Noise cancellation of white pulsed light with pulse-to-pulse observation of probe and reference pulses in spectral pump/probe measurement

In a pump/probe measurement, a sample is excited by the pump light (Pu), and the stimulus on the sample is detected using the probe light (Pr). Here, white Pr and a multi-wavelength detection through the spectral dispersion allow spectral pump/probe measurements. Although a supercontinuum (SC) light source with a highly nonlinear photonic crystal fiber (PCF) [1] is readily available for white Pr, the intensity noise of the SC light is generally formidable. The present study focuses on noise-cancellation techniques for the multi-wavelength measurement of spectrally dispersed SC Pr.

This research is motivated by the development of stimulated Raman spectroscopy (SRS) with multi-wavelength detection. Measurements of the stimulated Raman effect are categorized in the pump/probe measurement, where at least two light pulses at different wavenumbers are irradiated at the same position of the sample. When the difference in the wavenumber matches the wavenumber of a molecular vibration mode, a photon of a higher wavenumber is annihilated, whereas the counterpart photon is generated based on the stimulated Raman effect [2]. The result is sensitively measured through a lock-in detection of either photons serving as Pr, whereas the intensity of the counterpart is modulated as Pu. The signal-to-noise ratio (S/N) of SRS is enhanced by approximately three orders of magnitude compared to that of conventional spontaneous Raman spectroscopy [3] under typical measurement conditions. SRS is frequently employed in molecular imaging applications for biomedical imaging [4–6], polymer material texture imaging [7], and other imaging types [8].

The wavelength conversion of the light source using the PCF is a simple and relatively low-cost solution for the preparation of plural pulsed lights at different wavelengths. The synchronization of the pulse emission timings of Pu and Pr is readily obtained using a single light source [9–12]. Furthermore, wideband wavelength conversion is possible [1, 13]. Multi-channel lock-in detection after spectral dispersion of the SC Pr enables a one-shot spectral measurement without wavelength scanning [9]. However, these advantages are obtained at the expense of abundant noise [14, 15].

The intensity of Pr through the sample is given by the following:

$$\begin{eqnarray}&&\left(P\,+\,\delta P\right)\left(1\,+\,m\right)\simeq P\,+\,Pm\,+\,\delta P,\end{eqnarray} \tag{ 1 }$$

where P and δP are the average power and noise of Pr, respectively. The stimulus on the sample using Pu appears as the intensity modulation (m) of Pr, and generally, m ≪ 1. Here, the noise of Pr dominates the S/N. For instance, when δP is 10⁻⁴ with a given observation bandwidth, S/N is unity for m = 10^−4, which is a typical modulation depth in SRS microscopy.

The origins of δP are identified as shot noise and dynamical fluctuations of the light source state. The shot noise is a quantum nature of the photon and determines the principle of the maximum S/N. The shot noise current (i_sh) is given by the following:

$$\begin{eqnarray}&&{i}_{{\rm{sh}}}\,=\,\sqrt{2qSPB},\end{eqnarray} \tag{ 2 }$$

where q, S, and B are the elemental charge, responsibility of the employed photodetector, and detection bandwidth, respectively. Therefore, the S/N determined by the shot noise (S/N_sh) is

$$\begin{eqnarray}&&{\rm{S}}/{{\rm{N}}}_{{\rm{sh}}}\,=\,{\left(SPm\right)}^{2}/{i}_{{\rm{sh}}}^{2}\,=\,SP{m}^{2}/2qB.\end{eqnarray} \tag{ 3 }$$

It can be seen that S/N_sh is improved with a high average power of the light source.

The noise due to the dynamical fluctuation of the light source is canceled by subtracting the noise component in the reference light (Rf) from that in Pr [16–18]. This simple subtraction, however, does not cancel the noise of the white Pr for the multi-wavelength measurement. Because the noise differs among the wavelengths covered by the multi-channel detectors [19, 20], the observation center wavelength and wavelength bandwidth must be identical for Pr and Rf; otherwise, the noise correlation of Pr and Rf is disturbed, and hence the noise in Pr cannot be properly canceled through a simple subtraction. We developed two noise-cancellation techniques to fulfill this requirement: time-division noise cancellation (TDNC) and phase-detection noise cancellation (PDNC).

The outline of the optics system is shown in figure 1. A portion of a pulsed optical beam from a light source is coupled to the PCF and is converted into a white pulsed beam. The white pulse is split into Pr and Rf after the polarization of the white pulse is fixed by a polarizer (P). This stabilization of the polarization is essential because the polarization of the white pulse fluctuates, whereas the splitting ratio depends on the polarization. If P is not inserted, the noise correlation between Pr and Rf is disturbed. Rf undergoes a delay [21] of a half or quarter cycle of the pulse repetition through an additional optical path for TDNC or PDNC, respectively. The path length is ∼2 or ∼1 m for a pulse repetition frequency of 76 MHz. The counterpart from the light source is employed as Pu. The pulse arrival time of Pu in the sample is adjusted to match that of Pr. Pu undergoes intensity modulation for a lock-in detection. Then, Pr, Rf, and Pu are spatially overlapped and irradiated on the sample. The intensity of the Pr is modulated, reflecting the stimulus on the sample. Then, Pr and Rf are spectrally dispersed together using a single spectrograph (SP). A spectral component is input to a noise-canceling detector, and the noise-canceled result is lock-in detected. With both techniques, Pr and Rf are commonly detected using a single photodetector, which enables an identical spectral dispersion by a single SP. The center wavelength and wavelength bandwidth for a detection channel are the same for Pr and Rf. This feature is advantageous for conservation of the noise correlation between Pr and Rf.

