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In state-of-the-art laser ultrasonics (LU), the signal-to-noise ratio (SNR) is limited by the shot noise of the detected laser radiation. Further improving the SNR then requires averaging multiple signals or increasing generation and/or detection laser intensities. The former strategy is time consuming and the latter leads to surface damages. For signal-independent limiting noises, Hadamard multiplexing increases the SNR by averaging multiple signals in parallel using a single detector. Here we consider the use of Hadamard multiplexing in LU for the non-contact ultrasonic inspection of materials. By using 31 element Hadamard masks to modulate the spatial intensity distribution of the generation laser beam, the measured SNR is improved by a factor 2.8, in good agreement with the expected multiplexing or Fellgett advantage. In contrast to many other applications of Hadamard multiplexing, the SNR is improved for shot-noise-limited measurements since the shot noise level is independent of the signal in LU. The Hadamard multiplexing of the detection laser beam is also considered but can only lead to a throughput or Jacquinot advantage. However, for pulse-echo LU, the Hadamard multiplexing of both generation and detection laser beams allows using the synthetic aperture focusing technique (SAFT).
©2012 Optical Society of America
1. Introduction
Laser ultrasonics (LU) is a well-established optical technique allowing non-contact generation and detection of ultrasound [1]. LU is mostly applied to the non-destructive ultrasonic testing of industrial materials such as metals, plastics and polymer-matrix composites. An ultrasonic wave is laser generated at the surface of the material by ablation or thermoelastic expansion. Ultrasonic waves are then remotely detected using a single-frequency detection laser beam reflected/backscattered by the unprepared surface of the material. The reflected/backscattered light, which is phase modulated by the ultrasonic surface displacement, is demodulated with a large throughput interferometer such as a confocal Fabry-Perot interferometer (CFPI) [2,3] or a photorefractive interferometer (PRI) [4,5]. High detection sensitivity is obtained by using a high power detection laser and by operating in a shot-noise-limited detection regime.
Hadamard multiplexing is a generic method of measurement which has found numerous applications [6–14], the most prevailing one being probably Hadamard transform infrared spectroscopy. Hadamard multiplexing is interesting when N individual signals must be successively measured with a single detector. By measuring N linearly independent superpositions of the N individual signals, still using a single detector, it is possible to recover each individual signal with the additional benefit of an increased signal-to-noise ratio (SNR). The SNR is increased by a factor F = (N + 1) / (2N1/2) called the multiplexing or Fellgett advantage. This advantage is only achieved for measurements limited by signal-independent noise sources [10,11]. The multiplexing advantage is reduced, cancelled or even turn to a disadvantage for signal-dependent noise sources such as shot noise. In LU, multiplexing the generation laser beam has no impact on the detected shot noise. Consequently, in contrast to many other techniques, the detected shot noise is signal-independent and this can be exploited advantageously. In some applications, Hadamard multiplexing can also lead to a throughput advantage known as the Jacquinot advantage in spectroscopy [15].
In this work, we show that, for some applications, the multiplexing advantage can be used to reduce the optical power density applied to materials inspected by LU. Even for shot-noise-limited measurements, the multiplexing advantage is achieved by applying the multiplexing strategy to the spatial distribution of the generation laser beam. In this first implementation, linear Hadamard masks with N = 31 elements are used for an expected multiplexing advantage F = 2.87. Such an increase in SNR would require averaging 8 elementary (non-multiplexed) measurements since F ≅ 81/2 = 2.83. The multiplexing strategy thus allows decreasing the acquisition time and/or reducing the applied power density. Even though this strategy may require a significant power scaling of the generation laser since multiple measurements are done in parallel, the laser power density on the material is reduced, for a given SNR, compared to that necessary for elementary measurements. In LU, multiplexing is particularly interesting in situations where laser power density must be voluntarily reduced to alleviate surface damages and averaging then becomes necessary to recover an acceptable SNR. These situations are often encountered when high spatial resolution is required on easily damaged materials.
The Hadamard multiplexing of the spatial distribution of the detection laser beam is also considered. Although, in this case, there is no direct multiplexing advantage for shot-noise-limited detection, a throughput or Jacquinot advantage is achieved [15]. More importantly, multiplexing simultaneously generation and detection laser beams brings both a multiplexing advantage and a compatibility with the synthetic aperture focusing technique (SAFT) for pulse-echo LU measurements [16].
The present approach increases the SNR in LU by multiplexing measurements without relying on phased array laser generation [17–19] or multichannel detection [20,21], which are both expensive in terms of hardware. Combining Hadamard multiplexing with LU is attractive for the non-contact inspection of parts for which surface damages are unacceptable and the thermoelastic regime does not generate large enough ultrasonic wave amplitudes. Examples are small welded parts in microelectronics or polymer-matrix composites used in aerospace applications. Biomedical imaging applications [22], for which laser exposure safety limits restrict the sensitivity [23], could also benefit from the present approach.
