Pulsed laser source digital holography efficiency measurements

Steven A. Owens; Mark F. Spencer; Mark F. Spencer; Douglas E. Thornton; Glen P. Perram

doi:10.1364/AO.453344

1. INTRODUCTION

Digital holography (DH) systems use the interference of light to boost a weak signal above the noise floor of a camera. They do so with the use of a strong reference. In turn, DH systems provide access to robust estimates of the complex optical field [1]. These benefits make DH systems advantageous in long-range imaging scenarios [2–9]. Such scenarios are often plagued with deep turbulence conditions and low SNRs [10–13], which limit the effective ranges of DH systems. Thus, it is convenient to quantify system performance in terms of the total system efficiency. This efficiency, in practice, is comprised of “component efficiencies” that speak to the individual sources that cause SNR loss.

Past efforts to quantify these aforementioned component efficiencies made use of continuous-wave (CW) laser sources and visible cameras [14–17]. Recall that with CW laser sources, the coherence length also limits the effective ranges of DH systems since the system performance depends on the interference between the signal and reference. In turn, the temporal-coherence requirements for analog holography systems were evaluated in the 1960s (shortly after the invention of the laser) [18]. Also recall that, with DH systems, the interference between the signal and reference is detected and digitized by the camera pixels, adding additional considerations, especially for applications involving atmospheric turbulence [19]. As such, recent work quantified the temporal coherence efficiency for a DH system with a phase-modulated CW laser source and a visible camera [15]. The results show that small changes in the temporal coherence between the signal and reference can drastically change the total system efficiency. This outcome is less concerning for laboratory applications like microscopy and medical imaging [20–23], but is most concerning when using DH systems for field applications like long-range imaging [2–9].

Using pulsed laser sources, as opposed to CW laser sources, introduces additional considerations for DH systems, such as the temporal delay between the signal and reference pulses, since the interference of light in this case requires that the pulses overlap in time. While pulsed laser sources have been used in the microscopy and medical-imaging communities since at least the late 1990s [23–25], there has been little published quantification of the system performance in terms of the total system efficiency. This paper addresses this shortcoming by formulating a new component efficiency—one that accurately characterizes system performance as a function of temporal delay between the signal and reference pulses. This new efficiency is formulated from the ambiguity function, which is well known within the radar community [26,27]. Consequently, this new efficiency is referred to here as the “ambiguity efficiency.”

It is important to note that this paper uses a DH system with a 1064 nm pulsed laser source and a short-wave IR (SWIR) camera. In effect, SWIR wavelengths provide better transmission through the atmosphere [26,28,29]; thus, this switch in wavelength (compared to previous experiments [14–17]) moves DH systems that much closer to being ready to be used in the field. It also presents new challenges that were previously not an issue at visible wavelengths.

The results of this paper ultimately show that the total system efficiency and its component efficiencies accurately characterize system performance. They do so even with multimode behavior from the pulsed laser source and substantial dark current noise from the SWIR camera. For example, at a zero path-length difference (ZPD) between the signal and reference pulses, the results show that the measured total system efficiency is 15.9%. Such results are consistent with those previously obtained with a 532 nm CW laser source and a visible camera [14]. In addition, the results show that, as a function of temporal delay between the signal and reference pulses, the total system efficiency is accurately characterized by the ambiguity efficiency. To the best of our knowledge, such results are novel, yet build on previous experiments in a meaningful way [14–17].

Section 2 formulates expressions for the SNR in terms of the total system efficiency and its component efficiencies, including the ambiguity efficiency. Section 3 then discusses the data collection and processing to obtain measured values for the total system efficiency and its component efficiencies. In Section 4, these measured values are compared to predicted values in two ways: (1) at ZPD between the signal and reference pulses and (2) as a function of the temporal delay between the signal and reference pulses. The paper concludes in Section 5 with a summary of these comparisons.

2. THEORETICAL SETUP

Multiple DH recording geometries exist and, in practice, each has its own benefits [1,10–12,19]. Because of the simplicity of its setup, the off-axis image plane recording geometry (IPRG) was used in this paper [1,10]. An example of the off-axis IPRG is shown in Fig. 1.

Fig. 1. Example of a DH system in the off-axis image plane recording geometry.

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For the off-axis IPRG, the laser light from a master oscillator is split into two optical paths. One path creates a signal by flood illuminating an optically rough, extended object. As shown in Fig. 1, the scattered signal light is then collected by a lens and imaged onto the focal plane array (FPA) of a camera. The other path creates a reference by flood illuminating the FPA of a camera. By means of a local oscillator (LO), the strong reference light is injected off axis relative to the pupil of the resulting imaging system.

It is important to note that the SNR of a DH system in the off-axis IPRG is dominated by the total system efficiency. This efficiency can be seen in the following SNR formulation and in greater detail elsewhere [1,10]. In practice, the total system efficiency is the product of many component efficiencies, each of which quantifies a source of degradation in terms of system performance. Many component efficiencies have been developed and analyzed for DH systems with CW laser sources [14–17,19], but a new efficiency, called the ambiguity efficiency, is required to fully characterize DH systems with pulsed laser sources. Thus, the ambiguity efficiency is formulated from the ambiguity function in this section.

A. SNR

This paper uses the power definition of the SNR [1,10], such that

(1)$${\rm{SNR}}\!\left({x,y} \right) = {\eta _{\rm{tot}}} \!\left({x,y} \right)\frac{{4q_I^2}}{\pi}\frac{{{{\bar m}_S} \!\left({x,y} \right){{\bar m}_R}}}{{{{\bar m}_S} \!\left({x,y} \right) + {{\bar m}_R} + \sigma _n^2}},$$

where $({x,y})$ are the estimated image plane coordinates, ${\eta _{\rm{tot}}}$ is the total system efficiency, ${q_I}$ is the image plane sampling quotient, ${\bar m_S}$ is the per-pixel mean number of signal photoelectrons, ${\bar m_R}$ is the mean number of reference photoelectrons, and $\sigma _n^2$ is the camera noise variance, which is comprised of various camera noise sources, such as read noise and dark current noise.