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In TDNC, the Pr and Rf pulses are distinguishably detected by a photodiode (PD) interfacing a high-speed transimpedance amplifier (TIA) (figure 2(a)). The detected pulses are discriminated based on the time, and the intensity noise in the Rf pulse is subtracted from that in Pr [22]. However, for the response speed of the TIA required to distinguish the Pr and Rf pulses, thermal noise overwhelms the shot noise. The thermal noise effect is mitigated by multiplication of the photocurrent in an avalanche photodiode (APD), accompanied by excess noise during multiplication. In our previous study [22], the performance achieved was 2.8 × 10⁻⁵ Hz^−1/2 in terms of modulation equivalent noise density (MEND) in the Pr with an average input power of 0.26 μW. The shot noise was estimated to be 1.6 × 10⁻⁶ Hz^−1/2, where the responsibility of the employed APD at unity multiplication factor was 0.45 A W⁻¹. The experimental results indicate further room for improvement of ∼25 dB. However, the recent advance in the gain-band product frequency (f_GBP ) of the operational amplifiers (op-amps) offers an acceptable tradeoff between the thermal noise and response speed for microwatt detection with APD. For example, the f_GBP of the op-amp employed in the previous study [22] was 240 MHz, whereas that in this study was 8 GHz. This is a 33-times advancement in reducing the thermal noise for the same response speed, and there is still further room for investigation.

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For PDNC, figures 3(a) and (b) show the complex amplitudes of Pr and Rf at the pulse repetition frequency. The horizontal arrow indicates the Pr amplitude. The upward arrow is the Rf amplitude whose phase is shifted by π/2 through the delay of the quarter-cycle of the pulse repetition. When the Pr intensity is modulated by the sample response (figure 3(a)), the intensity modulation is translated into the phase modulation of the superposition of Pr and Rf (broken arrow). At the same time, the phase is unresponsive to the intensity noise because the intensity noises on Pr and Rf are parallel (figure 3(b)). Therefore, the S/N is improved in the phase detection of the overlap of Pr and Rf. Here, the overlapped signal functions as the phase carrier. Note that PDNC requires a frequency band of only around the pulse repetition frequency and is adaptive to a resonant photodetector (figure 2(b)). If the same op-amps are employed for the TIA and resonant photodetectors, the latter is advantageous for suppressing the thermal noise effect with a large resistance [22–25]. This configuration also does not suffer from the noise amplification effect that emerges in the TIA topology [23]. Therefore, in a previous study [22], a superior performance was demonstrated with the resonant photodetector than with the TIA-photodetector. With a 0.26-μW optical input power, MEND was 1.0 × 10⁻⁵ Hz^−1/2, and the separation from the shot noise was reduced to ∼16 dB. Note that the separation was due to the excess noise in the multiplication in the APD, not the thermal noise. However, when the input power was further increased [26], the performance was further separate from the shot-noise limited performance despite the lower magnification factor. MEND was ∼5 × 10⁻⁶ Hz^−1/2 with an average input power of 3 μW, while the shot noise was estimated to be 4.9 × 10⁻⁷ Hz^−1/2. The experimental results indicate that the room improvement deteriorated to ∼20 dB.

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Therefore, for a low-power optical input (∼0.1 μW), PDNC adaptive to a resonant photodetector with a high resonant impedance is advantageous. The cancellation performance, however, does not improve proportionally with a high-power input. By contrast, as the incident power increases, TDNC with a direct observation of pulses with TIA may achieve a higher performance because the thermal noise effect is relatively hindered. To reduce the contribution of shot noise, which determines the principle limit, it is important to improve the noise cancellation performance with high power. Further, the noise-cancellation performance in our previous study was evaluated with only a few tens hertz band around a center frequency of a few kilo-hertz, and the wide band noise cancellation performance required for high-speed detection remains to be evaluated. In this study, for a wide band shot noise limited performance with a high power input, performance deterioration mechanisms are researched with wide band noise spectra for PDNC with resonant photodetectors. In addition, based on the findings in the aforementioned investigation, a near-shot-noise-limited performance is demonstrated by TDNC with a TIA of a modern high-speed op-amp. To the best of our knowledge, this is the best noise cancellation result of highly noisy SC light of ∼μW. For further a high-power input and wide-band noise cancellation for a rapid pump/probe measurement, the emerging limiting factors are also discussed, and countermeasures are provided.

Although the implementations were described in previous studies [22, 26, 27], the authors assumed only ideal conditions. Let us describe the implementations for noise cancellation, as well as the merits and demerits in TDNC and PDNC related to the explicitly expressed response of the photodetector, which is not always ideal. For both TDNC and PDNC, the Pr and Rf optical pulses of the average amplitude of A and the fluctuation of A_k are assumed to be trains of the Dirac delta function for simplicity [26]. The Pr and Rf are given through the following:

$$\begin{eqnarray}&&a\displaystyle \sum _{k=-\infty }^{\infty }\left(1\,+\,m\right)\left({A}_{k}\,+\,A\right)\delta \left(t\,-\,kT\right)\,{\rm{and}}\,b\displaystyle \sum _{k=-\infty }^{\infty }\left({A}_{k}\,+\,A\right)\delta \left(t-kT-T^{\prime} \right),\end{eqnarray} \tag{ 4 }$$

where a and b are the splitting ratios for Pr and Rf (a + b = 1), which are determined by optics. In addition, T and T' are the pulse repetition cycle and delay for Rf, respectively, and δ(x) is Dirac's delta function. When these pulses are observed, the pulse output timing from the optical input has been found to depend on the incident power [28]. Therefore, when the response of the photodetector is linear and time invariant, and is expressed by G(t), the detected signal is as follows:

$$\begin{eqnarray}&&a\displaystyle \sum _{k=-\infty }^{\infty }\left(1\,+\,m\right)\left({A}_{k}\,+\,A\right)G\left(t-kT-{\tau }_{p,k}\right)\,+\,b\displaystyle \sum _{k=-\infty }^{\infty }\left({A}_{k}\,+\,A\right)G\left(t-kT-T^{\prime} -{\tau }_{r,k}\right),\end{eqnarray} \tag{ 5 }$$

where τ_p,k and τ_r,k are the timing deviations. Here, G(t) = 0 for t < 0 (causality), and

$$\begin{eqnarray}&&\displaystyle {\int }_{-\infty }^{\infty }G\left(t\right)dt\,=\,{G}_{0}.\end{eqnarray} \tag{ 6 }$$

