [6]. Nonlinear ultrasound methods often measure the second harmonic generation (SHG) to obtain a nonlinear parameter \(\beta\), defined as the ratio of the second harmonic amplitude to the square of the fundamental wave amplitude, and draw conclusions about the material state. Nonlinear parameters can be measured, using longitudinal waves [7, 8, 9, 10, 11], Rayleigh waves [12, 13, 14], and Lamb waves [15, 16, 17]. Rayleigh waves require sufficient propagation distance to measure SHG; the measured nonlinearity is averaged over distance and cannot be localized. In addition, it is sometimes difficult to obtain pure, nonlinear Rayleigh waves because a certain setup can introduce source nonlinearity. Lamb waves have limited use because they propagate in very thin plates and there are few modes that meet the requirement for cumulative propagation. In contrast, longitudinal waves have been widely used for nonlinear characterization of materials. However, experiments are usually performed in the through-transmission (TT) mode requiring access to both sides, and this technique may be limited in field applications where access is only possible on the external surface of the component. A practical method for nonlinear ultrasonic measurements, using the normal incidence of longitudinal waves is pulse-echo (PE) testing, which uses only one side of the specimen. Biomedical imaging [18; 19] and fluid nonlinearity [20; 21] were studied in the PE mode, using heavy metals as reflectors to resemble the rigid boundary condition. However, the problem of PE nonlinear testing of solids with stress-free surfaces is that such boundaries suppress the SHG process and make it hard to acquire reliable nonlinear parameters. In the case of pure plane waves, the harmonic component generated during forward propagation is reflected off the stress-free boundary and decreases to zero after returning to the initial position [22; 23]. This is because the two second harmonic components--the reflected and newly generated second harmonics--cancel each other out due to their phase difference of \(\pi\) after being reflected from the stress-free boundary. According to theory and experiment, however, the received second harmonic amplitude reflected from the stress-free boundary is not really zero due to the finite size transducer and material attenuation. The currently available pulse-echo nonlinear testing method basically requires the test sample to be thick enough and may not be accurate for thin samples in nonlinear parameter measurements [24; 25]. An acoustic beam focused on a stress-free boundary was proven to improve SHG during back propagation from the stress-free boundary [26]. A recent study showed that the phase difference is about \(\pi/2\) between the two second harmonic components when a focused beam is used in water [27].