2. Hadamard multiplexing
2.1 Basic principles
For the sake of completeness, Hadamard multiplexing is briefly introduced in this subsection. A thorough description of the theoretical background can be found in Refs [6–11]. Hadamard multiplexing consists in measuring N Hadamard-multiplexed values {hi}, each of which is a linear superposition of N elementary (non-multiplexed) values {ei}. The elementary values are then retrieved from the Hadamard-multiplexed measurements. The benefit of this procedure is a decrease of the noise level by a factor F called the multiplexing of Fellgett advantage. By construction, elementary and Hadamard-multiplexed values can be grouped, respectively, in vectors e and h. The relationship between e and h is represented by a square multiplexing matrix M of order N. More specifically, h = M e, where the elements {mi,j} of M are equal to 0 or 1 for amplitude multiplexing or to ± 1 for phase multiplexing. In practice, the expectation values e and h cannot be measured without their corresponding random noise fluctuations ne and nh. Elementary and Hadamard-multiplexed experimental values are thus given, respectively, by e′ = e + ne and h′ = h + nh. Elementary experimental values r′ retrieved from the Hadamard-multiplexed measurements are given by
Since r′ = r + nr, the right-and side of Eq. (1) shows that r = e, which means that the retrieved expectations values are equal to the elementary expectation values, and that nr = M−1 nh. This last expression is used to compare nr with nh. Knowing that all elements of M−1 are equal to ±1, with a common normalizing factor K−1, the root-mean-square (rms) noise amplitudes, given by σk ≡ (nk ∙ nk / N)1/2, are linked bywhere the factor N1/2 comes from the superposition of N random noise contributions and the additive law of variances. The value and physical meaning of K are given in the following paragraphs. The multiplexing advantage F is defined by the ratio of the SNR of the retrieved values, SNRr∙≡∙r∙/∙σr, to that of the elementary values, SNRe∙≡∙e∙/∙σe, where r and e are, respectively, the retrieved and elementary expectation values. Knowing that r = e from Eq. (1) and using Eq. (2), the multiplexing advantage is given byFor amplitude Hadamard multiplexing (mi,j = 0,1), a maximum multiplexing advantage is obtained with M = S, where S is a Sylvester matrix [6,10]. S is obtained by a procedure which is particularly simple for N = 2n − 1, where n is an integer. The first row of S is then given by a pseudo noise (PN) bit sequence calculated using a primitive polynomial of order n. The first n elements of the PN bit sequence can be set arbitrarily (except not all zeros) and the following elements are easily calculated [10]. Once the first row of S is known, the following rows are obtained in a left (or right) cyclic fashion, each row being shifted by one element to the left compared to the previous one. A PN bit sequence containing K = (N + 1) / 2 elements equal to 1, each amplitude Hadamard-multiplexed measurement is the sum of K elementary signals. It can be shown that S−1 = K∙-1∙(2∙ST∙-∙J) where J is a square matrix of ones [6].
For phase Hadamard multiplexing (mi,j = ± 1), the maximum multiplexing advantage is obtained with M = H, where H is a Hadamard matrix [6,10]. The elements of H are all equal to ± 1. Each phase Hadamard-multiplexed measurement is thus a signed superposition of K = N elementary signals. For any Hadamard matrix, H HT = N I, I being the identity matrix [6]. Consequently, H−1 = K −1 HT where K = N.
For both amplitude and phase Hadamard multiplexing, K is physically interpreted as the number of elementary signals which are processed in parallel by the single detector for each multiplexed measurement. When measurements are limited by a signal-independent noise source, σh = σe . For shot-noise-limited measurements, σh = K1/2 σe, the shot noise level being proportional to the square root of the power [3]. The multiplexing advantage depends on the type of Hadamard multiplexing (amplitude or phase) and the type of limiting noise (signal-independent or shot). Four important possible values of F are given in Table 1 . It is seen that, for shot-noise-limited measurements, there is no multiplexing advantage (F ≤ 1). This is why Hadamard multiplexing is usually considered only for measurements limited by noise sources which are independent of the signal.
Table 1. Multiplexing or Fellgett advantage F. Approximate values are valid for large N.
Usually, in optical applications such as Hadamard transform spectroscopy, each row of M physically corresponds to a mask of N elements which are either opaque or transparent to represent, respectively, 0s and 1s (amplitude Hadamard multiplexing). Light transmitted by all transparent elements is processed in parallel by a single detector and, instead of N successive elementary measurements, N Hadamard masks are used successively. Phase Hadamard multiplexing is less commonly applied since it requires either phase sensitive measurements or differential detection [6].
2.2 Construction of the augmented Hadamard mask
In this work, amplitude Hadamard multiplexing is considered. The case n = 5, corresponding to N = 31 elements was chosen to use the cyclic construction described in Subsection 2.1. The PN bit sequence defining the elements is constructed by setting the first 5 bits to 1 and by calculating the remaining bits of the sequence with the primitive polynomial of order 5, P5 (x) = x5 + x2 + 1, using the procedure described in Ref [10]. The obtained PN bit sequence corresponds to the first row of S and the following rows are obtained by the left cyclic procedure described previously. The cyclic character of S is useful when realizing the N masks physically since a single augmented Hadamard mask of 2N − 1 elements sliding in front of an N-element-wide aperture can be used to generate the N different Hadamard masks. The bit sequence defining the augmented mask is obtained by concatenating the N − 1 first bits of the PN bit sequence at the right of the PN bit sequence. The resulting sequence is given in Fig. 1(a) with a schematic representation of the augmented mask in which opaque elements (gray squares) represent 0s and transparent elements (empty squares) represent 1s. Figure 1(b) shows a schematic representation of the resulting augmented mask (gray) in front of the vertical aperture (black blades) limiting the transmission to only N = 31 elements. The 31 Hadamard masks are obtained by sliding the augmented mask, one element at a time, in the direction of the arrow in Fig. 1(b).
Fig. 1 (a) Bit sequence of 2N − 1 = 61 elements defining the augmented Hadamard mask with the corresponding schematic representation of opaque (gray squares) and transparent (empty squares) elements. (b) Schematic representation of the augmented mask (gray) in front of the N = 31 element wide vertical aperture (black blades).
2.3 Analogy between spectroscopy and laser ultrasonics
The use of Hadamard multiplexing in LU was inspired by an analogy between spectroscopy and LU. Although many similarities arise from this analogy, differences can also be noticed. The analogy is illustrated schematically in Fig. 2 .
Fig. 2 Schematic diagrams illustrating elementary measurements (a) in spectroscopy and (b) in LU and Hadamard-multiplexed measurements (c) in spectroscopy and (d) in LU.