2.1. TDNC

A block diagram of TDNC is shown in figure 4(a). In TDNC, T' is set to T/2. The Pr and Rf pulses appear alternatively every half cycle (figure 4(b)). The pulses are discriminated in terms of time using a fast switch synchronous to the pulse repetition. The detected signal is then discriminated using a high-speed switch, such as a diode switch. The discriminated Rf signal (the second term of equation (5)) is subtracted from the Pr signal (the first term) after the weight for the Rf signal is equated using an electrical circuit. This operation is equal to the multiplication of a rectangular wave with an offset of (b −a)/(a + b) (figure 4(b)), that is,

$$\begin{eqnarray}&&\begin{array}{l}2b\left[a\displaystyle \sum _{k=-\infty }^{\infty }\left(1+m\right)\left({A}_{k}+A\right)G\left(t-kT-{\tau }_{p,k}\right)\right.\\ \left.+b\displaystyle \sum _{k=-\infty }^{\infty }\left({A}_{k}+A\right)G\left(t-kT-T/2-{\tau }_{r,k}\right)\right]\,{\rm{for}}\,lT\leqslant t\lt \left(l+\displaystyle \frac{1}{2}\right)T,\\ {\rm{and}}\\ -2a\left[a\displaystyle \sum _{k=-\infty }^{\infty }\left(1+m\right)\left({A}_{k}+A\right)G\left(t-kT-{\tau }_{p,k}\right)\right.\\ \left.+b\displaystyle \sum _{k=-\infty }^{\infty }\left({A}_{k}+A\right)G\left(t-kT-T/2-{\tau }_{r,k}\right)\right]\,{\rm{for}}\,\left(l+\displaystyle \frac{1}{2}\right)T\leqslant t\lt \left(l+1\right)T.\end{array}\end{eqnarray} \tag{ 7 }$$

Here, l is a natural number. An adjustment of the offset is possible with feedback control based on the final result (see the supplemental material for the TDNC circuity available online at stacks.iop.org/JPCO/4/125009/mmedia). In general, a measured value is the time average of the observed physical quantity. For example, the integration time is determined as the time constant in the lock-in detection, bandwidth in the spectrum analysis, and so on. The time average of equation (7) over τ = nT + t' (0 ≤ t' < T) is given by the following:

$$\begin{eqnarray}&&\begin{array}{l}-\displaystyle \frac{2}{\tau }\left(1+m\right)\displaystyle \sum _{k=-\infty }^{\infty }\left({A}_{k}\,+\,A\right)\displaystyle \sum _{l=0}^{n-1}\left[ab\displaystyle {\int }_{lT}^{\left(l+1/2\right)T}{\rm{d}}t{\rm{G}}\left(t-kT-{\tau }_{p,k}\right)-{a}^{2}\displaystyle {\int }_{\left(l+1/2\right)T}^{\left(l+1\right)T}{\rm{d}}t{\rm{G}}\left(t-kT-{\tau }_{p,k}\right)\right]\\ -\displaystyle \frac{2}{\tau }\displaystyle \sum _{k=-\infty }^{\infty }\left({A}_{k}\,+\,A\right)\displaystyle \sum _{l=0}^{n-1}\left[ab\displaystyle {\int }_{\left(l+1/2\right)T}^{\left(l+1\right)T}{\rm{d}}t{\rm{G}}\left(t-kT-T/2-{\tau }_{r,k}\right)-{b}^{2}\displaystyle {\int }_{lT}^{\left(l+1/2\right)T}{\rm{d}}t{\rm{G}}\left(t-kT-T/2-{\tau }_{r,k}\right)\right],\\ +\displaystyle \frac{1}{\tau }F\left(t^{\prime} \right)+\sigma \left(t\right)\end{array}\end{eqnarray} \tag{ 8 }$$

where F(t') is the residual term out of the multiple of T, and σ is noise other than light source fluctuation (e.g., shot noise and thermal noise).

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Here, the response speed of the photodetector is sufficiently fast to relax within T/2, that is,

$$\begin{eqnarray}&&G\left(t\right)=0,\,{\rm{for}}\,t\gt T/2-{\tau }_{p{\rm{and}}r{,}^{\forall }k},\end{eqnarray} \tag{ 9 }$$

and equation (8) is as follows:

$$\begin{eqnarray}&&\displaystyle \frac{2m}{\tau }\displaystyle \sum _{l=0}^{n-1}\left[\left({A}_{l}\,+\,A\right)ab{G}_{0}\right]+\displaystyle \frac{1}{\tau }F\left(t^{\prime} \right)\,+\,\sigma \left(t\right).\end{eqnarray} \tag{ 10 }$$

The first term is the signal of the sample, and A_l functions as the multiplicative noise. The additive noises in Pr and Rf are cancelled by each other. By setting τ sufficiently larger than T, the residual contribution by the final term is diminished. Note that under the condition of equation (9), the pulse timing jitter has no effect on the output. The TDNC is robust to jitter (τ_p,k , τ_r,k ) caused by the photodetector and subsequent electrical circuity.

When the response speed of the photodetector is slow compared to the pulse repetition, however, the signal component (line 1 in equation (8)) is also cancelled. Assuming that an extreme case of G(t) is nearly constant and a = b, the signal disappears. Therefore, a high-speed photodetector that is sufficient to observe an optical pulse is preferred for TDNC. However, the response speed and the thermal noise of the TIA are in a trade-off relation imposed by resistance (R) in the feedback loop. The 3-dB cutoff frequency is given by the following:

$$\begin{eqnarray}&&1.4\sqrt{\displaystyle \frac{{f}_{{\rm{GBP}}}}{2\pi R\left({C}_{{\rm{D}}}\,+\,{C}_{{\rm{i}}}\,+\,{C}_{{\rm{fb}}}\right)}},\end{eqnarray} \tag{ 11 }$$

where f_GBP, C_D, C_i, and C_fb are the gain-bandwidth product of an employed op-amp, the junction capacitance of the PD, the input capacitance of the op-amp, and the phase-compensation capacitor used to stabilize the feedback in TIA (details of the response are given in the supplemental material for TIA) [29], respectively. Meanwhile, the thermal noise current (i_th) is determined as follows:

$$\begin{eqnarray}&&{i}_{{\rm{th}}}=\sqrt{4kBT/R},\end{eqnarray} \tag{ 12 }$$

where k, B, and T are the Boltzmann constant, frequency bandwidth, and absolute temperature, respectively. The S/N for the thermal noise (S/N_th) is given by the following:

$$\begin{eqnarray}&&{\rm{S}}/{{\rm{N}}}_{{\rm{th}}}={\left(SPm\right)}^{2}/{i}_{{\rm{th}}}^{2}={\left(SPm\right)}^{2}R/4kBT.\end{eqnarray} \tag{ 13 }$$

Therefore, an increase in R improves the S/N while limiting the response speed. Moreover, TIA exhibits a noise amplification effect for the voltage noise of the op-amp owing to a reduction in the feedback ratio at high frequencies [29].