Assuming the use of a strong reference, ${\bar m_R} \gg {\bar m_S} + \sigma _n^2$. In turn, Eq. (1) simplifies into [1]

(2)$${\rm{SNR}}\!\left({x,y} \right) = {\eta _{\rm{tot}}} \!\left({x,y} \right)\frac{{4q_I^2}}{\pi}{\bar m_S}\!\left({x,y} \right).$$

Such an expression says that the DH system is operating in a shot-noise-limited regime [1,17].

B. Total System Efficiency

Ideally, the total system efficiency ${\eta _{\rm{tot}}}$ is comprised of several multiplicative terms. Thus, it is assumed that no coupling exists between the various component efficiencies that make up the total system efficiency. Each multiplicative term is then an independent source for SNR loss, such that

(3)$${\eta _{\rm{tot}}} \!\left({x,y,\tau} \right) = {\eta _{\rm{ern}}}{\eta _{\rm{snl}}}\!\left({x,y} \right){\eta _{\rm{mix}}}\!\left(\tau \right),$$

where ${\eta _{\rm{ern}}}$ is the excess reference noise efficiency, ${\eta _{\rm{snl}}}({x,y})$ is the shot noise limit efficiency, ${\eta _{\rm{mix}}} (\tau)$ is the mixing efficiency, and $\tau$ is the temporal delay between the centers of the signal and reference pulses. It should be noted other component efficiencies could exist; however, they are beyond the scope of the present analysis.

The excess reference noise efficiency, ${\eta _{\rm{ern}}}$, is a measure of the excess noise present in the reference [14,17]. If the reference was perfectly uniform (with no excess amplitude noise [30]), then ${\eta _{\rm{ern}}} = 100\%$. For simplicity, this paper assumes that ${\eta _{\rm{ern}}} = 100\%$ for predicted values of ${\eta _{\rm{ern}}}$.

Fig. 2. Overview of the experimental setup.

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The shot noise limit efficiency, ${\eta _{\rm{snl}}}({x,y})$, is a quantification of the strong reference assumption made in Eq. (2) [14,17]. In a perfect experimental setup, ${\eta _{\rm{snl}}}({x,y}) = 100\%$ but, in practice, excess signal noise caused by the pupil-autocorrelation term in the Fourier plane [see Fig. 3(b)], as well as camera noise, degrades this component efficiency. Thus, this paper assumes that

(4)$${\eta _{\rm{snl}}}\!\left({x,y} \right) = \frac{{{{\bar m}_R}}}{{{{\bar m}_R} + {{\bar m}_S}\!\left({x,y} \right) + \sigma _n^2}}$$

for predicted values of ${\eta _{\rm{snl}}}$.

The mixing efficiency, ${\eta _{\rm{mix}}} (\tau)$, is a measure of how well the signal and reference pulses interfere as a function of $\tau$. For DH systems using pulsed laser sources,

(5)$${\eta _{\rm{mix}}} \!\left(\tau \right) = {\eta _{{\rm pol}}} {\eta _{\rm{mod}}} {\eta _{{\rm amb}}}\!\left(\tau \right),$$

where ${\eta _{\rm{pol}}}$ is the polarization efficiency [14], ${\eta _{\rm{mod}}}$ is the modulation efficiency [19], and ${\eta _{\rm{amb}}}(\tau)$ is the novel ambiguity efficiency.

For optically rough, extended objects with dielectric substrates, this paper assumes that ${\eta _{{\rm pol}}}$ is 50% because only half of the completely unpolarized scattered signal light interferes with the completely polarized strong reference light.

As discussed in [19], ${\eta _{\rm{mod}}}$ is a measure of how accurately the interference between the signal and reference is detected and digitized by the camera pixels. In this paper, ${\eta _{\rm{mod}}}$ is calculated using the square pixel modulation transfer function, viz

(6)$${\eta _{\rm{mod}}} = \left\langle {P\!\left({{f_x} - \alpha ,{f_y} - \beta} \right){\sin}{{\rm{c}}^2}\!\left({p {f_x},p {f_y}} \right)} \right\rangle ,$$

where $\langle \cdot \rangle$ denotes spatial average, $P$ is a shifted pupil filter function, $({{f_x},{f_y}})$ are the Fourier plane coordinates, $({\alpha ,\beta})$ are the shifts, and $p$ is the square pixel width. In this paper, ${\rm{sinc}}(x) = 1$ when $x = 0$ and ${\rm{sinc}}(x) = {{\sin ({\pi x})/({\pi x})}}$ otherwise [31,32]. Additionally, $({\alpha = 0.196,\beta = 0.198})$ in units of inverse pixels and $p = 15\;{\rm{\unicode{x00B5}{\rm m}}}$. These properties, along with an assumed 100% pixel fill factor, results in ${\eta _{\!\rm{\bmod}}} = 75\%$.