The elementary (non-multiplexed) measurement schemes are illustrated in Fig. 2(a) for spectroscopy and in Fig. 2(b) for LU. It is important to mention that numerous experimental configurations exist in both spectroscopy and LU. The configurations illustrated here were chosen and simplified for the benefit of the analogy. In Fig. 2(a), the input light beam (LB) coming from an extended light source is first transmitted by a single aperture in the entrance plane (E) of the spectrometer. A light dispersing element (LD), such as a grating or a prism, distributes angularly the spectral content (represented by different colors) of the input beam. The spectral content is then distributed spatially by light propagation (LP) in the spectrometer. Light collection (LC) through a single aperture then selects a given spectral component up to the detection plane (D) where is located the photodetector (PD). Sliding the single aperture in E allows successive elementary measurements of the spectral components of the extended light source. In Fig. 2(b), the generation laser beam (GB) is first limited in area by a single aperture in the entrance plane (E), ultrasonic generation (UG) then occurs on a limited area of the surface of the material specimen (S). Ultrasonic propagation (UP), which is accompanied by ultrasound diffraction spreading and scattering by the microstructure of the material or any defect, results in a spatial distribution of the angular spectrum components of ultrasound waves. Ultrasound “collection” (UC) is limited to the area of the detection laser beam (DB) and detected, in this case, by an optical demodulator (OD).
Hadamard-multiplexed spectroscopy is illustrated in Fig. 2(c). It is seen that the amount of light is multiplied by the number of apertures K in E (in this illustration, K = 3). This leads to the superposition of K spectral components on the single PD. Similarly, in Hadamard-multiplexed LU, illustrated in Fig. 2(d), the area of the generation laser beam is multiplied by the number of apertures in E (again, K = 3). The linear superposition of K angular spectrum components of the ultrasonic waves is detected by the single OD, which also has a linear response with the phase modulation.
An important difference between spectroscopy and LU stems from the fact that light detected in spectroscopy comes directly from the extended light source whereas light detected in LU is independent of the generation laser beam. In spectroscopy, the link between the light source and the detected light eliminates the multiplexing advantage for shot-noise-limited detection (see Table 1). On the contrary, in LU, increasing the generation laser light power increases the signal but not the detected laser light power, thus leaving the shot noise level unchanged. Consequently, when Hadamard multiplexing is applied to the generation laser beam, the shot noise of the detected laser light is independent of the signal, which is the necessary condition to preserve the multiplexing advantage. The multiplexing advantage is thus achievable for shot-noise-limited LU measurements as long as the multiplexing strategy is applied to the generation laser beam. This reasoning is generally applicable because, in most practical applications of LU, the intensity noise of the generation laser is too weak to be the limiting source of noise. LU measurements are thus generally limited by the detection noise. The above reasoning shows that, even in the best case of shot-noise-limited detection, multiplexing the generation laser beam preserves the multiplexing advantage in terms of SNR.
However, when Hadamard multiplexing is applied to the detection laser beam (not shown in Fig. 2), the accompanying increase of detected laser light power increases the shot noise level and eliminates the multiplexing advantage (for shot-noise-limited measurements). In subsections 4.2 and 4.3, it will be seen that multiplexing the detection laser beam in LU can, however, provide other advantages.
Another difference between spectroscopy and LU stems from the type of detection. In spectroscopy, a square law PD is used, which generally forces the use of amplitude Hadamard masks. In LU, the OD being phase sensitive, amplitude and phase Hadamard multiplexing are both applicable when multiplexing the detection laser beam. It is clear, however, that only amplitude Hadamard multiplexing is applicable for the generation laser beam.
3. Materials and methods
3.1 Generation and detection lasers
The generation laser is a frequency-doubled Q-switched Nd:YAG laser emitting 10 ns pulses at a wavelength of 532 nm. The output pulse energy is adjustable from 0 to 25 mJ with a computer-controlled attenuator.
The detection laser is composed of a continuous-wave single-frequency Nd:YAG master oscillator (MO) emitting at a wavelength of 1064 nm. The MO is followed by a multipass diode-pumped Nd:YAG power amplifier (PA). For detection on sensitive materials, limiting the detection laser pulse duration to the ultrasound propagation time is essential. For this purpose, an electro-optic intensity modulator is added between the MO and the PA. See Ref [24]. for a similar laser source. The temporal profile of a detection laser pulse is shown in Fig. 3 . The pulse duration is about 6 µs (FWHM: full width at half maximum) and the output pulse energy is about 700 µJ. The generation laser pulse, represented by the green line in Fig. 3, is launched approximately 2 µs after the beginning of the detection laser pulse. Light detected before the generation laser pulse is used for the evaluation of the shot noise level.
Fig. 3 Temporal profile of the detection laser pulse. The green line represents the arrival time of the generation laser pulse.
3.2 Augmented Hadamard mask
Following the design described in Subsection 2.2, an augmented Hadamard mask was cut in a 25 µm thick copper foil. Each element was chosen to be 100 µm × 100 µm in size. Only 16 apertures had to be cut since many transparent elements are adjacent to each other. The different spatial intensity distributions exiting from the sliding augmented mask followed by the limiting aperture were imaged on a CCD camera. The results are shown in Fig. 4 where the mask #1 is repeated below the mask #31 to illustrate the cyclic character of the series.
Fig. 4 Measured spatial intensity distributions at the exit of the augmented mask followed by the limiting aperture. Size of square elements: 100 µm × 100 µm.
Hadamard masks with elements of 100 µm × 1000 µm were also cut using the same design. The purpose of these masks is explained in detail in Subsection 4.3. Essentially, using rectangular elements (slits) allows controlling the vertical extent of all elements by adjusting the spatial extent of the incoming laser beam.
3.3 Scanning and detection systems
In the present study of Hadamard multiplexing in LU, three different scanning setups are successively used. Those setups are schematically shown in Fig. 5 . The first setup, shown in Fig. 5(a), is used to study the spatial multiplexing of the generation laser beam for through-thickness LU measurements. The second setup, shown in Fig. 5(b), is used to study the spatial multiplexing of the detection laser beam for through-thickness LU measurements. The third setup, shown in Fig. 5(c), is used to study the simultaneous multiplexing of both generation and detection laser beams for pulse-echo LU measurements.
