Frequency response measurement of high-speed photodiodes based on a photonic sampling of an envelope-modulated microwave subcarrier

Mengke Wang; Shangjian Zhang; Ying Xu; Yutong He; Yali Zhang; Zhiyao Zhang; Yong Liu

doi:10.1364/OE.420662

1. Introduction

Frequency response of a photodiode (PD) is a critical parameter for photoelectric conversion in broadband radio-over-fiber links and optical fiber communication systems [1–4]. With the increasing requirement of bandwidth and resolution, frequency response of a PD must be precisely and subtly measured [5–7], especially for microwave photonics applications.

A large number of methods have been developed, such as all-optical wavelength-beating [8–12], all-optical noise-beating [13–16], electro-optic frequency sweeping [17–20], carrier-suppressed electro-optic frequency sweeping [21], and frequency-shifted heterodyne [22]. The all-optical methods employ the beat notes between different wavelength signals or noises, and enable ultra-wideband frequency response measurement without any electro-optic or microwave components. Whereas, the frequency resolution is generally low, due to wavelength drifting and linewidth broadening. In order to achieve high-resolution characterization, the electro-optic frequency sweeping methods are developed with the assistance of a broadband modulator and a vector network analyzer (VNA) or an electrical spectrum analyzer. In the VNA-based method, the main disadvantage is that extra calibration should be implemented to subtract the electro-optic frequency response of the auxiliary modulator. The carrier-suppressed electro-optic frequency sweeping method employs a pair of symmetrical modulation sidebands as the optical stimulus, which relives the frequency sweep range requirement of the microwave source. However, it requires a Mach-Zehnder modulator (MZM) with a high-extinction ratio, and is very sensitive to the phase bias of the MZM. The frequency-shifted heterodyne method avoids the bias drifting problem, and is capable of measuring frequency response of PDs without any extra calibration. However, this scheme is a little complicated. For the ultra-wideband measurement, we demonstrated a self-calibrating method based on low-repetition-rate photonic sampling [23]. Nevertheless, the measurement can only be implemented at discrete frequency points equal to multiple of repetition rate, which limits the resolution to be tens of MHz.

In this paper, a novel electro-optic approach to measuring the frequency response of PDs is proposed and experimentally demonstrated, which is realized through employing an ultra-short optical pulse train to sample an envelope-modulated microwave subcarrier. This method features high-resolution and ultra-wideband measurement. Hyperfine frequency response of a PD at any frequency is acquired by subtly varying the center frequency of the microwave subcarrier from 0 to half of the optical pulse repetition rate. Moreover, the ultra-short optical pulse train provides a broadband comb stimulus to extend the frequency measurement range beyond hundreds of GHz. A proof-of-concept experiment is carried out to verify the feasibility of the proposed method, where the measured frequency response of a high-speed PD agrees well with those obtained by using the conventional methods.