The ambiguity efficiency, ${\eta _{\rm{amb}}}(\tau)$, is a measure of the coherence between the signal and reference pulses and is formulated from the ambiguity function, $\chi (\tau)$. In practice [26,27],

(7)$$\begin{split}\chi \!\left({\tau ,{\nu _D}} \right) &= \int_{- \infty}^\infty {U_R}\!\left(t \right)U_S^*\!\left({t - \tau} \right){e^{j2\pi {\nu _D} \tau}}{\rm{d}}t \\&= \int_{- \infty}^\infty \tilde U_R^*\!\left(\nu \right){\tilde U_s}\!\left({\nu - {\nu _D}} \right){e^{- j2\pi \nu \tau}}{\rm{d}}\nu ,\end{split}$$

where ${\nu _D}$ is the Doppler frequency shift (caused by a moving object), ${U_R}$ and ${U_S}$ are the complex optical fields of the reference and signal pulses (in the temporal domain), respectively, $t$ is time, ${\tilde U_R}$ and ${\tilde U_S}$ are the complex optical fields of the reference and signal pulses (in the frequency domain), respectively, and $\nu$ is frequency. However, this paper assumes that ${\nu _D} = 0$ because the optically rough, extended object is stationary. Therefore, $\chi ({\tau ,{\nu _D}})$ can be simplified to $\chi (\tau)$.

In accordance with the power definition of the SNR [see Eq. (1)], ${\eta _{\rm{amb}}}(\tau) = {| {\chi (\tau)} |^2}$. Thus, ${\eta _{\rm{amb}}}(\tau)$ simplifies to

(8)$$\begin{split}{\eta _{\rm{amb}}}\!\left(\tau \right) &= {\left| {\int_{- \infty}^\infty {U_R}\!\left(t \right)U_S^*\!\left({t - \tau} \right){\rm{d}}t} \right|^2} \\&= {\left| {\int_{- \infty}^\infty \tilde U_R^*\!\left(\nu \right){{\tilde U}_s}\!\left(\nu \right){e^{- j2\pi \nu \tau}}{\rm{d}}\nu} \right|^2}.\end{split}$$

This formulation assumes that there is spatial uniformity in the signal and reference pulses.

3. EXPERIMENTAL SETUP

As shown in Fig. 2, the experimental setup made use of a 1064 nm pulsed laser source and a SWIR camera to create a DH system in the off-axis IPRG. The goal of this experimental setup was to measure the total system efficiency: (1) at the ZPD between the signal and reference pulses and (2) as a function of temporal delay between the signal and reference pulses. In turn, a custom-built NP Photonics coherent high-energy pulsed fiber laser system was used as the pulsed laser source [33].

The pulsed laser source was comprised of a CW seed laser with a vendor-specified 1064 nm center wavelength and 5 kHz linewidth, high-speed phase and intensity modulators to carve out pulses from the CW seed laser, and multiple ytterbium-doped fiber amplifier stages. Together, this configuration produced 10 ns pulses with average energies of 10 µJ. The average energy per pulse fluctuated ${\pm} 16\%$ over a 15-min time period, which was the amount of time required to collect one dataset.

As shown in Fig. 2, the pulse train from the pulsed laser source was sent through a half-wave plate before entering a polarized beam splitter (PBS). This PBS split the pulse train into reference and signal pulses. The average energy per pulse associated with the reference pulses was controlled by the half-wave plate before the PBS, while the average energy per pulse associated with the signal pulses was further reduced by a continuously variable neutral density filter with optical density values from 0.04 to 4.0. Such a filter allowed the experimental setup to avoid saturation of the camera pixels.

Fig. 3. Frame demodulation example. Here, the data processing involves: (a) the recorded digital hologram frame, (b) the associated Fourier plane, and (c) the associated image plane.

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After the PBS, the signal pulses were sent through an optical trombone to control the amount of temporal delay between the signal and reference pulses at the camera. Thereafter, the signal pulses were expanded by a 3x beam expander and scattered off a sheet of Labsphere Spectralon (i.e., the stationary, optically rough, extended object) with a vendor-specified 99% Lambertian reflectivity, depolarizing the light. This unpolarized, scattered-signal light was then imaged onto a camera with a 2.54 cm diameter lens, which gave rise to a circular pupil. The object and image distances were set to obtain a measured image-plane sampling quotient, ${q_I}$, of 3.35 [1,10]. By definition, ${q_I}$ represents the number of circular-pupil diameters that can fit across the Fourier plane.

The reference pulses were sent through a second half-wave plate and fiber-coupling optics. This second half-wave plate aligned the linear polarization of the reference pulses with the slow axis of a 3 m polarization-maintaining, single-mode optical fiber. In accordance with the off-axis IPRG, the back-end tip of this fiber was placed next to the imaging lens. Thereafter, the reference pulses flood illuminated the FPA of the camera.

An Allied Vision Goldeye G-033 SWIR TEC1 was used for the camera. This camera had a vendor-specified pixel well depth of 25,000 photoelectrons (pe), a pixel width of 15 µm, and a quantum efficiency of 77% at 1064 nm. This camera also had a measured unstable gain region for integration times less than 25 µs, which resulted in more than 25% of the pixel well depth being filled by dark current noise. In turn, the experiment was set up for the signal and reference pulses to arrive near the 27 µs integration time mark with a total frame integration time of 30 µs. The dark current noise was still the dominant factor in the camera noise variance, $\sigma _n^2$. In total, $\sigma _n^2 = 6415\;{\rm{p}}{{\rm{e}}^2}$.

To avoid saturation of the camera pixels, the mean number of reference and signal photoelectrons were set such that ${\bar m_{\!R}} = 11,784\;{\rm{pe}}$ and ${\bar m_{\!S}} = 88\;{\rm{pe}}$. Because the camera noise variance was over half the mean number of reference photoelectrons generated (i.e., $\sigma _n^2 \gt 1/2{\bar m_R}$), the strong reference assumption made in Eq. (2) was not valid. As such, the DH system used in this experiment was not operating in a shot noise limited regime [1,17]. By design, the total system efficiency ${\eta _{\rm{tot}}}$ accounted for this shortcoming with its component efficiencies; in particular, the shot noise limit efficiency ${\eta _{\rm{snl}}}$ [see Eq. (4)].