Fig. 5 Schematic diagrams of scanning setups used for the spatial multiplexing of (a) the generation laser beam, (b) the detection laser beam, and (c) both detection and generation laser beams. All components are described in the text.
In the first setup, shown in Fig. 5(a), the output beam of the generation laser (GL) is sent through the generation beam shaper (GBS) to get a rectangular top hat beam with dimensions adapted to the useful area of the augmented mask, that is, slightly larger than 3.1 × 0.1 mm2. This beam illuminates the augmented mask (H) with square elements (100 µm × 100 µm) located on a computer-controlled translation stage (T) allowing scanning along the axis x. The mask H is followed by a fixed 3.1-mm-wide aperture (A) limiting the number of effective elements to N = 31. Relay lenses L1,2 (focal lengths: f1,2 = 150 mm) are used to image the transmitted beam on the surface of the test specimen (S) with a unit magnification. Imaging the spatially modulated laser beam eliminates the need to put the mask in the immediate vicinity of S, thereby preserving the remote-measurement advantage of LU. The nearly Gaussian output beam of the detection laser (DL) is first sent through a polarizing beam splitter (PBS) and a quarter-wave plate (QW) before being focused on a 60 µm spot (diameter at 1/e2) by lens L3 (f3 = 60 mm) on the surface of S. Reflected and/or backscattered light collected by L3 exits from the side of the PBS. This light is injected in a multimode optical fiber (F: 0.4 mm core diameter, 0.39 numerical aperture) with lens L4 (f4 = 50 mm). The phase demodulation of the collected light is done with a 50-cm-long actively-stabilized confocal Fabry-Perot interferometer (CFPI) used in reflection mode to obtain a flat demodulation response above 5 MHz. The highest accessible frequency was limited to about 80 MHz by the pulse duration of the GL. A beam sample is used at the input of the CFPI to eliminate the intensity noise of the detection laser by differential detection in order to achieve shot-noise-limited measurements. The generation beam shaper (GBS) is schematically shown in Fig. 6 . The main element is the Gauss to square-top-hat beam converter (model GTH-4-2.2FA from Eksma) which is followed by a spherical lens Ls (fs = 2000 mm) and a cylindrical lens Lc (fc = 300 mm). Lenses L1 (f1 = −75 mm) and L2 (f2 = 200 mm) are used to adjust the Gaussian beam diameter (at 1/e2) of the generation laser to 4 mm in the plane of the GTH.
Fig. 6 Schematic diagram of the generation beam shaper GBS. Components are described in the text. Lenses L1,2 are not present in the DBS which is otherwise identical to the GBS.
The second scanning setup, shown in Fig. 5(b), is used for the spatial multiplexing of the detection laser beam. The components are essentially the same, except for the detection beam shaper (DBS) and the lenses’ focal lengths: f1,2 = 200 mm, f3 = 150 mm, and f4 = 20 mm. Again, an augmented Hadamard mask (H) with square elements (100 µm × 100 µm) is used to spatially modulate the detection laser beam intensity distribution and the plane of the mask is imaged with a unit magnification on S. The detection beam shaper (DBS) is similar to the GBS shown in Fig. 6, except that lenses L1,2 are not necessary, the quasi-Gaussian output beam of the detection laser having the right input diameter (4 mm).
The third scanning setup, shown in Fig. 5(c), is used for the spatial multiplexing of both generation and detection laser beams. In this case, two identical augmented Hadamard masks (Hs) are located on the translation stage to ensure a simultaneous scanning. However, the masks are made of rectangular elements (vertical slits: 100 µm × 1000 µm). Both multiplexed beams are combined using a mirror (M) and a dichroic mirror (D). In this third setup, the focal lengths are: f1,2 = 200 mm, f3 = 120 mm, and f4 = 20 mm. Both masks are thus imaged on S with an optical magnification of 0.6. In principle, a single mask could be used to modulate both beams. However, this approach leads to spurious phase modulation of the detection laser beam by ultrasonic waves generated on the mask by the generation laser beam. The additional benefit of using two masks is the independent control of the height of the elements for generation and detection laser beams. For instance, this allows using Hadamard-multiplexed line generation in combination with Hadamard-multiplexed point detection.
3.4 Experimental procedure
The experimental procedure followed with each experimental setup in Fig. 5 is intended to determine the multiplexing advantage and to verify if there is any impact of the multiplexing approach on the spatial resolution. In LU terminology, the signal as a function of time, for a given position of the beam on the test specimen, is called an A-scan. All A-scans acquired along a line on the test specimen can be grouped in a B-scan image which gives the signal as a function of the lateral position and time. B-scan images are often numerically processed using the synthetic aperture focusing technique (SAFT) [16]. This procedure, also referred to as delay-and-sum method or beamforming, uses the known velocity of the ultrasonic wave (typically, the longitudinal wave) to obtain a SAFT image in which the lateral resolution is improved compared to that of a B-scan image.
For each setup presented in Fig. 5, measurements are carried using a sliding augmented mask for the Hadamard measurements and a single aperture for the elementary measurements. Both Hadamard and elementary measurements are done using the same laser power density on the test specimen. The noise levels of the Hadamard A-scans, the Hadamard-retrieved A-scans and the elementary A-scans are then compared to determine the impact of multiplexing on the SNR. In principle, multiplexing has no effect on the spatial resolution. With setups in Figs. 5(a) and 5(b), this is experimentally tested by comparing SAFT images of the point spread function (PSF) of Hadamard-retrieved and elementary measurements. With the setup in Fig. 5(c), SAFT images of the edge spread function (ESF) are compared for the same purpose.