2.2. PDNC

In PDNC, T' is set to T/4. The block diagram for the PDNC is shown in figure 5. In general, during the phase detection, a phase reference signal is multiplied, and the result is time-averaged. In our PDNC, the synchronous signal (Sync.) to the pulse repetition is employed as the phase reference, and the phase of Sync. is modulated (feedback controlled) to ensure that the multiplied result is nulled. In this way, the phase deviation in the photodetected signal is obtained as the phase control signal for Sync., while the amplitude fluctuation is rejected.

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As the dominant merit of PDNC, the required frequency band is only within the vicinity of the repetition frequency. Therefore, we first assume that the response of the photodetector is that of the parallel resonator (figure 2(b)), that is,

$$\begin{eqnarray}&&\begin{array}{l}G\left(t\right)\,=\,\displaystyle \frac{L{\omega }_{0}^{2}}{\sqrt{1-{\zeta }^{2}}}U\left(t\right)\exp \left(-\zeta {\omega }_{0}t\right)\cos \left(\omega ^{\prime} t+\beta \right);\\ \omega ^{\prime} \,=\,\sqrt{1-{\zeta }^{2}}{\omega }_{0},\,{\rm{and}}\,b\,=\,{\tan }^{-1}\displaystyle \frac{\zeta }{\sqrt{1-{\zeta }^{2}}},\end{array}\end{eqnarray} \tag{ 14 }$$

where U(t) is the unit step function used to express the causality, and the resonant angular frequency (ω₀) and bandwidth, or the Q-value, is given by the following:

$$\begin{eqnarray}&&{\omega }_{0}^{2}\,=\,\displaystyle \frac{1}{L\left(C+{C}_{{\rm{D}}}+{C}_{{\rm{i}}}\right)},{\rm{and}}\,2\zeta =\displaystyle \frac{1}{Q}=\displaystyle \frac{1}{R}\sqrt{\displaystyle \frac{1}{C+{C}_{{\rm{D}}}+{C}_{{\rm{i}}}}},\end{eqnarray} \tag{ 15 }$$

respectively. Here, C_D and C_i of the buffering amplifier contribute to the capacitance in the resonator. The detected signal is as follows:

$$\begin{eqnarray}&&\displaystyle \sum _{k=1}^{\infty }\left[{A^{\prime} }_{k}\left(t;m\right)\cos \left\{\omega ^{\prime} \left(t-kT\right)+\beta -\displaystyle \frac{\omega ^{\prime} }{2}\left(\displaystyle \frac{T}{4}+{\tau }_{r,k}+{\tau }_{p,k}\right)+{\psi }_{k}\left(m\right)\right\}\right]+{F}_{k=0}\left(t\right),\end{eqnarray} \tag{ 16 }$$

where F_{k = 0}(t) is a residual term of k = 0, and

$$\begin{eqnarray}&&\begin{array}{l}\omega ^{\prime} \,=\,\sqrt{1-{\zeta }^{2}}{\omega }_{0}\\ {\psi }_{k}\left(t;m\right)\,=\,{\tan }^{-1}\left[\displaystyle \frac{{a}_{k}\left(1+m\right)-{b}_{k}}{{a}_{k}\left(1+m\right)+{b}_{k}}\,\tan \,\displaystyle \frac{\omega ^{\prime} }{2}\left(\displaystyle \frac{T}{4}+{\tau }_{r,k}-{\tau }_{p,k}\right)\right]\\ {a}_{k}\,=\,a\,\exp \left(\zeta {\omega }_{0}{\tau }_{p,k}\right)\\ {b}_{k}\,=\,b\,\exp \left\{\zeta {\omega }_{0}\left(T/4+{\tau }_{r,k}\right)\right\}\\ {A^{\prime} }_{k}\left(t;m\right)\,=\,\displaystyle \frac{L{\omega }_{0}^{2}}{\sqrt{1-{\zeta }^{2}}}\left({A}_{k}\,+\,A\right)\exp \left\{-\zeta {\omega }_{0}\left(t-kT\right)\right\}\\ \,\times \,{\left[{a}_{k}^{2}{\left(1+m\right)}^{2}+{b}_{k}^{2}+2{a}_{k}{b}_{k}\left(1+m\right)\cos \left\{\omega ^{\prime} \left(\displaystyle \frac{T}{4}+{\tau }_{r,k}-{\tau }_{p,k}\right)\right\}\right]}^{\displaystyle \frac{1}{2}}.\end{array}\end{eqnarray} \tag{ 17 }$$

The Sync. is

$$\begin{eqnarray}&&D\,\sin \left({\omega }_{c}t\,+\,\phi \left(t\right)\right),\end{eqnarray} \tag{ 18 }$$

and the multiplied result of a low-frequency component is the following:

$$\begin{eqnarray}&&\displaystyle \frac{D}{2}\displaystyle \sum _{k=1}^{\infty }\left[{A}_{k}^{{\prime} }\left(t;m\right)\sin \left\{\phi \left(t\right)-{\psi }_{k}\left(t;m\right)+\left({\omega }_{c}-\omega ^{\prime} \right)t+k\omega ^{\prime} T-\beta +\displaystyle \frac{\omega ^{\prime} }{2}\left(\displaystyle \frac{T}{4}+{\tau }_{r,k}+{\tau }_{p,k}\right)\right\}\right],\end{eqnarray} \tag{ 19 }$$

where ω_c is the pulse repetition angular frequency. Note that F_{k = 0}(t) is omitted because it varies rapidly and contributes little to the low-frequency component. In the feedback phase modulation of Sync., this output is integrated with time. Because ϕ(t) is controlled to ensure that this result is zero, equation (19) nearly converges to zero. Therefore,

