3.1.

Signal and Noise Sources in Transient Absorption Microscopy

To interpret our results and understand the limits of adaptive noise canceling, we first outline the various signal and noise sources. This discussion is inspired by the detailed account of signal and noise in coherent Raman spectroscopy by Zhang et al.;¹⁵ our focus, rather than comparing SNR of different techniques, is to highlight signal and noise sources in terms of their handling by the signal processing chain. An additional resource for signal and noise considerations in transient absorption microscopy can be found in the review by Fischer et al.² We start by considering the optical signal and noise sources, adding electronic noise (amplifier, ADC) later. The photocurrent generated from the transmitted probe intensity through the sample, $i_0(t)$, can be expressed as

In the frequency domain, we can rewrite Eq. (1) as

We have neglected the product between probe RIN and the pump, as it is significantly weaker than the other terms. Only the third term contains the desired pump-probe signal, $\mathrm{\Sigma }(f,\tau )$, which is to be demodulated by the LIA. The first term $\propto T_s$ conveys the linear transmissivity, and its bandwidth depends on the line scan rate, diffraction-limited spot size, scan area, and the spatial frequency extents of the sample. This term can be rejected with large $f_m$ and lock-in detection, but for high-speed scanning, $T_s(f)$ can overlap with the lock-in band, resulting in residual background signals in the pump-probe measurement [see Figs. 6(j), 7, and related discussion]. The last term, $I_{\text{shot}}$, can be mitigated by having sufficient pump-probe power product for an acceptable signal-to-shot noise ratio:

The second term in Eq. (3), $R{P}_{\mathrm{RIN}}(f)\otimes T_s(f)$, usually handled by modulating the pump at a frequency where RIN is absent.^17,18 However, for fiber-based laser sources with non-$1/f$, broad-spectrum noise,¹³ it is impossible to find a lock-in detection band absent of RIN. When transmitted probe RIN leaks into the lock-in detection band, the net effect is to contaminate the transient absorption image with the linear transmitted light image (see Fig. 6). This effect worsens with scanning speed, as the increased lock-in detection bandwidth gathers more RIN. Effective RIN cancellation can be done with balanced detection^4,9,10 (referred to as ratiometric detection in low repetition rate experiments¹¹). To cancel this term with balanced detection, the reference must be modulated to match the time-varying sample transmissivity $T_s$, imposing a serious limitation on imaging speed without custom electronics.^4,19 Gambetta et al. utilized an unmodulated reference to achieve the conventional balanced detection for SRS signal enhancement. However, because of the persistence of the intensity mismatch between reference and signal, a 500-ms integration time is required to attain the desired sensitivity.¹⁹ This RIN term is what we seek to eliminate with digital ANC.

Next, we consider electronic noise sources. The built-in photodiode transimpedance amplifier converts the overall photocurrent $I_0(f)$ to voltage. The transimpedance amplifier and the analog-to-digital conversion contribute additional noise. Thus, in the frequency domain, the overall signal before the adaptive filter can be expressed as follows (assuming a $50\text{-}\mathrm{\Omega }$ load):

A summary of these processes is sketched in Fig. 2. After superposition of all the noise sources, the pump-probe signal, marked as red, becomes buried underneath the unwanted components.

A balanced detector can eliminate noise that is correlated between the signal and reference arms (common-mode noise) such as laser RIN. When the electronic noise floor is below the shot noise floor, a balanced detector will reduce the overall noise floor to 3 dB above the shot noise floor.²⁰ This limit arises from the separate shot noise contributions of the two detectors. Unlike the RIN, these add together at the balanced detector’s output because they are uncorrelated. Likewise, electronic noise (photodiode amplifier noise, ADC input noise) is also uncorrelated. Therefore, when the electronic noise floor is above the shot noise floor, a balanced detector will reduce the overall noise floor to 3 dB above the electronic noise floor. We take this theoretical limit into consideration when tuning the adaptive filter’s performance.

3.2.

Digital Adaptive Filtering

A digital adaptive filter, in the configuration shown in Fig. 1, searches for filter coefficients that transform the reference, $x\to y$, to match the desired probe signal, $d$.²¹ In this work, we use an LMS adaptive filter, which is an FIR filter that adjusts its coefficients to minimize mean-square error between $d$ and $y$ through gradient descent (we refer readers to any textbook on adaptive signal processing, e.g., Ref. 22, for details). In a noise canceling configuration, $d$ is a noisy signal and $x$ is an independent sampling of the noise. The filter output $y$ estimates the noise component of $d$ and subtracts the two to produce an estimate of the underlying signal, $e$. This technique has been widely used in areas such as telephone echo cancellation, noise cancellation, equalization of communication channels, active noise control, and adaptive control systems.²² In our experiment, the laser RIN common to the probe ($d$) and reference ($x$). is correlated, allowing the adaptive filter to produce an estimate of the laser RIN in the probe ($d$) for direct cancellation.