A. Data Collection and Processing

Data collection occurred for temporal delay values from $\tau = - 6.5\;{\rm{ns}}\;{\rm{to}}\;{{+ 4.5}}\;{\rm{ns}}$ in 1 ns increments and from $\tau = - 1.2\;{\rm{ns}}\;{\rm{to}}\; + 0.5\;{\rm{ns}}$ in 0.1 ns increments. The negative values of $\tau$ correspond to a delay of the reference pulse with respect to the signal pulse. Conversely, the positive values of $\tau$ correspond to a delay of the signal pulse with respect to the reference pulse. For each increment of $\tau$, the Labsphere Spectralon sheet was rotated to generate 10 distinct speckle realizations (for averaging during data processing). Furthermore, for each speckle realization, 10 digital hologram frames, 10 signal-only frames, and 10 reference-only frames were collected, totaling 300 frames for each dataset (i.e., 30 total frames per speckle realization, for 10 speckle realizations). After the datasets were collected, 100 background frames were also taken, so that the background and camera noise could be appropriately accounted for during efficiency calculations.

The aforementioned frames were imported to Matlab for data processing. The first step was frame demodulation. Figure 3 shows an example using a DH frame.

The real-valued DH frame in Fig. 3(a) underwent a discrete inverse Fourier transform to obtain the associated complex-valued Fourier plane in Fig. 3(b). In accordance with the off-axis IPRG, the Fourier plane contained four distinct terms:

(1) The signal term (the data in the top-right circular pupil);
(2) The complex conjugate signal term (the data in the bottom-left circular pupil);
(3) The pupil autocorrelation term (the circularly symmetric data centered at DC); and
(4) The LO autocorrelation term (the noncircularly symmetric data centered at DC).

Provided (1)–(4), a pupil-filter function was used to filter the desired signal term. To complete the frame demodulation, the filtered data was first centered in the Fourier plane, then subsequently underwent a discrete Fourier transform to obtain the associated complex-valued image plane in Fig. 3(c).

Frame demodulation was performed on each individual frame to avoid any piston phase mismatch introduced on a frame-to-frame basis. The energies, or square magnitudes, of the demodulated frames in ${\rm pe}^2$ were subsequently calculated in accordance with the power definition of the SNR [see Eq. (1)]. Then, the mean of all 100 demodulated energy frames was computed for each pulse delay, $\tau$, to produce an average demodulated energy frame. This process was repeated for the collected signal-, reference-, and background-only frames using the same pupil-filter function that was used for the DH frames. Doing so ensured the noise collected by the hologram frames was appropriately accounted for within the analysis.

B. Measured Total System Efficiency

The average energy frames at each pulse delay, $\tau$, were used to calculate the measured total system efficiency and its component efficiencies. For this purpose, it is convenient to define the following quantities:

(9)$$\begin{split}{E_H}\!\left({x,y,\tau} \right) &= \left[{{{\bar m_R^\prime}}\!\left({x,y,\tau} \right) - {{\bar m_B^\prime}}\!\left({x,y,\tau} \right)} \right]\\&\quad \times \left[{{{\bar m_S^\prime}}\!\left({x,y,\tau} \right) - {{\bar m^\prime}_B}\!\left({x,y,\tau} \right)} \right],\end{split}$$

(10)$${E^\prime _N}\!\left({x,y,\tau} \right) = {E^\prime _{D - R}}\!\left({x,y,\tau} \right) + {E^\prime _{D - S}}\!\left({x,y,\tau} \right) - {E^\prime _{D - B}}\!\left({x,y} \right),$$

and

(11)$${E^\prime _H}\!\left({x,y,\tau} \right) = {E^\prime _{D - H}}\!\left({x,y,\tau} \right) - {E^\prime _N}\!\left({x,y,\tau} \right),$$

where ${E_H}$ is the hologram energy; ${\bar m_R^\prime }$, ${\bar m_S^\prime }$, and ${\bar m_B^\prime }$ are the measured mean number of reference, signal, and background photoelectrons, respectively; ${E_N^\prime }$ is the measured noise energy; ${E^\prime _{D - R}}$, ${E^\prime _{D - S}}$, and ${E^\prime _{D - B}}$ are the measured reference, signal, and background energies after frame demodulation, respectively; ${E^\prime _H}$ is the measured hologram energy; and ${E^\prime _{D - H}}$ is the measured hologram energy after frame demodulation. Note that for Eqs. (9)–(11), the dependence on $\tau$ is caused by pulse energy fluctuations within the datasets, not the pulse delay itself. Also note that the substantial dark current noise from the SWIR camera is accounted for with $\bar m_B^\prime $ and $E_{D - B}^\prime$. In practice, both ${\bar m_R^\prime }$ and ${\bar m_S^\prime }$ contain this dark current noise, which must be removed with background subtraction to calculate the desired energies in Eqs. (9) and (10). This background subtraction is explicit in Eq. (9). For Eq. (10), the measured noise energy is the sum of the reference, signal, and background energies. Since the measured reference and signal energies each contain the measured background energy, it is only subtracted once in Eq. (10).