4. Results
4.1 Hadamard multiplexing of the generation laser beam
Hadamard multiplexing is first applied to the generation laser beam alone using the setup shown in Fig. 5(a). Instead of focusing the generation laser on a single generation spot and scanning its position at N successive locations, the generation laser is used to illuminate the augmented mask. This mask is then imaged on a 1-mm-thick aluminum plate specimen. The N Hadamard-multiplexed measurements are then obtained by sliding the augmented mask, one element at a time, that is, by 100 µm steps. Each measurement is thus the result of the linear superposition of K ultrasonic waves – the number of open elements on each mask – detected with a single optical demodulator. The total number of multiplexed measurements is, again, equal to N. Note that, for a given power density on the material, this procedure requires generation laser pulses with an energy of at least K times that necessary for a non-multiplexed measurements (in our case, about N times, due to opaque elements on the mask). The Hadamard A-scan associated with each mask can be grouped in what we call a Hadamard B-scan. It should be noted that, for any Hadamard B-scan, the “lateral position” cannot be associated to a specific lateral position on the test specimen but, more specifically, to the position of the augmented mask. The corresponding retrieved B-scan is obtained using Eq. (1) where “elements” of h′ are A-scans of the Hadamard B-scan and “elements” of r′ are A-scans of the retrieved B-scan. A typical Hadamard B-scan and its corresponding retrieved B-scan are shown in Figs. 7(a) and 7(b) (upper images) where the green lines represent the arrival time of the generation laser pulses.
Fig. 7 (a) Hadamard B-scan, (b) corresponding retrieved B-scan, and (c) elementary B-scan. The averaging number was 16 for both, Hadamard and elementary measurements. Each graph on the lower row shows the central A-scan (between white arrows) of the corresponding B-scan. Green line: generation laser pulse. L and S stand for longitudinal and shear waves.
To evaluate the multiplexing advantage, the augmented mask is replaced by a square aperture (100 µm × 100 µm) corresponding to a single open element. The power density of the generation laser beam is left unchanged as well as the level of detection laser light collected by the CFPI. An elementary B-scan is then measured by sliding the single square aperture in steps of 100 µm. The resulting elementary B-scan is shown in Fig. 7(c) (upper image). Both retrieved and elementary B-scans show the hyperbolic shaped signals associated with the longitudinal ultrasonic wave (upper hyperbola), identified from its known propagation speed in aluminum, vL = 6.37 mm/µs, and the transverse or shear ultrasonic wave (lower hyperbola), also identified from its known propagation speed in aluminum, vS = 3.11 mm/µs. However, it is seen that the retrieved B-scan in Fig. 7(b) has a better SNR compared to the elementary one in Fig. 7(c). Central A-scans (between white arrows) of B-scans in Fig. 7 are shown in their respective graph (lower row). It is clearly seen that the Hadamard-multiplexed A-scan has the largest amplitude since it is associated to the superposition of K = 16 ultrasonic waves. However, its interpretation is not intuitive. As expected, the retrieved and elementary A-scans have essentially the same amplitude but the noise level is significantly lower for the retrieved A-scan, by about a factor F. This is why the longitudinal wave (L) is hardly seen in the elementary A-scan. All measurements in Fig. 7 were obtained by averaging 16 A-scans for each position of the augmented mask (or the single square aperture).
The noise level of all A-scans in Fig. 7 can be determined using the signal from each A-scan before the occurrence of the generation laser pulse, that is, for t < 0 µs. Figure 8 shows the rms noise of all A-scans in Fig. 7 once normalized to the average rms noise (green line) of the 31 elementary A-scans (green dots). Clearly, the rms noise (black dots) of the multiplexed A-scans is very similar to that of the elementary A-scans with an average value σh / σe = 1.01 (black line). This confirms that the noise level is the same for both multiplexed and elementary measurements. The average value (red line) of the rms noise values (red dots) of the retrieved A-scans is σr / σe = 0.355, which corresponds to F = 2.82. This is essentially equal to the theoretical multiplexing advantage F = 2.87.
Fig. 8 (a) Rms noise of A-scans in Fig. 7 normalized to the average rms noise of elementary A-scans (green line): elementary A-scans (green dots), Hadamard A-scans (black dots) and retrieved A-scans (red dots). Each horizontal line gives the average value of corresponding series of dots (same color).
Retrieved and elementary B-scans obtained with different averaging numbers were further processed by the synthetic aperture focusing technique (SAFT) to recover the point spread function (PSF) [16]. These results are shown in Fig. 9 . The SAFT reconstruction is done with a synthetic aperture half-angle of 50° and a bandwidth from 10 to 80 MHz. Time-domain SAFT with time derivatives is used for the reconstruction [24]. The generation laser power density was chosen to the lowest value allowing a single shot Hadamard measurement of the PSF with an acceptable SNR. The SAFT images are shown in Fig. 9(a) for results using Hadamard multiplexing and in Fig. 9(b) for results obtained by elementary measurements. The averaging number is shown in parenthesis aside each SAFT image. Figure 9 also shows horizontal and vertical profiles passing through each PSF (between the corresponding arrows). These profiles allow the appreciation of both the SNR and the spatial resolution. As expected, it is easily seen that the spatial resolution is essentially the same for both Hadamard-multiplexed and elementary measurements (for profiles of acceptable SNR). However, without averaging, the PSF is not recovered with elementary measurements. In fact, with elementary measurements, averaging 8 measurements was necessary to recover the PSF with an SNR comparable to that of the single shot Hadamard-multiplexed measurement. The acquisition time was thus reduced by a factor of 8 using Hadamard multiplexing. As expected, this factor is almost equal to F 2 = 8.26. It should be noted that the increase of SNR was obtained although the measurements were limited by the shot noise since the multiplexing was applied to the spatial intensity distribution of the generation laser beam, which has no incidence on the shot noise level. It is also interesting to note that the theoretical multiplexing advantage is essentially reached although the rectangular top hat laser beam impinging on the Hadamard mask was not perfectly uniform. The strategy is thus rather insensitive to intensity fluctuations of about ± 15% along the mask (for single shot measurements). Consequently, no active control of the intensity of each open element was necessary and no correction of the multiplexing matrix was used for data processing.
Fig. 9 PSFs obtained (a) by Hadamard multiplexing of the generation laser beam and (b) by elementary measurements for averaging numbers indicated in parenthesis. Horizontal and vertical profiles passing through each PSF are shown, respectively, on the top and left sides of each image.