$$\begin{eqnarray}&&0\simeq \displaystyle \sum _{k=1}^{\infty }\left[{A}_{k}^{{\prime} }\left(t;m\right)\left\{\phi \left(t\right)-{\psi }_{k}\left(m\right)+{L}_{k}\left(t;\omega ^{\prime} \right)-\beta +\displaystyle \frac{\omega ^{\prime} }{2}\left(\displaystyle \frac{T}{4}+{\tau }_{r,k}+{\tau }_{p,k}\right)\right\}\right],\end{eqnarray} \tag{ 20 }$$

where

$$\begin{eqnarray}&&{L}_{k}\left(t;\omega ^{\prime} \right)=\left({\omega }_{c}-\omega ^{\prime} \right)t+k\omega ^{\prime} T-2n\pi ,\end{eqnarray} \tag{ 21 }$$

and n is an integer used to ensure that

$$\begin{eqnarray}&&0\leqslant {L}_{k}\left(t;\omega ^{\prime} \right)\lt 2\pi .\end{eqnarray} \tag{ 22 }$$

Therefore,

$$\begin{eqnarray}\begin{array}{lll}\phi \left(t\right) & \,\simeq \, & \displaystyle \frac{\displaystyle {\sum }_{k=1}^{\infty }\left\{{A^{\prime} }_{k}\left(t;m\right){\psi }_{k}\left(m\right)\right\}}{\displaystyle {\sum }_{k=1}^{\infty }{A^{\prime} }_{k}\left(m\right)}\\ & & -\,\displaystyle \frac{\omega ^{\prime} }{2\displaystyle {\sum }_{k=1}^{\infty }{A^{\prime} }_{k}\left(m\right)}\displaystyle {\sum }_{k=1}^{\infty }\left\{{A^{\prime} }_{k}\left(t;m\right)\left({\tau }_{r,k}+{\tau }_{p,k}\right)\right\}\\ & & -\,\displaystyle \frac{\displaystyle {\sum }_{k=1}^{\infty }{A^{\prime} }_{k}\left(t;m\right){L}_{k}\left(t;\omega ^{\prime} \right)}{\displaystyle {\sum }_{k=1}^{\infty }{A^{\prime} }_{k}\left(m\right)}+\beta -\displaystyle \frac{\omega ^{\prime} }{8}T.\end{array}\end{eqnarray} \tag{ 23 }$$

The term ψ_k (m) is the phase modulation owing to the sample response to be measured. We found that a convolution of the jitter by A'_k (t) additively affects ψ_k (m), and L_k (t;ω') is a cause of the intensity noise leak. Here, by adjusting ω' to ω_c , L_k (t;ω') is nulled. That is,

$$\begin{eqnarray}&&{\omega }_{0}\,=\,{\omega }_{c}/\sqrt{1-{\zeta }^{2}}.\end{eqnarray} \tag{ 24 }$$

However, the effect of jitter is not rejected. Further, if τ_p,k and τ_r,k are zero, a_k and b_k are independent of k, i.e.,

$$\begin{eqnarray}&&\phi \left(t\right)\simeq {\tan }^{-1}\left[\displaystyle \frac{a\left(1+m\right)-b^{\prime} }{a\left(1+m\right)+b^{\prime} }\right]+\beta -\displaystyle \frac{\pi }{4},\end{eqnarray} \tag{ 25 }$$

and when ∣m∣ ≪ 1 and b' is close to a,

$$\begin{eqnarray}&&\phi \left(t\right)\simeq \displaystyle \frac{1}{1+{\left(a-b^{\prime} \right)}^{2}}\cdot \displaystyle \frac{2ab^{\prime} }{a+b^{\prime} }m\,+\,{\tan }^{-1}\left(a-b^{\prime} \right)+\beta -\displaystyle \frac{\pi }{4}.\end{eqnarray} \tag{ 26 }$$

where

$$\begin{eqnarray}&&b^{\prime} \,=\,b\exp \left(\zeta {\omega }_{0}T/4\right).\end{eqnarray} \tag{ 27 }$$

Therefore, the intensity of the noise is clearly cancelled with the resonant photodetector if τ_p,k and τ_r,k are zero. Because a large R is permitted independently of the pulse repetition frequency, the thermal noise effect is suppressed with a large R. However, the demerits are sensitive to both the resonant frequency and timing jitter. The resonant frequency shift causes an intensity noise leak, and the jitter directly causes phase noise.

Second, let us consider the TIA-photodetector for PDNC. We assume a response of

$$\begin{eqnarray}&&{\rm{G}}\left(t\right)=0,\,{\rm{for}}\,t\gt T-{\tau }_{p{\rm{and}}r{,}^{\forall }k}.\end{eqnarray} \tag{ 28 }$$

In this case, G(t) is expanded using a Fourier series, as follows:

$$\begin{eqnarray}&&\begin{array}{l}{\rm{G}}\left(t\right)=\left\{{\rm{U}}\left(t\right)-{\rm{U}}\left(t-T\right)\right\}\\ \,\,\,\times \,\left[\displaystyle \frac{{a}_{0}}{2}+\displaystyle \sum _{n=1}^{\infty }\left({a}_{n}\,\cos \,n{\omega }_{c}t+{b}_{n}\,\sin \,n{\omega }_{c}t\right)\right].\end{array}\end{eqnarray} \tag{ 29 }$$

where

$$\begin{eqnarray}&&{a}_{0}\,=\,\displaystyle \frac{2{G}_{0}}{T},{a}_{n}=\displaystyle \frac{2}{T}\displaystyle {\int }_{0}^{T}{\rm{G}}\left(t\right)\cos \left(n{\omega }_{c}t\right){\rm{d}}t,{b}_{n}\,=\,\displaystyle \frac{2}{T}\displaystyle {\int }_{0}^{T}{\rm{G}}\left(t\right)\sin \left(n{\omega }_{c}t\right){\rm{d}}t.\end{eqnarray} \tag{ 30 }$$