Using Eqs. (9)–(11), the measured total system efficiency, as well as the measured excess reference noise, shot noise limit, and mixing component efficiencies, respectively, were quantified as

(12)$$\begin{split}{\eta ^\prime _{\rm{tot}}}\!\left(\tau \right) &= \left\langle {\frac{{{\rm{SNR^\prime}}\!\left({x,y,\tau} \right)}}{{{\rm{SNR}}\!\left({x,y,\tau} \right)}}} \right\rangle \\&= \frac{\pi}{{4 q_I^2}}\left\langle {\frac{{{{E_H^\prime}}\!\left({x,y,\tau} \right)/{{E_N^\prime}}\!\left({x,y,\tau} \right)}}{{{{\bar m_S^\prime}}\!\left({x,y,\tau} \right) - {{\bar m_B^\prime}}\!\left({x,y,\tau} \right)}}} \right\rangle ,\end{split}$$

(13)$${\eta ^\prime _{\rm{ern}}}\!\left(\tau \right) = \frac{\pi}{{4 q_I^2}}\frac{{\left\langle {{{\bar m^\prime}_R}\!\left({x,y,\tau} \right) - {{\bar m_B^\prime}}\!\left({x,y,\tau} \right)} \right\rangle}}{{\left\langle {{{E^\prime_{D - R}}}\!\left({x,y,\tau} \right) - {{E^\prime_{D - B}}}\!\left({x,y,\tau} \right)} \right\rangle}},$$

(14)$${\eta ^\prime _{\rm{snl}}}\!\left(\tau \right) = \frac{{\left\langle {{{E^\prime_{D - R}}}\!\left({x,y,\tau} \right) - {{E^\prime_{D - B}}}\!\left({x,y} \right)} \right\rangle}}{{\left\langle {{{E^\prime_N}}\!\left({x,y,\tau} \right)} \right\rangle}},$$

and

(15)$${\eta ^\prime _{\rm{mix}}}\!\left(\tau \right) = \left\langle {\frac{{{{E^\prime_H}}\!\left({x,y,\tau} \right)}}{{{E_H}\!\left({x,y,\tau} \right)}}} \right\rangle .$$

As with Eqs. (9)–(11), Eqs. (12)–(15) gain a dependence on $\tau$ due to pulse energy fluctuations, not the temporal delay itself. The $\pi /4q_I^2$ term in Eq. (14) is necessary to account for the ratio of the pupil filter function area to the total Fourier plane area [1,10]. In accordance with the off-axis IPRG, ${q_I}$ accounts for the portion of the noise that is filtered from the Fourier plane by the pupil filter function.

4. RESULTS

This section compares the measured values [see Eqs. (9)–(15) in Section 3] to the predicted values [see Eqs. (3)–(8) in Section 2] for the total system efficiency and its component efficiencies. It does so in two ways: (1) at ZPD between the signal and reference pulses and (2) as a function of temporal delay between the signal and reference pulses. Even with multimode behavior from the pulsed laser source and substantial dark current noise from the SWIR camera, the results ultimately show that: (1) at the ZPD, they are consistent with those previously obtained with a 532 nm CW laser source and visible camera [14], and (2) as a function of temporal delay, they are accurately characterized by the ambiguity efficiency.

A. Results at ZPD

The results at the ZPD with the 1064 nm pulsed laser source and SWIR camera were compared to those previously obtained with a 532 nm CW laser source and visible camera [14]. This comparison can be found in Table 1, which also shows the measured values that were obtained using Eqs. (9)–(15) compared to the predicted values that were obtained using Eqs. (3)–(8) for the total system efficiency and its component efficiencies. It is important to note that the ambiguity efficiency, ${\eta _{\rm{amb}}}$, is 100% at the ZPD, so it is not included as a component efficiency in Table 1.

Table 1. Comparison of Measured and Predicted Values for the Total System Efficiency and Its Component Efficiencies at ZPD

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As shown in Table 1, the measured values for the total system efficiency and its component efficiencies were lower with the 1064 nm pulsed laser source and SWIR camera than those previously obtained with a 532 nm CW laser source and visible camera [14]. These differences were attributed to the multimode behavior from the pulsed laser source and substantial dark current noise from the SWIR camera.

Appendix A uses pulse diagnostic measurements to show that the 1064 nm pulsed laser source produced at least two modes within the reference and signal pulses. Recall that with the off-axis IPRG, the reference pulses diverge onto the FPA of the camera, whereas the signal pulses converge (see Fig. 1). Therefore, the multiple modes in the signal and reference pulses were not aligned at the point of detection and digitization by the camera pixels. This modal mismatch degraded the interference between the signal and reference, and contributed to the measured mixing efficiency being 8.3% less than the predicted mixing efficiency. Additionally, a pixel fill factor of 100% was assumed for the predicted mixing efficiency. If the fill factor was less than this ideal value, then the measured mixing efficiency would decrease accordingly.

Most SWIR cameras, including the one used in this paper, use indium gallium arsenide FPAs. These FPAs have inherently different noise properties than the silicon FPAs typically used in visible cameras. For example, in the CW experiment reported previously [14], the visible camera had a measured camera-noise variance of $5.5 \; {\rm pe}^2$ compared to a vendor-specified well depth of 10,482 pe. In the pulsed experiment reported here, the SWIR camera had a measured camera-noise variance of $6415 \; {\rm pe}^2$ compared to a vendor-specified well depth of 25,000 pe. As a result, the DH system in the CW experiment was operating in a shot noise limited regime [1,17], whereas the DH system in the pulsed experiment was not (i.e., both the measured and predicted shot noise limit efficiencies were significantly less than 100% for the DH system in the pulsed experiment).

The 15.4% increase between the measured and predicted values for the shot noise limit efficiency was primarily caused by the effects of excess reference noise. Recall that such effects result from a nonuniform reference. Also recall that the predicted values obtained using Eq. (4) assume that only shot noise from the reference is present, whereas the measured values obtained using Eq. (14) also takes into account the effects of excess reference noise. As such, when measured values for the excess reference noise and shot noise limit efficiencies are multiplied together, the 53.3% obtained in the pulsed experiment reported here is less than the 74.5% obtained in the previously reported CW experiment. This outcome is due to substantial dark current noise from the SWIR camera, which leads to the aforementioned increase in camera noise variance. A secondary cause for the 15.4% increase was probably due to the previously discussed modal mismatch between the signal and reference pulses. In general, this modal mismatch was not accounted for in the theoretical and experimental setups.