4.2 Hadamard multiplexing of the detection laser beam
As mentioned in Subsection 2.1, when amplitude Hadamard multiplexing is applied to the detection laser beam in shot-noise-limited regime, the multiplexing advantage F ≅ 2-1/2 turns to a slight disadvantage (Table 1). This disadvantage is however accompanied by a throughput or Jacquinot advantage since the optical etendue is multiplied by K for a given numerical aperture of the collecting optics. More importantly, it can be advantageous to multiplex both detection and generation laser beams since this allows further processing by SAFT for pulse-echo LU measurements (see Subsection 4.3).
The setup shown in Fig. 5(b) is used to test the spatial multiplexing of the detection laser beam only. Results obtained with a 1-mm-thick aluminum plate specimen are shown in Fig. 10 . The Hadamard B-scan and the retrieved B-scan are shown, respectively, in Figs. 10(a) and 10(b) (upper images). Again, the augmented mask (H) is replaced by a single square aperture (100 µm × 100 µm), the power density on the specimen being left unchanged. The resulting elementary B-scan is shown in Fig. 10(c) (upper image). The central A-scan (between white arrows) is shown below each B-scan in Fig. 10. It is easily seen that, multiplexing the detection laser beam leads to a slight disadvantage.
Fig. 10 (a) Hadamard B-scan, (b) corresponding retrieved B-scan, and (c) elementary B-scan. The averaging number was 16 for both, Hadamard and elementary measurements. Each graph on the lower row shows the central A-scan (between white arrows) of the corresponding B-scan. Green line: generation laser pulse. L and S stand for longitudinal and shear waves.
The multiplexing disadvantage is confirmed, in Fig. 11 , by comparing the rms noise of all A-scans in Fig. 10. Again, all the rms noise values are normalized to the average rms noise of the elementary measurements (green). Still in Fig. 11, it is seen that multiplexing the detection laser beam increases the rms noise of Hadamard measurements (black) by σh / σe = 3.93, which is almost equal to 4, the value expected from the shot noise scaling law σh = K1/2 σe. We also see that the rms noise of the retrieved A-scans (red) is σr / σe = 1.43. This corresponds to a noise reduction, compared to Hadamard measurements, by a factor 3.93 / 1.43 = 2.75 due to the multiplexing strategy alone. The result is a multiplexing disadvantage F = σe / σr = 1.43−1, near from the expected value of F = [(N + 1) / (2N)]1/2 = 1.39−1 (Table 1). Figure 11 illustrates the physical origin of the multiplexing disadvantage for shot-noise-limited measurements: the increase of the shot noise level due to the increased throughput (by illuminating K spots simultaneously) is not completely offset by the multiplexing strategy.
Fig. 11 (a) Rms noise of A-scans in Fig. 10 normalized to the average rms noise of elementary A-scans (green line): elementary A-scans (green dots), Hadamard A-scans (black dots) and retrieved A-scans (red dots). Each horizontal line gives the average value of corresponding series of dots (same color).
From a technical standpoint, care must be taken to couple the K detection spots in the optical fiber with the same efficiency. This was somewhat problematic due to the high aspect ratio of the linear mask. In Fig. 11, the lower rms noise of elementary measurements (green dots) near the limits of the scan confirms a slightly lower coupling efficiency for these detection spots.
In Fig. 12 , PSFs reconstructed using SAFT processing confirm the slight multiplexing disadvantage and also confirm that spatial resolution is essentially unaffected when comparing multiplexed and elementary measurements shown, respectively, in Figs. 12(a) and 12(b). The multiplexing strategy is thus also applicable with multiple simultaneous detection laser beams coupled in a single optical demodulator which, in this case, is a CFPI. Figure 12 also shows horizontal and vertical profiles passing through each PSF (between the corresponding arrows). Again, these profiles allow the appreciation of both the SNR and the spatial resolution.
Fig. 12 PSFs obtained (a) by Hadamard multiplexing of the detection laser beam and (b) by elementary measurements for averaging numbers shown in parenthesis. Horizontal and vertical profiles passing through each PSF are shown, respectively, on the top and left sides of each image.
4.3 Simultaneous Hadamard multiplexing of generation and detection laser beams
In LU, probing the material in reflection geometry (pulse-echo mode) is often preferred because only one side of the specimen is accessible in many practical applications. In such cases, SAFT reconstruction is achieved by simultaneously scanning superimposed generation and detection laser beams, the only drawback being the strong signal, at the beginning of each A-scan, due to thermoelastic or ablative generation. Consequently, when applying Hadamard multiplexing in pulse-echo mode, both generation and detection laser beams have to be multiplexed simultaneously. The Hadamard-retrieved B-scan then corresponds to the usual B-scan. However, for pulse-echo LU measurements, Hadamard multiplexing leads to some crosstalk between detection channels due to Rayleigh waves (surface waves) which are simultaneously generated with the bulk longitudinal and shear waves. This crosstalk leads to unwanted signals at the beginning of A-scans retrieved from Hadamard measurements. These unwanted signals define what we call the crosstalk zone (more details below).
The setup shown in Fig. 5(c) is used for pulse-echo LU measurements. Two identical augmented Hadamard masks with rectangular elements (100 µm × 1000 µm) are imaged and superimposed on the test specimen with an optical magnification of 0.6. Consequently, on the test specimen, each element is w = 60 µm wide and the scan line is l = 1.86 mm long. The test specimen is an aluminum plate with a 0.5 mm step on the back side. The purpose of the experiment is the detection of this step in order to estimate both the SNR and the spatial resolution through the ESF. The thickness of the specimen is 3 mm on the left hand side and 2.5 mm on the right hand side. A schematic side view is shown in Fig. 13 with the region of interest (ROI) of 1.86 mm wide by 0.9 mm deep inside the dotted rectangle. Figure 13 also schematically shows Rayleigh (R), shear (S) and longitudinal (L) waves generated on one element of the scan line by the generation laser beam (GB).