Fig. 2

Signal and noise processes in transient absorption microscopy: (a) DC, 1 MHz, and RIN convolve with sample structural baseband to produce the 1-MHz pump-probe signal contaminated by laser RIN. (b) The superimposed electronic noise from transimpedance amp of the photodetector further degrades SNR. (c) Adding ADC electronic noise floor directly to generate the overall noise floor. The overall noise floor prohibits the shot noise-limited detection.

Figure 3 shows typical inputs and outputs the digital adaptive filter. Figure 3(a) shows the measured reference $x(n)$ and probe $d(n)$ a line scan across BGO crystal particles, covering both forward and backward motion of the resonant scanner. Figure 3(b) shows that the adaptive filter’s transformation of the reference $x(n)$ to $y(n)$ tracks the probe amplitude $d(n)$. The difference, $e(n)$, contains the pump-probe signal, with RIN canceled, plus some residual tracking error. Figure 3(c) plots $e(n)$ with time on a logarithmic scale, showing convergence in about $1\mu \mathrm{s}$.

Fig. 3

(a) Acquired reference and probe signals from one line scan across the BGO particles. (b) The adaptive filer alters $x$ to become $y$, tracking the amplitude of $d$. Subtraction of $y$ from $d$ produces the adaptive filter output, $e$. (c) Plot of $e$ with respect to logarithmic time scale.

For a given adaptive filter length $L$, we select a step size $\mu$ such that the noise floor of $e=y-d$ is at 3 dB above the electronics noise floor (the theoretical limit for an electronic noise-limited balanced detector, as discussed in Sec. 3.1), which we estimate from the power spectrum at $f>15\mathrm{MHz}$, above the photodetector cutoff. This turns out to be $\mu \approx 0.8/L$ for our conditions. As can be seen in Fig. 4, larger $\mu$ has an advantage in tracking transmitted probe intensity more closely, but also erodes the signal and increases high-frequency noise. This increased noise for large $\mu$ is also a symptom of the LMS gradient descent overshooting and oscillating about the optimum filter coefficients.²² The best SNR enhancement appears to be at $L=8$ and $\mu =0.1$. Filters with $L\ge 16$ (not shown) were found to be less stable under our conditions and fail to further enhance SNR.

Fig. 4

ANC-enhanced PSDs for BGO particle imaging experiments. PSDs shown are max projections across all scan lines of the image, for a single repetition. The adaptive filter noise canceling performance is evaluated with respect to different filter lengths $L$ and step sizes $\mu$.

3.3.

Sampler Bit Width, Oversampling, and Voltage Resolution

An important consideration is whether the ADC provides enough quantization steps to resolve weak pump-probe signals.¹² We use a 14-bit ADC, operating in a $\pm 1\mathrm{V}$ range, resulting in a quantization step of $\sim 122\mu \mathrm{V}$. A CIC filter with $R=128$, $M=4$, $N=2$ is used for decimation. The total bit width of the decimated result is given as¹⁴

4.1.

Single Scan Line, Uniform Sample, and Power Spectral Density Results

Figure 5(b) shows PSD plots of the transmitted probe ($d$), reference ($x$), the noise canceling output of the adaptive filter ($e$), along with the ADC noise floor and calculated shot noise limit. (These data were acquired on a single, uniform, BGO crystal, and so the LMS parameters $L=3$, $\mu =0.2$ were set differently than for the case of a heterogeneous structured sample.) PSD units are in decibels relative to the carrier (dBc), referenced to the DC power of the reference ($\propto 0.1225{\mathrm{V}}^2$). As can be seen from probe and reference PSDs, the ADC noise floor is about 15 dB above the shot noise limit and the laser RIN contributes an additional $\sim 20\mathrm{dB}$ to the overall noise floor. A $1/f$ character of the RIN can be seen from DC to 1 MHz. Above 1 MHz, a gradual increase starts to appear, hitting a maximum at 8 MHz, which we attribute to the photodetector transimpedance amplifier.²⁴ Above 15 MHz, the noise from both probe and reference photodetectors converges to the ADC noise floor.

When inspecting the PSD of adaptive filter output $e$, the noise floor in the 1-MHz region has been decreased and the 1-MHz pump-probe signal can be seen clearly. The total noise reduction in the vicinity of this signal is $\sim 10\mathrm{dB}$. This highlights that the electronic noise is currently the limiting factor, and that additional averaging is required to see a clear pump-probe image. We show later that averaging alone is not sufficient to recover a pump-probe image under these conditions (Figs. 6 and 7), but first we discuss what it will take to reach the shot noise limit and estimate how much averaging is needed in the meantime.

Fig. 6

Imaging results. (a) The wide-field transmission image is shown on the top, with laser scanned area ($70\mu \mathrm{m}\times 100\mu \mathrm{m}$) highlighted by the red box. (b) Laser-scan transmissivity, (c) lock-in output without ANC, and (d) lock-in output with ANC, at 0-ps pump-probe delay. (d) Laser-scan transmissivity, (e) lock-in output without ANC, and (f) lock-in output with ANC, for a $z$-section offset by $5\mu \mathrm{m}$. (h) Laser-scan transmissivity, (i) lock-in output without ANC, and (j) lock-in output with ANC, for 2-ps pump-probe delay.