Despite the multimode behavior from the pulsed laser source and substantial dark current noise from the SWIR camera, the predicted values for the total system efficiency were within the uncertainty bounds of the measured values. This outcome serves as an indication that no coupling existed between the various component efficiencies that make up the total system efficiency. In turn, the results at the ZPD were consistent with those previously made with a 532 nm CW laser source and a visible camera [14].

B. Results as a Function of Temporal Delay

The results as a function of temporal delay were first compared to the predicted values for the total system efficiency. This comparison can be found in Fig. 4, which shows the measured values, obtained using Eq. (12), to the predicted values, obtained using Eq. (3), for the total system efficiency. Here, the measured values, obtained using Eqs. (13)–(15), served as the component efficiencies in Eq. (3); i.e., ${\eta _{\rm{tot}}} = {\eta _{\rm{ern}}^\prime }{\eta _{\rm{snl}}^\prime }{\eta _{\rm{mix}}^\prime}$.

Fig. 4. Comparison of the measured (-$\blacklozenge$) and predicted (··×) values for the total system efficiency as a function of temporal delay (top), with residuals (•) and measured uncertainties ($\blacklozenge$) also as a function of temporal delay (bottom).

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As shown in Fig. 4, the residuals were less than the measured uncertainties at each temporal delay value. This outcome served as another indication that no coupling existed between the various component efficiencies that make up the total system efficiency. If a coupling between component efficiencies had been introduced within the experiment, then the total system efficiency would not be the multiplicative product of the component efficiencies and the residuals would have exceeded the measured uncertainties.

With the outcomes of Table 1 and Fig. 4 in mind, the results as a function of temporal delay were then compared to the predicted values for the ambiguity efficiency. This comparison can be found in Fig. 5, which shows the measured values, obtained using Eq. (12), to the predicted values, obtained using Eq. (B3) in Appendix B, for the total system efficiency. To formulate Eq. (B3) in Appendix B, the pulse diagnostic measurements from Appendix A were used to inform a multimode fit to the ambiguity efficiency formulated in Section 2 [see the right-most term in Eq. (8)]. For convenience, Eq. (B3) is repeated here, viz

(16)$${\eta _{\rm{amb}}}\!\left(\tau \right) = A_{1}^2 \Delta {\nu ^4}\exp \!\left({- 4\pi \Delta \nu \left| \tau \right|} \right) + A_{2}^2 \Delta {\nu ^4}\exp \!\left({- 4\pi \Delta \nu \left| \tau \right|} \right),$$

where ${A_{1}}$ is the amplitude of the fundamental mode, $\Delta \nu$ is the half width, half maximum of the assumed Lorentzian line shape, $\tau$ is again temporal delay, and ${A_2}$ is the amplitude of the transverse mode. See Table 2 in Appendix B for the multimode fit parameters. For simplicity, the spectrum was assumed to be comprised of two modes of the same width, and each mode was assumed to have a Lorentzian line shape. To characterize the total system efficiency as a function of $\tau$, the predicted values from Eq. (16) for the ambiguity efficiency were multiplied by the measured total system efficiency at the ZPD, as shown in Table 1.

Fig. 5. Comparison of the measured values for the total system efficiency (-$\blacklozenge$) to the predicted values for the total system efficiency (--) using a multimode fit to the ambiguity efficiency.

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Table 2. Multimode Fit Parameters

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As shown in Fig. 5, the measured values for the total system efficiency matched the predicted values for the total system efficiency well for $| \tau | \lt 1\;{\rm{ns}}$. There were many possibilities for what might have caused the overestimation for $| \tau | \gt 1\;{\rm{ns}}$. One strong possibility is attributed to improperly accounting for multimode effects in the predicted values for the ambiguity efficiency.

It is important to remember that Fabry–Perot interferometers have system line shapes so any measurement reported by these devices is the convolution of this system line shape and the spectral line shape of the incident pulse train. The model used in this paper assumed two modes existed, but the convolution with the Fabry–Perot interferometer line shape may have hidden spectral features, such as additional modes or phase interruptions present within the incident pulse train. Such features could have been masked by small misalignments while measuring the spectral line shape with the Fabry–Perot interferometer.

The Lorentzian line shapes used to represent the spectrum were also not completely accurate. For example, Fig. 8(b) in Appendix B shows how different spectral line shapes affect the ambiguity efficiency curve. A more accurate fit to the spectral data would most likely result in more accurate predicted values for the ambiguity efficiency, particularly for $| \tau | \gt 1\;{\rm{ns}}$. In addition, the modes were assumed to be independent from one another, but such independence was not verified in the experiment. Dependence between modes would have caused oscillations within the envelope of the predicted values for the ambiguity efficiency, and the depth of modulation within the envelope would have been proportional to the degree of dependence and the mode amplitudes.

The multimode pulsed laser source also resulted in a pulse train without perfect transverse coherence, meaning the spectral content in the signal and reference pulses were different across their respective wavefronts. Because the Fabry–Perot interferometer was a 1D measurement device, it did not captured this lack of transverse coherence. As mentioned concerning Eq. (8), spatial nonuniformity was not captured in the predicted values for the ambiguity efficiency and would negatively affect the predicted values for the total system efficiency. The variation in intensity is also greater than the mean intensity for multimode sources [30,34]. This last point was supported by the variation in measured total efficiency near $\tau = 0$ and could have caused a disproportionate decrease in SNR as the hologram energy approached the total noise floor of the camera.

5. CONCLUSION

In this paper, a 1064 nm pulsed laser source and a SWIR camera were used to measure the total system efficiency associated with a DH system in the off-axis image plane recording geometry. At the ZPD between the signal and reference pulses, the measured total system efficiency (15.9%) and its component efficiencies, including the excess reference noise efficiency (66.8%), shot noise limit efficiency (79.8%), and mixing efficiency (29.2%), were consistent with those previously obtained with a 532 nm CW laser source and a visible camera [14]. In addition, as a function of temporal delay between the signal and reference pulses, the total system efficiency was accurately characterized by a new component efficiency, which was formulated from the oft-used ambiguity function from the radar community. Even with multimode behavior from the pulsed laser source and substantial dark current noise from the SWIR camera, system performance was accurately characterized by the resulting ambiguity efficiency.

