There are multiple methods for solving the diffraction formula (Kreis et al., 1997) to reconstruct the complex field at the object plane. Figure 1 displays the mathematical procedure (algorithm) used for reconstruction and extraction of the optical phase. In this paper, the Fresnel approximation method with the chirp function is used. The transmission function calculated from the phase-shifted holograms is first multiplied by the chirp function, ψ . Then, it is transformed into the frequency domain through a Fourier Transform (FT). Finally, it is multiplied by the quadratic phase factor, Q , Γ n , m = Q n , m × FT t k , l × ψ k , l .

The reconstructed complex field is denoted by Γ n , m , where n and m are the digitized coordinates of x ′ , y ′ at the object plane. The quadratic phase factor, Q n , m , and chirp function, ψ k , l , can be described, respectively, by, Q n , m = ⁡ exp − i π λ d 0 n 2 N 2 Δ ξ 2 + m 2 N 2 Δ η 2 , and ψ k , l = ⁡ exp − i π λ d 0 k 2 Δ ξ 2 + l 2 Δ η 2 , where N 2 is the total number of pixels, Δ ξ and Δ η are the pixel dimensions of the CCD sensor, and k , l are the digitized coordinates of ξ , η at the CCD plane. For computational efficiency, the quadratic phase factor, Q , is neglected as it is not needed when computing intensities or phase differences (Kreis et al., 1997).

The reconstructed field is complex in nature, Γ = A e i Φ , and contains both the amplitude, A n , m , and phase, Φ n , m , of the object wavefront. The parameter of interest, for purposes in metrology, is the optical phase. Since Γ can be rewritten as, A ⁡ cos ⁡ Φ + i   A ⁡ sin ⁡ Φ , the optical phase can be extracted by, Φ n , m = ⁡ arctan Im Γ n , m Re Γ n , m   . where Re … and Im … denote the real and imaginary components, respectively.

The optical phase calculated from a single reconstructed field would yield static information, and thus, changes to the optical phase are measured interferometrically via a double exposure (DE) method. In this method, a hologram of an “undeformed” reference state is recorded and subsequently subtracted from “deformed” states of the object. This operation yields the phase difference, which for simplification of algorithm, can be directly calculated by, Φ def − Φ ref = ⁡ arctan Re Γ def ∙ Im Γ ref − Re Γ ref ∙ Im Γ def Im Γ def ∙ Im Γ ref − Re Γ ref ∙ Re Γ def .

Due to the nature of the four-quadrant inverse tangent, the phase values are wrapped within − π , π (mod2 π ). Therefore, the phase difference field needs to be unwrapped, Ω n , m = unwrap Φ def − Φ ref , where Ω n , m is the fringe-locus function of the DE hologram.

Conventional holographic methods, such as electro-speckle pattern interferometry (ESPI) and Stetson holographic interferometry, use imaging lenses which have a positive effective focal length to project the object plane onto the CCD plane (Kreis et al., 1997). Due to this, either the lenses or the CCD plane must be physically moved in order to modify the object plane’s distance. In lensless digital holography, as described in Section 2.1.1, the complex field is numerically reconstructed at the object plane, and therefore, multiple fields at different distances can be reconstructed instantaneously, as shown in Figure 2. This is especially useful in circumstances where, for example, the sample’s geometry extends beyond the system’s depth-of-field (DOF) at the object plane, or there are multiple components at various distances to be inspected for comparative metrology purposes.

The chirp function, ψ k , l , used in the reconstruction is a 2D oscillatory signal whose frequency linearly varies with the spatial coordinates. Therefore, the computed complex field is expanding with increasing reconstruction distances. Due to this, the spatial parameters will vary with the distance as well as other parameters to be discussed.

As mentioned in Section 1, an important in-situ and industrial metrology capability is the measurement of large FOV components. To fit such components within the maximum permissible angle of the object wavefront, described in Section 2.1.4.1, the working distance would need to be increased. This solution, however, will decrease the signal-to-noise ratio (SNR) and further expose the stability of the system to random external environmental fluctuations.