Fig. 7

Imaging results along with power spectral densities of each scan line. (a) Laser-scan transmissivity, (b) lock-in output with ANC, (c) PSD without ANC, and d) PSD with ANC for 0-ps pump-probe delay. Note the pump-probe signal at 1 MHz in both PSDs. (e) Laser-scan transmissivity, (f) lock-in output with ANC, (g) PSD without ANC, and (h) PSD with ANC for 2-ps pump-probe delay. Note the absence of pump-probe signal at 1 MHz in both PSDs.

From Fig. 5, we notice that the superimposed noise floor of the ADC and the photodetector (shown as the black solid curve) is 24 dB larger than the shot noise limit. This shot noise limit is equivalent to $27\mathrm{pW}/\sqrt{\mathrm{Hz}}$. By comparison, the NEP of our photodetector is $100\mathrm{pW}/\sqrt{\mathrm{Hz}}$ and the ADC raises the NEP of the entire detection system to $1000\mathrm{pW}/\sqrt{\mathrm{Hz}}$. We note that commercial systems that incorporate a low-noise transimpedance amplifier and ADC can reach NEP of $10\mathrm{pW}/\sqrt{\mathrm{Hz}}$ (e.g., DPD80, Resolved Instruments), indicating that digital adaptive RIN canceling can in principle achieve shot noise-limited detection with a suitable low-noise detector and ADC.

4.2.

Imaging Results

Here, we perform pump-probe imaging, with and without ANC, on BGO crystal particles. The specimen was prepared by crushing a fragment of BGO on a glass slide with a metal pick. Then, a spacer was placed around the particles, with a coverslip on top. Images were acquired with $100\times$ line repetitions. Subsequent lines were acquired by stepping the sample stage in $1\text{-}\mu \mathrm{m}$ increments. Given the 50-MHz sample rate ($20\mathrm{ns}/\text{sample}$) and $128\times$ downsampling after the lock-in mixers, this $100\times$ averaging is equivalent to a lock-in integration time constant of $256\mu \mathrm{s}$. If, instead of translating the sample on the $y$ axis, we were to use the 2-Hz frame rate y-scanner on the EOPC system and implement the signal processing chain on a real time DSP processor or FPGA (future work), including forward and backward scans, this $100\times$ averaging time would translate to a $25\text{-}\mathrm{s}/\text{frame}$ pump-probe frame rate. The field of view is $75\mu \mathrm{m}\times 100\mu \mathrm{m}$.

Two images were acquired at 0 ps delay, at z-sections $5\mu \mathrm{m}$ apart. The third image was acquired at the lower z-slice, with 2 ps delay. As the delay scan in Fig. 5(a) indicates, we expect a strong pump-probe signal at 0 ps, and no signal at 2 ps delay. We will show this is the case only after adaptive filtering. Results are shown in Fig. 6 for transmitted light (no averaging of line scans), $100\times$ averaging of the conventional lock-in, and $100\times$ averaging of lock-in after noise canceling.

Without adaptive filtering (Fig. 6, center column), the pump-probe output merely follows the transmitted intensity, with only a slight reduction in brightness at 2 ps delay. In this scenario, the lock-in primarily detects laser RIN within its bandwidth; the result is a noisy copy of the transmitted light image. On the other hand, with adaptive filtering (Fig. 6, right column), the RIN is reduced and the lock-in algorithm produces pump-probe images that are sensitive to z-section, and also sensitive to pump-probe time delay, as expected. We note in Fig. 6(j) that, even though BGO is not expected to have long-lived excited states for our pump/probe wavelengths, there is still a signal at 2 ps. We attribute this to a $\tau$-independent background caused by leakage of the linear transmitted image $T_s$ into the lock-in detection band, as will be discussed next.

Figure 7 shows the $z=-5\mu \mathrm{m}$ results along with corresponding PSD of the signal from each line, just before lock-in detection. The 1-MHz pump-probe signal is clearly present in the PSD plot of the 0 ps delay image and disappears at 2 ps delay, as expected for BGO [see Fig. 5(a)]. It can also be seen that the background within the lock-in bandwidth (magenta box) is much cleaner after ANC. Finally, we observe leakage of the linear transmission image $T_s$ (centered at 0 MHz on the spectrograms) into the lock-in band (magenta box) in Figs. 7(c), 7(d), 7(g), and 7(h). This is especially pronounced around scan lines 15 and 70 to 85, where the presence of signal within the lock-in detection band results in bright spots even in the 2 ps image [Fig. 7(f)], even though there is no pump-probe signal. This can potentially be resolved by increasing the pump modulation frequency $f_{\mathrm{m}}$, slowing the scanner, or using RF phase-sensitive lock-in (the lock-in for these experiments was outputting the phase-insensitive magnitude; limitations with the DAQ digital buffers prevented simultaneous acquisition of a pump synchronization signal).