The components in the photodetected signal in the vicinity of the fundamental component are given by the following:

$$\begin{eqnarray}&&\begin{array}{l}{A}_{k}^{{\prime} }\left[a\left(1+m\right)\cos \left\{{\omega }_{c}\left(t-{\tau }_{p,k}\right)-\displaystyle \frac{\pi }{4}+\beta \right\}\right.\\ +\,\left.{\rm{U}}\left(t-kT-T/4\right)b\,\sin \left\{{\omega }_{c}\left(t-{\tau }_{r,k}\right)-\displaystyle \frac{\pi }{4}+\beta \right\}\right],\\ +\,{A}_{k-1}^{{\prime} }\left\{1-{\rm{U}}\left(t-kT-T/4\right)\right\}b\,\sin \left\{{\omega }_{c}\left(t-{\tau }_{r,k-1}\right)-\displaystyle \frac{\pi }{4}+\beta \right\}\end{array}\end{eqnarray} \tag{ 31 }$$

for kT ≤ t < (k + 1)T, where

$$\begin{eqnarray}&&{A}_{k}^{{\prime} }=\left({A}_{k}+A\right)\sqrt{{a}_{1}^{2}+{b}_{1}^{2}},\end{eqnarray} \tag{ 32 }$$

and

$$\begin{eqnarray}&&\beta \,=\,{\tan }^{-1}\left[\displaystyle \frac{{a}_{1}-{b}_{1}}{{a}_{1}+{b}_{1}}\right].\end{eqnarray} \tag{ 33 }$$

The low-frequency component in the multiplied result of the photodetected signal and the phase reference signal is given by

$$\begin{eqnarray}&&\displaystyle \frac{D}{2}{\left\langle {A^{\prime} }_{k}\sin \left[\phi \left(t\right)-{\psi }_{k}\left(m\right)+\displaystyle \frac{1}{2}{\omega }_{c}\left({\tau }_{r,k}+{\tau }_{p,k}\right)\right]\right\rangle }_{\tau };k\in \tau ,\end{eqnarray} \tag{ 34 }$$

where 〈f〉τ is the time average over τ in the integral feedback control. Therefore, through the feedback control of ϕ(t),

$$\begin{eqnarray}&&\begin{array}{l}\phi \left(t\right)={\left\langle {\tan }^{-1}\left[\displaystyle \frac{a\left(1+m\right)-b}{a\left(1+m\right)+b}\tan \left\{\displaystyle \frac{\pi }{4}-\displaystyle \frac{{\omega }_{c}\left({\tau }_{r,k}-{\tau }_{p,k}\right)}{2}\right\}\right]\right\rangle }_{\tau }\\ \,\,\,+\,\displaystyle \frac{1}{2}{\omega }_{c}\left\langle {\tau }_{p,k}+{\tau }_{r,k}\right\rangle +\beta -\displaystyle \frac{\pi }{2}\\ \,\,\,\simeq \,\displaystyle \frac{2ab}{1+{\left(a-b\right)}^{2}}m+\displaystyle \frac{1}{2}{\omega }_{c}{\left\langle {\tau }_{p,k}+{\tau }_{r,k}\right\rangle }_{\tau }\\ \,\,\,+\,\left(a-b\right){\omega }_{c}{\left\langle {\tau }_{p,k}-{\tau }_{r,k}\right\rangle }_{\tau }+{\tan }^{-1}\left(a-b\right)+\beta -\displaystyle \frac{\pi }{2},\end{array}\end{eqnarray} \tag{ 35 }$$

where b is close to a, and ∣m∣, ∣ω_c τ_p,k ∣, ∣ω_c τ_r,k ∣ ≪ 1 are assumed for the approximation. Therefore, in addition to the TIA-photodetector, the amplitude noise is cancelled. Similar to equation (23), the performance is sensitive to jitter, although the amplitude noise is rejected independently of the form of the response function.

3.1. Optics

3.2. Experimental setup for PDNC

3.3. Experimental operation for PDNC

3.4. Experimental setup for TDNC

3.5. Experimental operation for TDNC

Figure 6 shows the output dependence of the resonant photodetector on the optical input power. The values were normalized to zero at a 2-μW input. The phases were retarded in a larger input power with all of the resonant parameters, where the retardation was more sensitive with a larger R and L. The output phase is described with the parameters of the resonant load as follows:

$$\begin{eqnarray}&&\displaystyle \frac{R}{{\omega }_{c}L}\left\{1-L\left(C+{C}_{{\rm{D}}}+{C}_{{\rm{i}}}\right){\omega }_{c}^{2}\right\}=\displaystyle \frac{R}{{\omega }_{c}L}-R\left(C+{C}_{{\rm{D}}}+{C}_{{\rm{i}}}\right){\omega }_{c}\end{eqnarray} \tag{ 36 }$$

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The fact that the phase shift was more prominent with a larger R supports the change in the resonant parameter. The sensitivity enhancement in parallel with L in the phase shift indicates that the major origin is an increase in the C_D of the APD because its contribution to the capacitance of the resonator increases in the tuning. The increase in C_D for a high-power optical input was also observed in prior research [30]. This deviation of C_D is translated into a resonant-frequency shift.

The optimal resonant frequency (equation (24)) is essential for achieving the best performance. Therefore, the parameter dependence on the input power requires an optimization of ω₀ at any time along with the average input power. This explanation resumes a static and constant pulse intensity. In an actual situation, however, the intensity fluctuates. In addition, the aforementioned results are based on the input of only the Pr pulse. In the actual case, Pr and Rf, where the input timing shifts, are input. If Pr influences the gain (magnitude of b_k in equation (17)) and phase (τ_r,k ) of Rf, this effect also adds the phase noise of the carrier. A dynamic effect should be taken into account for a precise investigation of the deterioration factors in PDNC.

Therefore, noise spectra with the inputs of Pr and Rf of a white pulse were evaluated. Figure 7(a) shows the normalized noise spectrum of Pr. The peak at 76.35 MHz is the fundamental component of the pulse train, and the flat baseline is the noise floor. The noise floor was horizontal and constant within the bandwidth of the TIA. The relative intensity noise was −47.9 dBc with an observation bandwidth of 10 kHz. This value was converted into a modulation equivalent noise density (MEND) of 4.02 × 10⁻⁵ Hz^−1/2. This value indicates a 35% relative standard deviation of a pulse-to-pulse fluctuation. The gain curve of the resonant photodetector for PDNC is shown in figure 7(b). This curve is explained well by the Lorenz curve for the ideal parallel resonant circuit, and the 3-dB bandwidth was estimated to be 1.125 MHz.