Therefore, in the measurement of components that do not fit within the reconstructed FOV, a negative lens is placed in front of the CCD. The negative focal length modifies the object wave incident ray angles, and therefore, larger angles now fit within the permissible range. Due to the inclusion of a lens, the reconstruction distance is modified, as shown in Figure 3. Mathematically, a virtual object, which is smaller than the real object and whose size depends on the focal length of the lens, is created at a distance, b , from the lens. Thus, the modified reconstruction distance is now (Schnars et al., 1996), d 0 ′ = b + d lens , where d l e n s is the distance between the CCD and lens, and b = f − 1 − g − 1 − 1 with f denoting the focal length of the lens, and g denoting the distance between the virtual and real objects.

The stability of holographic systems relies, among other factors, on minimizing the absolute displacements, which are typically induced by external environmental disturbances, between the OH and the sample of interest along the sensitive direction. To correctly tackle the stability problem in digital holography, the sensitivity direction must be determined. This parameter depends on the sensitivity vector of the system, which is derived by the opto-geometric constraints between the OH setup and a sample of interest.

A schematic depicting the geometric constraints of the optical rays as well as their respective modifications during deformation of the sample is shown in Figure 5. The illumination of a point, P , on the sample can be described by a vector which originates from the endface position of the optical fiber. The observation vector is that which is reflected or scattered from P and is received by a pixel, ξ , η , in the CCD sensor. The sensitivity vector, K ⇀ , of the system at P can be described by the vectorial subtraction of illumination and observation vectors, b ⇀ − s ⇀ . When a local deformation is induced on the sample at P , the changes in the OPL of the light, also known as the optical path difference (OPD), can be determined by a relationship between the deformation direction and the sensitivity of the system, O P D P = d ⇀ ∙ K ⇀ . The out-of-plane component of the OPD can be simplified as, O P D ⊥ = d ⊥ 1 + ⁡ cos ⁡ θ , where d ⊥ is the out-of-plane component of the deformation vector, d ⇀ , θ is the angle between the illumination, s ⇀ , and observation, b ⇀ , vectors, and ⊥ denotes the out-of-plane component. The OPD induced by deformation of the sample is related to the phase change of the DE hologram by, Δ Ω ⇀ = O P D ∙ k ⇀ , where k ⇀ is the wave vector and its magnitude, k ⇀ = 2 π / λ , is the wavenumber. Therefore, the out-of-plane displacements that are measured with the designed holographic system are described by, d ⊥ = λ   Δ Ω ⊥ 2 π 1 + ⁡ cos ⁡ θ .

As discussed in the previous section, the stability of a holographic system is determined by its robustness to external factors which disturb the OPD between the reference and object beams. These disturbances can be categorized as either short- (transient) or long-term (steady-state). The stability of the system must be validated for these conditions in order to determine which applications it is suitable for.

The short-term stability of the system is studied by analyzing the mean optical phase, Δ Ω ¯ , of an ideal rigid sample. These types of instabilities affect mostly the measurement of absolute displacements, or when the boundary areas of harmonic deformations are not captured within the FOV. For example, the measurement of mode shapes of a cantilever beam is only validated against short-term instabilities as long as the instantaneous optical phase of a point on the boundary fixture is known. Long-term stability, which is the system’s robustness against steady-state disturbances, is validated by analyzing the decorrelation of speckles. This is done by computing the modulus of the complex coherence coefficient in time lapsed holograms, which is described by Dainty (1975), μ = Γ ref Γ i * Γ ref 2 Γ i 2 , where Γ ref and Γ i are the reconstructed complex fields for the reference “undeformed” and subsequent states of the sample, respectively, and … denotes the statistical average. The solution to μ has been previously studied, where it was used to model and predict speckle decorrelation to achieve denoising of the measured optical phase (Meteyer et al., 2021; Picart, 2021), and is used in this paper for stability validations.