APPENDIX A: PULSE DIAGNOSTIC MEASUREMENTS

To inform the results presented in Section 4, pulse diagnostic measurements were obtained. For this purpose, Fig. 6 shows the temporal profile and spectral line shape of the pulses. The temporal profile, as measured using a Thorlabs DET08C photodetector, is shown in Fig. 6(a), and the spectral line shape, as measured using a Thorlabs SA200-8B scanning Fabry–Perot interferometer with a 7.5 MHz resolution and 1.5 GHz free spectral range, is shown in Fig. 6(b) [35,36].

The temporal profile in Fig. 6(a) showed that the pulsed laser source produced 10 ns pulses with tails that were 10 s of nanoseconds long. Moreover, the measured spectral line shape in Fig. 6(b) appeared to contain two spectral peaks spaced approximately 50 MHz apart with the right peak at approximately 80% the amplitude of the left peak. A comparison between the measured spectral line shape and the Fourier transform of the temporal profile indicated that the pulsed laser source did not produce Fourier transform limited pulses [37]. Simply put, the measured spectral line shape was wider than the Fourier transform of the temporal profile. For this reason, the spectral formulation [see the right-most term of Eq. (8)] was used in Appendix B to obtain predicted values for the ambiguity efficiency.

Fig. 6. Pulse diagnostic measurements of (a) temporal profile and (b) spectral line shape, where the measured spectral line shape (–) is compared to the Fourier transform of the temporal profile ($\text{-}\text{-}$).

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Fig. 7. Pulse diagnostic measurement of (a) the entire spatial profile and (b) the spatial profile with the fundamental mode removed.

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The multipeaked nature of the measured spectral line shape in Fig. 6(b) strongly indicated multimode operation within the pulsed laser source. In particular, the CW seed laser had a longitudinal mode spacing on the order of 3 GHz and a corresponding transverse mode spacing on the order of 1.5 GHz, both of which were significantly larger than the measured 50 MHz separation [38]. However, if the 1.5 GHz free spectral range of the Fabry–Perot interferometer was taken into account, the observed double-peaked line shape could have been the result of two overlapping scanned spectra. For example, if two subsequent Fabry–Perot interferometer scans were labeled “a” and “b,” the secondary mode of scan “a” was overlapped with the fundamental mode of scan “b.” To investigate the nature of this potential secondary mode, the pulse train was expanded and the spatial profile was visually inspected using a Xenics Xeva-FPA-1.7-320 camera. A sample spatial profile measurement (with a normalized scale after computing the square root of the raw camera data) is shown in Fig. 7.

As shown in Fig. 7(a), the spatial profile indicated the presence of at least two modes. This is confirmed in Fig. 7(b), which shows the spatial profile of the beam after the fundamental mode [assumed to be a Gaussian (0,0) mode] was removed. As such, the analysis suggested that the pulsed laser source produced a secondary mode consistent with a Laguerre–Gauss (1,0) mode with an astigmatic phase shift [39,40]. This determination was made after taking into account the measured spectrum in Fig. 4(b), the mode spacing of the CW seed laser, the free spectral range of the Fabry–Perot interferometer, and the spatial profile in Fig. 7(b).

APPENDIX B: MULTIMODE FIT TO THE AMBIGUITY EFFICIENCY

The pulse diagnostic measurements from Appendix A were used to inform a multimode fit to the ambiguity efficiency formulated in Section 2 [see the right-most term in Eq. (8)]. For simplicity, the spectrum was assumed to be comprised of two modes of the same width, and each mode was assumed to have a Lorentzian line shape, since the pulsed laser source was assumed to be phase noise dominated [41,42]. Using these assumptions, a linear least squares regression fit was performed on the spectral line shape [see Fig. 6(b)] to calculate a fitted equation with the adjusted ${\rm R}^2$ fit value of 0.985 and standard error of 3.8%. The right peak of the resulting equation was then shifted 1.5 GHz to the positive frequency side to account for the overlapping spectra and the free spectral range of the Fabry–Perot interferometer. As a result,

(B1)$$f\!\left(\nu \right) = {A_{1}}\frac{{\Delta {\nu ^2}}}{{{{\!\left({\nu - {{\bar \nu}_1}} \right)}^2} + \Delta {\nu ^2}}} + {A_2}\frac{{\Delta {\nu ^2}}}{{{{\!\left({\nu - {{\bar \nu}_2}} \right)}^2} + \Delta {\nu ^2}}},$$

with the multimode fit parameters listed in Table 2. It is important to note that Gaussian and Voigt line shapes were also considered, but did not fit as well as the assumed Lorentzian line shape. The multimode fit parameters predicted by Eq. (B1) are shown in Fig. 8(a).

Fig. 8. (a) The predicted multimode fit and (b) the predicted ambiguity efficiency assuming the Lorentzian spectral line shape (–), as well as the measured spectral line shape (-·) and a Gaussian baseline ($\text{-}\text{-}$).