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The noise spectra with the PDNC are shown in figure 7(c). A valley structure formed, and its position shifted with the input power. The dip also moved when C was changed (not shown), which this indicates that the frequency for effective noise cancellation depends on the resonant frequency. The power dependence of the dip position was related to the static resonant frequency shift owing to the C_D change. The results also show that the shot-noise-limited performance cannot be achieved even when the bottom frequency of the valley is adjusted through engineering the resonant parameters. For a 1-μW input, for instance, the minimum noise was ∼5 × 10⁻⁶ Hz^−1/2, whereas the shot noise was estimated as 8.9 × 10⁻⁷ Hz^−1/2 for the 0.4 A W⁻¹ sensitivity of the present APD. The obstacle to the shot-noise-limited performance was considered to be the dynamical fluctuation of the C_D and/or jitter of the photocurrent; that is, the convoluted effect (caused by A_k '(t) in equation (23)) of the fluctuation of ω₀ (fluctuation of L_k (t)) and/or output timing disturbance of the Pr and Rf photocurrents (the second term) cause the phase noise. The analysis of these effects on the noise spectra is interlaced and complicated.

Therefore, to eliminate the effect of L_k (t) and the convolution effect of the jitter over one period, a TIA-photodetector with a shortened relaxation time within one cycle of the pulse repetition was employed. The output waveforms with 10 and 20 kΩ Rs are plotted in figure 8(a). The peaks of the Pr and Rf output pulses are distinguished with both Rs, whereas the Rf-pulse height-ratio was more extensive with 20 kΩ R. This fact indicates that the Rf optical pulse is input when the Pr photocurrent output is less relaxed with 20 kΩ R than with 10 kΩ R.

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Figure 8(b) shows the noise spectra of PDNC with a 1-μW input. Contrary to the dictation by equation (35), the performance depended on the response speed. The amount of noise with the 10 k Ω R was less abundant than that of the 20 kΩ R. From this result, it can be inferred that there is non-negligible nonlinearity concerning jitters of the Pr and Rf pulses. There is also the possibility of a nonlinearity in the correlation of the photocurrent intensities, which is expressed by the pulse-to-pulse fluctuation of a and b (note the fourth term in equation (35)). Prior studies [30, 31] have also referred to reductions in the response speed and quantum efficiency owing to an abundant photocurrent. Less disturbance was due to less nonlinear effects with a faster relaxation. Note that the minimum noise of the TIA-photodetector with 10 kΩ R was similar to that of the resonant photodetector. Therefore, the disturbance in one cycle of the pulse train dominated the performance; the nonlinearity by the nearest pulse in the time domain is critical in the performance. One can expect that the amount of noise cancellation will be modified with a rapid relaxation with a smaller R. In this way, however, one must compromise the thermal noise performance.

In our previous study, [28] another countermeasure was described. This method is based on the correlation of the phase noise with the intensity noise. The phase of the Sync. was modulated based on the intensity noise used to cancel the phase noise correlated with the intensity noise. This phase noise cancellation functions well for a TIA-photodetector, whereas it will not work for the resonant photodetector. In the TIA-photodetector, the phase and intensity fluctuations are determined commonly in one cycle of the pulse repetition, and the correlation of the fluctuations will be secured. Therefore, pulse-to-pulse modulation of ϕ(t) by − [ω_c (τ_p,k + τ_r,k )/2 + tan⁻¹(a_k − b_k )] based on A_k can cancel the phase noise (equation (35)). By contrast, the phase fluctuation in the resonant photodetector is a convolution of the jitters by A_k exp(−ζω₀ T), whereas the intensity noise is that of A_k by exp(−ζω₀ T) (equation (23)). The difference in the convolution weights for the phase and intensity noises destroys the correlation in the noises. Instead of executing the previous countermeasure against the phase noise with the TIA-photodetector compromising the thermal noise effect, in the following, let us investigate a more straightforward way to avoid the nonlinear effect of jitter and the responsibility of APD.

TDNC is robust against the jitter problem. Further, T' is set at the half-cycle of the pulse repetition rather than the quarter cycle to serve as an adequate time margin for the relaxation of the Pr photocurrent. The correlation in the Pr and Rf noises is expected to be conserved under a situation in which the Rf optical pulse inputs after the Pr photocurrent sufficiently relax to avoid a nonlinear effect.

5.1. Results and discussion for TDNC

The output waveform with a 1-μW average input is shown in figure 9(a). The pulse signals of Pr and Rf separately appeared and mostly did not overlap in the time domain. It can be verified that the Pr pulse sufficiently decayed at the input of the Rf pulse. The blue plot in figure 10(a) is the noise spectrum for TDNC with a 1-μW input. The shot-noise-limited performance (broken violet line) is 9.56 × 10⁻⁷ Hz^−1/2, based on the responsibility of 0.35 AW⁻¹. One can find that the optical source noise (the red plot) was canceled out to ∼1.6-times of the shot-noise-limited performance in a low-frequency region below 1 MHz. The power dependence of the noise spectra is shown in figure 10(b). The amount of noise was reduced in the higher power input. However, a bump, as one can recognize on the plot of the 5-μW input, appeared when the input power was more than ∼2 μW. The average noise from the 0.2–1 MHz region is plotted in figure 10(c). From these data, one can consider that a near-shot-noise-limited performance was achieved from an optical input of 0.6 to 2 μW. Therefore, an adequate time margin for the Pr pulse until decay was effective in maintaining the intensity noise correlation of Pr and Rf. Note that TDNC is also robust to jitter of the pulse signals compared to PDNC because switching is in principle insensitive to the time drift in the half-cycle of the pulse repetition. Because the output timing of the photocurrent pulse is affected by the optical input power [28, 31], this robustness also contributes to the performance.