Fig. 13 Schematic diagram (side view) of the aluminum plate specimen with a step on the back side. GB, generation laser beam; R, Rayleigh waves; S, shear wave; L, longitudinal wave. The region of interest (ROI: dotted box) is 1.86 mm wide by 0.9 mm deep. Dimensions are in mm.
For the pulse-echo LU measurements, Hadamard masks with rectangular elements were preferred mostly to facilitate the alignment procedure. With rectangular elements, the height of the elements is not limited by the physical height of each aperture (1000 µm) but by the height of the rectangular top hat beam in the plane of the mask. For the generation beam, the full height (at 1/e2) of the top hat beam was adjusted to about 600 µm in the plane of the mask (line generation). For the detection laser beam, the full height (at 1/e2) was adjusted to about 150 µm in the plane of the mask. These heights were set by adjusting the distance between the cylindrical lens of each beam shaper and the following mask. In this case, each mask was replaced by a single vertical slit (instead of a square aperture) for elementary measurements. In this way, mechanical adjustments were less critical along the slit direction.
According to the length l = 1.86 mm of the scan line, the crosstalk zone due to Rayleigh waves (propagation speed in aluminum: vR = 2.91 mm/µs) is expected to have a duration tR = l / vR = 0.640 µs. This corresponds to a propagation of the longitudinal wave (propagation speed: vL = 6.37 mm/µs) over a distance dL = vL tR = 4.08 mm. Taking into account the two way propagation in pulse-echo measurements, imaging at depths shallower than dL / 2 = 2.04 mm is expected to be affected by signals due to Rayleigh waves. This is a limitation of the multiplexing technique, when used for pulse-echo LU measurements. However, as discussed in Subsection 5.3, the Hadamard mask does not have to be linear and using a square array of elements instead of a linear one can reduce significantly the depth of the crosstalk zone.
To illustrate the impact of Rayleigh (surface) waves on Hadamard-multiplexed pulse-echo LU measurements, typical B-scan images and A-scans are shown in Fig. 14 . The Hadamard-multiplexed measurement is shown in Fig. 14(a) and the elementary measurement is shown in Fig. 14(b). For each image (curve), the amplitude scale is adjusted by an attenuation factor (indicated in a red box) above the horizontal red line. This allows the visualization of both the strong signals at shallow depths and the weaker signals in the ROI, where the step back side is located. In the B-scan and the A-scan of Fig. 14(a) relatively large amplitude signals can be seen above the horizontal red lines: these signals, which are due to Rayleigh waves, define the crosstalk zone. In the ROI, however, it is seen that the noise level is significantly lower for the Hadamard-multiplexed measurement in Fig. 14(a) compared to the elementary measurement in Fig. 14(b). The multiplexing advantage is thus confirmed.
Fig. 14 (a) Left: B-scan image retrieved from Hadamard measurements. Right: Corresponding A-scan at x = 0.5 mm (between arrows) (b) Left: B-scan image obtained from elementary measurements. Right: Corresponding A-scan at x = 0.5 mm (between arrows). For each image (curve), the amplitude scale is adjusted above the red line by an attenuation factor given in the red box.
SAFT reconstruction was used with a synthetic aperture half-angle of 20° corresponding to the ratio of l / 2 = 0.930 mm over the depth (2.5 mm) of the shallower back side. Different averaging numbers were used for both Hadamard and elementary measurements. The results are shown in Fig. 15 . For Hadamard measurements shown in Fig. 15(a), the step is easily seen for all averaging numbers (1, upper row; 4, middle row; 16, lower row), either for B-scan or SAFT images. For elementary measurements, shown in Fig. 15(b), the step on the back side of the aluminum specimen cannot be seen without averaging (upper row images). By averaging 4 measurements (middle row images), the step is barely noticeable in the B-scan image but visible in the SAFT image. Averaging 16 elementary measurements (lower row images) is necessary to clearly detect the step in both B-scan and SAFT images. The improvement, in terms of SNR, obtained by using Hadamard multiplexing, is thus confirmed.
Fig. 15 (a) Left: B-scan images retrieved from Hadamard measurements. Right: Corresponding SAFT images. (b) Left: B-scan images obtained from elementary measurements. Right: Corresponding SAFT images. In (a) and (b), averaging numbers are: 1 (upper row), 4 (middle row), and 16 (lower row).
When using Hadamard multiplexing, false horizontal lines can be seen, in B-scan and SAFT images in Fig. 15(a), aside the true horizontal lines corresponding to the stepped back side. These artifacts are mainly attributed to the experimental difficulty of coupling the light from all the detection elements in the optical fiber F, and that, with the same efficiency. This difficulty arises from the high aspect ratio of the line mask. Since the theoretical model assumes that all elements are equal to 0 or 1, any intermediate value leads to artifacts in the result. It should be noted, however, that imaging a stepped back side is more challenging when using Hadamard multiplexing since any non-uniformity in light collection will result in a horizontal artifact aside a true horizontal signal. Moreover, in all B-scan images in Fig. 15(a), each horizontal artifact is of opposite polarity compared to the adjacent true horizontal signal. Consequently, SAFT reconstruction increases the visibility of the true horizontal line except in the central part of the image where SAFT processing results in a destructive interference between the artifact and the true signal, thus leading to an extended ESF. A more uniform coupling into the optical fiber is expected to decrease this detrimental effect.
5. Discussion
5.1 Signal-to-noise ratio
An SNR improvement by a factor 2.82 has been obtained using amplitude Hadamard multiplexing of the spatial distribution of the generation laser beam. This improvement is essentially equal to the predicted value of the multiplexing advantage F = 2.87. The multiplexing advantage has been achieved despite intensity fluctuations of about ± 15% along the Hadamard mask (for single shot illumination).