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Substituting Eq. (B1) into the right-most term of Eq. (8), where $f(\nu) \approx \tilde U_R^*(\nu){\tilde U_s}(\nu)$, resulted in the following expression for the ambiguity efficiency:

(B2)$$\begin{split}{\eta _{\rm{amb}}}\!\left(\tau \right) &= A_{1}^2 \Delta {\nu ^4}\exp \!\left({- 4\pi \Delta \nu \left| \tau \right|} \right) + A_{2}^2 \Delta {\nu ^4}\exp \!\left({- 4\pi \Delta \nu \left| \tau \right|} \right)\\& \quad+ {A_{1}}{A_2} \Delta {\nu ^4}\exp \!\left({- i2\pi \tau \!\left({{\nu _1} - {\nu _2}} \right)} \right)\exp \!\left({- 4\pi \Delta \nu \left| \tau \right|} \right)\\&\quad + {A_{1}}{A_2} \Delta {\nu ^4}\exp \!\left({- i2\pi \tau \!\left({{\nu _2} - {\nu _1}} \right)} \right)\exp \!\left({- 4\pi \Delta \nu \left| \tau \right|} \right).\end{split}$$

For further simplicity, and in accordance with common practice, it was assumed the modes were statistically independent, meaning the modes did not interact with one another [30]. This assumption resulted in each peak from Eq. (B1) being independently substituted into the right-most term of Eq. (8) to generate independent terms. In turn,

(B3)$${\eta _{\rm{amb}}}\!\left(\tau \right) = A_{1}^2 \Delta {\nu ^4}\exp \!\left({- 4\pi \Delta \nu \left| \tau \right|} \right) + A_{2}^2 \Delta {\nu ^4}\exp \!\left({- 4\pi \Delta \nu \left| \tau \right|} \right).$$

As can be seen by comparing Eqs. (B2) and (B3), treating the modes as being statistically independent removes the complex cross terms. The multimode fit parameters to the ambiguity efficiency predicted by Eq. (B3) are shown in Fig. 8(b).

As shown in Fig. 8(b), Eq. (B3) was plotted alongside the predicted ambiguity efficiency assuming the measured spectral line shape [see Fig. 6(b)] and a baseline 10 ns, Fourier transform limited Gaussian pulse [37]. In general, the predicted ambiguity efficiency associated with the Lorentzian spectral line shape was narrower than that associated with the measured spectral line shape for $| \tau | \le 4.5$ ns and the Gaussian baseline for $| \tau | \le 12.5$ ns. At $| \tau | = 1$ ns, the predicted ambiguity efficiency associated with the Lorentzian spectral line shape was approximately 74%, which is nearly 20% worse than that predicted with the measured spectral line shape and 24% worse than the Gaussian baseline.

Acknowledgment

The authors would like to thank M. Akbulut for many insightful discussions regarding the results presented in this paper, as well as the Joint Directed Energy Transition Office for sponsoring this research.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	CW Source	Pulsed Source	Pulsed Source
Efficiency	ZPD,Measured	ZPD,Measured	ZPD, Predicted
Total System, $η_{t o t}$	$25.6 % \pm 6.3 %$	$15.9 % \pm 10.3 %$	24.2%
Excess Reference	$74.5 % \pm 2.0 %$	$66.8 % \pm 1.6 %$	100.0%
Noise, $η_{e r n}$
Shot Noise	$100 % \pm 0.0 %$	$79.8 % \pm 15.6 %$	64.4%
Limit, $η_{s n l}$
Mixing, $η_{m i x}$	$36.8 % \pm 10.2 %$	$29.2 % \pm 14.2 %$	37.5%

Variable	Symbol	Value	Units	95% Confidence Bounds
Amplitude, Fundamental Mode	$A_{1}$	0.88	A.U.	(0.869, 0.883)
Center Frequency, Fundamental Mode	${\bar{ν}}_{1}$	−4.26	MHz	(−4.48, −4.03)
Amplitude, Transverse Mode	$A_{2}$	0.57	A.U.	(0.56, 0.58)
Center Frequency, Transverse Mode	${\bar{ν}}_{2}$	1546.88	MHz	(1546.54, 1547.21)
Half-Width, Half Maximum	$Δ ν$	27.87	MHz	(27.60, 28.14)

	CW Source	Pulsed Source	Pulsed Source
Efficiency	ZPD,Measured	ZPD,Measured	ZPD, Predicted
Total System, $η_{t o t}$	$25.6 % \pm 6.3 %$	$15.9 % \pm 10.3 %$	24.2%
Excess Reference	$74.5 % \pm 2.0 %$	$66.8 % \pm 1.6 %$	100.0%
Noise, $η_{e r n}$
Shot Noise	$100 % \pm 0.0 %$	$79.8 % \pm 15.6 %$	64.4%
Limit, $η_{s n l}$
Mixing, $η_{m i x}$	$36.8 % \pm 10.2 %$	$29.2 % \pm 14.2 %$	37.5%

Variable	Symbol	Value	Units	95% Confidence Bounds
Amplitude, Fundamental Mode	$A_{1}$	0.88	A.U.	(0.869, 0.883)
Center Frequency, Fundamental Mode	${\bar{ν}}_{1}$	−4.26	MHz	(−4.48, −4.03)
Amplitude, Transverse Mode	$A_{2}$	0.57	A.U.	(0.56, 0.58)
Center Frequency, Transverse Mode	${\bar{ν}}_{2}$	1546.88	MHz	(1546.54, 1547.21)
Half-Width, Half Maximum	$Δ ν$	27.87	MHz	(27.60, 28.14)

Pulsed laser source digital holography efficiency measurements

Abstract

1. INTRODUCTION

2. THEORETICAL SETUP

A. SNR

B. Total System Efficiency

3. EXPERIMENTAL SETUP

A. Data Collection and Processing

B. Measured Total System Efficiency

4. RESULTS

A. Results at ZPD

B. Results as a Function of Temporal Delay

5. CONCLUSION

APPENDIX A: PULSE DIAGNOSTIC MEASUREMENTS

APPENDIX B: MULTIMODE FIT TO THE AMBIGUITY EFFICIENCY

Acknowledgment

Disclosures

Data availability

REFERENCES

Data availability

Cited By

Figures (8)

Tables (2)

Equations (19)

Applied Optics