The spatial resolution of the system is validated with the use of a standardized resolution test target (USAF 1951). The target is positive, using a chrome pattern with clear background, and a white glass is placed behind to use as a background for improvement of contrast. The specific target has no birefringence and is therefore not polarization dependent. It can be used to characterize spatial resolutions from 1 to 57 lp/mm (Groups 0–5). The system is tested for working distances (WD’s) of 200, 500 and 1,000 mm. In order to achieve high uniformity in illumination distribution at each WD, the top-hat diffuser, described in Section 2.3, was used. The results are depicted in Figure 7, where Figure 7A shows the intensities of the reconstructed fields at each WD, and Figure 7B contains a plot comparing the cross-sectional distribution profiles of each field. The reconstructed intensities show groups 0 and 1, and zoomed-in versions show groups 2 and 3. The plot shows the intensity profiles for elements 1 through 6 in groups 0, 1 and 2 for all three WD’s. The validated resolutions determined are approximately 12.7, 9, and 5 lp/mm for WD’s 200, 500 and 1,000 mm, respectively.

The DOF of the system is characterized using a contrast target designed with one row of line pairs at 5 lp/mm. The line pairs are imaged perpendicularly with respect to the optical axis of the OH. Similar to the validations of spatial resolution, the system’s DOF is tested for WD’s of 200, 500 and 1,000 mm. At each WD, the OH is displaced over an array of stepped positions relative to the focal plane in order to measure the variability of defocusing. The results are depicted in Figure 8, where Figure 8A shows the intensities of the reconstructed fields at three chosen OH positions for each WD, Figure 8B depicts a plot of the intensity profiles of each field, and Figure 8C depicts the degree of defocusing trend calculated for each WD. The degree of focus is quantified by a Gaussian derivative filter and curve fitted with a Gaussian function. The DOF, which is twice that of the half-width at half-maximum (HWHM), is computed for each WD to be approximately 10, 50 and 360 mm for 200, 500 and 1,000 mm, respectively.

As discussed in Section 2.3.2, both transient and steady-state disturbances are characterized for validation of stability. The testing target is chosen to be a mechanically rigid artificial sample with negligible time-varying behaviors. Considering the variety of positioning requirements for in-situ metrology, the stability of the system will be validated for two setup configurations, first with a pitch angle of 0° (forward), and second with a pitch angle of 90° (top-down). Given the structural design of the mechanical positioner, the external disturbances have a varying effect on the stability of the system depending on the measurement orientation. For simplicity, throughout this section, the first setup configuration will be referred to as forward optical head (FOH), and the second as downward optical head (DOH). The results are shown in Figure 9, where Figures 9A, B display the FOH and DOH configurations, respectively.

Short-term stability validations are shown in Figure 9C. This set of measurement considers mainly transient disturbances and is done by computing the mean optical phase distribution in a recorded video. The video is approximately 30 s long and contains 300 frames of the extracted optical phase. As seen from the plot, the FOH configuration proves more stable than DOH since it deviates between − 3.7 π , 1.7 π , compared to − 0.9 π , 23.5 π from DOH, and the mean slope of deformation is lower. Therefore, setup can only be used as long as the relative displacements are of interest. For example, in the measurement of vibration mode shapes, the transient disturbances will only affect the absolute displacements and not the relative mode shape which can be correctly quantified as long as a known fixed boundary point is within the reconstructed field.

Long-term stability validations are shown in Figure 9D. This set of measurement considers mainly steady-state disturbances and is done by studying the decorrelation of speckles in DE holograms, as described in Section 2.3.2, as long as the reference hologram is unchanged for the complete measurement dataset. The degree of speckle decorrelation is computed by solving for the magnitude of the modulus of complex coherence coefficient during a time lapse measurement that lasts 4 h. As seen from the plot, the mean coherence factor computed for FOH is linearly decreasing at a slope of approximately − 0.14/hour and was higher than 0.5 in general throughout the first 2.5 h. The mean coherence factor computed for DOH is relatively linear and above 0.5 for the first 0.5 h and then drops to approximately 0.2 for the remainder of the measurements. Therefore, applications requiring time lapse measurements of steady-state phenomena, such as vibrations and acoustics, for longer than 30 min will require the FOH setup.