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The origin of the deterioration in the low-power region (figure 10(c)) is considered to be the excessive noise of the APD [32]. However, the estimated MEND with an interpolation at an input of 0.26 μW was ∼5 × 10⁻⁶ Hz^−1/2. This value is a 6-dB modification from the previous study [22] because of the larger R and smaller excess noise with reduced multiplication factor of APD. The value of R can be increased to 10 kΩ from 510 Ω relying on the present high-speed op-amp. In the high-power region, the bump is accounted for by the degradation and limits of the dynamic range. Prior studies in [30] and [31] demonstrated that a high power input resulted in a slow response speed because of the space-charge effect in the depletion layer. Thus, abundant photogenerated charges were left in the prolonged decay over the half cycle of the pulse repetition owing to the high-power input. When the optical pulse is input under this situation, its quantum efficiency is adversely influenced by a carrier recombination [30]. The gain for the optical pulse after an intense optical pulse is diminished by this mechanism. The input power dependency of the 512 averaged waveforms shown in figure 9(b) support this mechanism. With a 1-μW input, Pr and Rf pulses were identical, whereas at 5 μW, the gain for the Rf pulse was clearly reduced. The difference in τ_p,k and τ_r,k was not prominent compared in gain. Therefore, the intensity noise fluctuates the space-charge effect and results in a disturbance against the ratio of gains for Pr and Rf. The fluctuation of the gain ratio destroys the correlation between the noises of Pr and Rf. It may be practical to reduce the incident light intensity per unit area against this deterioration mechanism [31, 33]. One can optically divide the Pr and Rf lights after the sample using a fiber bundle and detect them with the same photodetectors. The detected signals are then merged for processing. Another way is to employ a large area PD. Although C_D increases in parallel with the light-receiving area, it can be compensated in the bootstrapping topology [29, 34–36].

The deterioration of the noise cancellation can be seen within a high-frequency region (figures 10(a) and (b)). For a fast measurement, a wideband noise cancellation is desirable. Let us calculate the noise power spectral density (PSD) to investigate the degradation mechanism. Herein, we assume an infinite response speed of the photodetector and that no jitter is induced. In this case, the photodetected signal is the same as in equation (4). PDS is calculated based on the difference between the first and second terms in equation (4) (see supplemental material). When A_k is white noise, as displayed in figure 7(a), the PSD is given by the following:

$$\begin{eqnarray}&&{\rm{PSD}}\left(\omega \right)\,=\,{\left|{\rm{LPF}}\left(\omega \right)\right|}^{2}\left[\displaystyle \frac{{a}^{2}+{b}^{2}-2ab\,\cos \left(\omega T^{\prime} \right)}{T}\left\{{s}^{2}+\displaystyle \frac{2\pi {A}^{2}}{T}\displaystyle \sum _{q=-\infty }^{\infty }{\rm{\delta }}\left(\omega -{\omega }_{c}q\right)\right\}+{\sigma }^{2}\right]+{\sigma ^{\prime} }^{2},\end{eqnarray} \tag{ 37 }$$

where LPF(ω), s², σ², and σ'² are the response of the LPF; the variance of the pulse intensity; noise before the LPF, including shot noise; and noise after the LPF, respectively. Therefore, for the best performance within the low-frequency region, a and b should be equal to 1/2. In this case, one can also see that the optical source noise leaks with a factor of sin²(ωT'/2) as ω increases.

The theoretical curve for the 1-μW input predicted by equation (37) is drawn by the pink line in figure 10(a). In the theoretical calculation, σ² was assumed to be the shot noise of the 1-μW input, and the ant T' value was the half-cycle delay. The term σ' ² was ignored. It can be seen that the experimental result supports well the theoretical curve and deterioration mechanism of T'. Therefore, the deterioration in the high-frequency region was caused not by the nonlinearity but by a timing mismatch in the Pr and Rf pulses (T') because equation (37) assumes a linear time-invariant system. It is preferable that T' be as small as possible for wideband noise cancellation. Here, if T' is shrunken by shortening the optical delay, there is a tradeoff between the noise leak owing to T' and the conservation of the noise correlation. Instead, one can delay the Pr signal after separation in the circuity. This delay is readily provided electronically with a coaxial cable or a passive delay line circuity. When the Pr signal is delayed by T', the optical source noise other than the shot noise is canceled independently of the probing frequency. To reduce the shot noise contribution, enhancing the dynamic range and expanding the noise cancellation bandwidth is essential for a fast measurement with a high-power light source [37, 38]. We will verify the effectiveness of the discussed countermeasures in the near future.

The performance-limiting factors were investigated using a resonant photodetector and TIA-photodetector for PDNC. In the case of the resonant photodetector, the static C_D dependence on the optical intensity was found to cause a deviation of the optimal parameter setting for the resonator. Even under the optimal conditions, the dynamical C_D fluctuation and photocurrent jitter responding to the pulse-to-pulse intensity swing induced a phase noise and deteriorated the noise-cancellation performance. To investigate the pulse-to-pulse jitter effect apart from the parameters of the resonator, the PDNC performance was also evaluated using the fast TIA-photodetector, in which the photocurrent decayed within one cycle of the pulse repetition. From the results, it can be inferred that the disturbance in the one-cycle ratio of the Pr and Rf photocurrent intensities and/or output delays dominated the phase noise. The performance was modified with a shorter relaxation time of the photodetector, where Rf was input after a sufficient decay of the Pr photocurrent. The nonlinear effects owing to the space charge effect and a quantum efficiency reduction in the photodetector were found to have a non-negligible impact on the performance. In other words, the gain and output delay for Pr and Rf were conserved at the same level when the interval of the inputs was sufficiently long for the photocurrent to decay and avoid nonlinear effects.

Based on these findings, to maintain the intensity noise correlation between Pr and Rf, the delay for Rf was set at the half-cycle of the pulse repetition, which was the maximum delay to avoid the nonlinear effect between Pr and Rf. TDNC was applied for this optics configuration. The near-shot-noise-limited performance was demonstrated within the low-frequency region. However, the performance deteriorated for a high-power input. It may be effective to reduce the optical intensity per unit area for a PD with multiple photodetectors after optical splitting or a large-area PD interfacing the bootstrap circuity. In addition, the delay for the Rf signal from the Pr induced an intensity noise leak within the high-frequency region. The passive delay element in the Pr signal branch after the discrimination of Pr and Rf is able to compensate for the delay effect. We believe that these countermeasures for the present TDNC enable spectral pump/probe measurements of a high-dynamic range and wide bandwidth using a high-power SC-light source.

This study was financially supported by JSPS KAKENHI, Grant No. JP17K18145.