As expected, a slight multiplexing disadvantage by a factor 1.43−1 has been observed experimentally by applying Hadamard multiplexing to the detection laser beam. Again, this value is in good agreement with the theoretical value F = 1.39−1. This slight disadvantage is essentially due to the use of amplitude Hadamard multiplexing. Using phase Hadamard multiplexing would eliminate this disadvantage (see Table 1) since the multiplexing would, in this case, exactly offset the increase of shot noise accompanying the increased throughput. Multiplexing the detection laser beam cannot lead, in principle, to an SNR improvement for shot-noise-limited measurements (Table 1). However, the throughput or Jacquinot advantage, which has been observed experimentally for detection-multiplexed measurements (Fig. 11), will facilitate reaching a shot-noise-limited detection regime by increasing the number of detected photons compared to elementary measurements. Consequently, the throughput or Jacquinot advantage will increase the SNR only when multiplexing K detection elements will be sufficient to reach a shot-noise-limited detection regime whereas elementary measurements would otherwise be dominated by a signal-independent noise sources such as a technical detection noise or Johnson noise.
Multiplexing both generation and detection laser beams also leads to a multiplexing advantage coming, essentially, from the amplitude multiplexing of the generation laser beam. However, the increase of SNR is slightly reduced when amplitude multiplexing is used for the detection laser beam. Consequently, the maximum multiplexing advantage, when both beams are multiplexed, would be obtained by combining amplitude Hadamard multiplexing of the generation laser beam with phase Hadamard multiplexing of the detection laser beam. This is applicable, in principle, by using the link between Sylvester and Hadamard matrices [6].
From a more technical standpoint, although the Hadamard-multiplexed signals are less intuitive to interpret, the direct observation of any signal is facilitated in multiplexed measurements. Consequently, first observations and setup optimization can be facilitated by the multiplexing approach.
5.2 Spatial resolution
For through-thickness LU measurements, no significant effects have been observed on the spatial resolution of the reconstructed PSFs although some slight crosstalk between adjacent elements was certainly present due to the nominal mechanical tolerance of about 5 µm (5%) in the masks realization. The tolerance of the technique to the imperfect imaging of the mask(s) due to optical aberrations also seems to be good considering these first results. Again, this seems to indicate a good tolerance of the technique (see chapter 6 in Ref [6].).
For pulse-echo LU measurements, the impact of Hadamard multiplexing on the spatial resolution, due to the observed artifacts (Fig. 15), is not a fundamental limitation of the technique. In this first implementation, coupling the detection laser light coming from all the elements into the optical fiber was somewhat difficult to achieve due to aberrations of the collecting optical system and to the high aspect ratio of the scan line. This could however be overcome by using a square distribution of the elements instead of a linear distribution, as explained in the next subsection.
Replacing a physical mask with a spatial light modulator (SLM), a digital micro-mirror device (DMD) or a deformable mirror would clearly be an interesting alternative by limiting the losses, by allowing the correction of the intensities of the elements and by eliminating the need for the scanning of a physical mask.
5.3 Geometrical considerations
The proof-of-principle experiment described here was based on a linear augmented Hadamard mask for simplicity. However, the elements could be grouped in a square configuration to obtain large generation and detection spots with sub-elements determining the final resolution. Grouping elements in a square configuration would facilitate the optical design of the system and, in particular, the fiber optic coupling of the detection laser light.
For pulse-echo LU measurements, grouping the elements of the Hadamard masks in a square configuration would also reduce the crosstalk zone due to the concomitant propagation of Rayleigh and bulk waves. The crosstalk zone can be effectively limited by folding the scan line [6]. If we suppose a linear mask of N square elements of width w, the length of the scan line is l = N w and the scanned surface area is A = N w2. If the line is folded to get a square area, the width of the square area is w' = N1/2 w, which corresponds to a diagonal length d = (2N)1/2 w. The maximum length leading to crosstalk by Rayleigh waves is thus reduced by a factor l / d = (N/2)1/2. This means that, in the experiment considered here (ignoring the fact that 31 is not a square number), a factor of about (31/2)1/2 ≅ 4 could have been obtained, thus reducing the crosstalk zone from the observed ~2 mm depth to only ~0.5 mm depth. A square distribution of the elements, by allowing a better optical coupling of all detection beam elements into the optical fiber, would also reduce artifacts in images retrieved from Hadamard measurements. However, the approach based on a sliding augmented Hadamard mask would no more be directly applicable. Again, replacing a physical mask with a SLM, a DMD or a deformable mirror would be a more versatile alternative.
6. Conclusions
Results presented herein give a first experimental demonstration of the benefits of Hadamard multiplexing in LU. We have shown that a multiplexing or Fellgett advantage, in terms of SNR, is obtained by modulating the spatial intensity distribution of the generation laser beam. In contrast to many other applications of Hadamard multiplexing, the multiplexing advantage is observed in shot-noise-limited detection regime.
We have also shown that modulating the spatial intensity distribution of the detection laser beam leads to a slight multiplexing disadvantage in shot-noise-limited detection regime. This disadvantage could, however, be eliminated by using phase Hadamard modulation of the detection laser beam instead of amplitude Hadamard modulation.
We have also shown that Hadamard multiplexing is compatible with SAFT reconstruction for pulse-echo LU measurements when generation and detection laser beams are multiplexed simultaneously. In this case, however, the best approach would be the combination of amplitude multiplexing of the generation laser beam and phase multiplexing of the detection laser beam.
In LU, the power scaling required by the multiplexing approach will limit the number N of elements of the Hadamard masks and, consequently, the achievable multiplexing advantage. Experimental setups used in this first demonstration would gain in simplicity and versatility by using a SLM, a DMD or a deformable mirror to eliminate beam shapers and physical Hadamard masks. Even though this first demonstration was done using a CFPI, the same procedure should be applicable with a PRI. Combining Hadamard multiplexing with LU is attractive for high resolution applications on low laser-damage-threshold materials. The present method also keeps major advantages of LU such as remote generation and detection of ultrasound on unprepared surfaces which are not required to be flat.
Acknowledgments
The referees and Daniel Lévesque are gratefully acknowledged for their careful reading of the manuscript.
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