A particular interest in the field is the measurement of leading and trailing edges of a turbine blade. For such blades with complex geometries, it is challenging for most conventional techniques given the variability of measurement sensitivity that arises from a highly diversified surface normal vector. A solution previously used is the placement of a mirror to inspect both edges with maximized sensitivity in each. In DIC and ESPI methods, if the DOF is not high enough to cover both projections, at least two exposures must be taken for each edge due to their dependance on the projection of the object plane onto the sensor plane via a positive lens configuration. In lensless digital holography, the numerical focusing capabilities allow for the instantaneous reconstruction of fields at the leading and trailing edges.

The individual blade that was tested with in the previous section is one of multiple that form part of a turbomachinery component called a rotor. In the aerospace industry, aside from characterizing the individual mode shapes of each blade, the measurement of a complete rotor is of high interest. Characterizing the mode shape of a rotor provides an understanding of the dynamic behavior of the entire assembly and is especially important in the avoidance of superimposed resonance in which every individual blade resonates with the same loading frequency.

The sample used for this experiment is an aluminum heat sink designed for packaged electronics. The experimental configuration is shown in Figure 13, along with intensity reconstructions focused on different planes throughout the length of the heat sink fin. In the setup, as depicted in Figure 13A, the OH is aligned with respect to the heat sink sample, which is attached via a thermal paste onto a thermo-electric heater (TEH). The TEH is externally controlled from software and has a temperature stabilization range of 0.1°C. An infrared (IR) camera is used to inspect the temperature distribution throughout the length of the heat sink, and thus, is placed in perpendicular fashion. Figures 13B, C, and Figure 13D show the reconstructed intensities of the heat sink from a single phase-shifted hologram numerically focused on the front-plane, mid-plane, and back-plane of the heat sink, respectively.

Transient thermal deformations are measured throughout the interior surface of a heat sink fin, as shown in Figure 14. The experimental procedure consists of thermally loading the heat sink by increasing the top surface temperature of the TEH, where the sample’s bottom surface is thermal pasted, from 25 to 27°C. Throughout the temperature increase, a video of the optical phase is recorded at approximately 10 fps. A selected number of frames throughout the measurement are shown in Figure 14A, alongside the temperature field quantified by the IR camera. The DE holograms are divided into multiple sections at different distances along the heat sink fin and are each individually numerically reconstructed and unwrapped to compute the fringe-locus function, Δ Ω i . Each of the fields are stitched together in the Z-direction, as shown in Figure 14B, for each selected time step.

The test sample is a synthetic left pinna with anthropometric concha and ear canal made of soft silicone material. The sample was designed and fabricated to resemble the human pinna both in structure and performance to aid in the development of supra-aural and circum-aural earphones. Therefore, the frequency response and acoustic characteristics of this sample closely mimic that of a real human pinna. The results are shown in Figure 15. Figure 15A shows the experimental configuration used for these measurements as well as results. An acoustic speaker with a frequency range of 0–20 kHz is placed on the top surface of the optical table and below the sample. A mirror is placed behind the sample at a particular position and orientation such that the posterior of the ear is projected within the FOV of the holographic system.

Acoustically induced deformations on the synthetic pinna are measured in full-FOV. The reconstructed intensity of the anterior of the sample is shown in Figure 15B, and that of the posterior is shown in Figure 15C. The sample was stimulated with the use of the acoustic speaker at 350 Hz, and the deformation field for both anterior and posterior sides are instantaneously measured, as shown in Figure 15D.