2. Operation principle

Figure 1 shows the schematic diagram of the proposed method. The operation principle can be described as follows. Firstly, an ultrashort optical pulse train with a center frequency of f₀ and a repetition rate of f_r [i.e., a frequency comb centered at f₀ and with an interval of f_r, as shown in Fig. 1(a)] from a mode-locked laser diode (MLLD) is injected into an MZM, where it samples an envelope-modulated microwave subcarrier with three frequency components (i.e., a center frequency of f_s and two frequency sidelobes of f_s±f_e, f_e<<f_s≤f_r/2). The output optical spectrum from the MZM is shown in Fig. 1(b), where each frequency cluster is centered at f₀+lf_r (l is from –N to N), and involves seven frequency components. The two modulation sidebands, each of which involves three frequency components corresponding to the envelope-modulated microwave subcarrier, are symmetrically located at both sides of the comb tooth at f₀+lf_r, and their intensity are proportional to that of the corresponding comb tooth. Then, the sampled optical pulse train is detected by the PD under test, and the spectrum of the output signal from the PD is acquired by an electrical spectrum analyzer (ESA). There are a large number of frequency components in the output electrical spectrum due to the heterodyne beating between any combinations of two frequency clusters in Fig. 1(b). The most important two groups of components are located at f_e (i.e., a fixed low frequency recovered from heterodyne beating between the microwave subcarrier and its envelope sidebands) and nf_r+2f_s-f_e (i.e., a varying frequency determined by n and f_s) as presented in Fig. 1(c1). When the microwave subcarrier is set at a single frequency in the sweeping range of 0<f_s≤f_r/2, a fixed low frequency of f_e and n+1 frequencies of nf_r+2f_s-f_e (n=0, 1, 2, …, 2N) can be obtained in one measurement process as shown in Fig. 1(c1). Through measuring the relative amplitude ratio between the two groups of frequency components at f_e and nf_r+2f_s-f_e, the response of the PD at the frequencies of nf_r+2f_s-f_e, namely R(nf_r+2f_s-f_e), can be calculated by taking the response at f_e, namely R(f_e), as a reference. It is noted that the relative amplitude includes not only the frequency response of PD itself, but also the uneven comb intensity of MLLD. In order to obtain the uneven comb intensity p₀/p_n, a microwave subcarrier at the specific joint-frequency of f_s’ is chosen to let the assumption of if_r+2f_s’-f_e≈(i+1)f_r-2f_s’+f_e (i=0, 1, 2, …, n-1) hold, and the uneven comb intensity p_i/p_i+₁ of two adjacent comb teeth can be calculated as illustrated in Fig. 1(c2). Then, the uneven comb intensity p₀/p_n of the MLLD is solved by multiplying the coefficient p_i/p_i+₁ (i is from 0 to n-1). Finally, through subtracting the uneven comb intensity of the ultrashort pulse train, the self-calibrated measurement of the PD at any frequency of nf_r+2f_s-f_e can be achieved. The following mathematical description will present the measurement process and the calibration process in detail.

Fig. 1. Schematic diagram of the proposed method, MLLD: mode-locked laser diode, PC: polarization controller, MZM: Mach-Zehnder modulator, EM-MS: envelope-modulated microwave source, PD: photodiode, DUT: device under test, ESA: electrical spectrum analyzer.

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Mathematically, for a MLLD with a center frequency of f₀ and a repetition rate of f_r, the output optical field can be expressed as [24]

(1)$${E_1}(t )= \sum\limits_{l ={-} N}^N {{q_l}{e^{j2\pi ({{f_0} + l{f_r}} )t}}}$$

where 2N+1 is the effective tooth number of the optical comb, and q_l is the amplitude of the corresponding comb tooth. The optical intensity can be written as

(2)$$P(t )= {|{{E_1}(t )} |^2} = {p_0} + 2\sum\limits_{n = 1}^{2N} {{p_n}\cos ({2\pi n{f_r}t} )}$$

where p_n represents the comb intensity of the optical pulse train, and is calculated as

\sum\limits_{l ={-} N}^{N - n} {{q_l}{q_{l + n}}} = {p_n},({n = 0,1,2,\ldots ,2N} )

The ultrashort optical pulse train samples an envelope-modulated microwave subcarrier, whose center frequency is f_s, and envelope is slowly varying at the low-frequency of f_e (f_e<<f_s≤f_r/2)

(3)$$v(t )= ({1 + {a_e}\cos 2\pi {f_e}t} ){V_s}\cos 2\pi {f_s}t$$

Thereinto, a_e and V_s are the envelope varying factor and the amplitude, respectively. After photonic sampling, the output optical field of the MZM can be given by [25]

(4)$${E_2}(t )= {E_1}(t )\cdot \left[ {{e^{ - j\frac{{m({1 + {a_e}\cos 2\pi {f_e}t} )\cos 2\pi {f_s}t}}{2}}} + \gamma {e^{j\frac{{m({1 + {a_e}\cos 2\pi {f_e}t} )\cos 2\pi {f_s}t}}{2} + j{\varphi_b}}}} \right]$$

where γ, φ_b and m are the asymmetric factor, the phase bias and the modulation index of the MZM, respectively.

The sampled optical pulse train is finally detected by the PD under test, and the output photocurrent is calculated as

(5)$$\begin{array}{l} i(t )= R \cdot {|{{E_2}(t )} |^2} = R\left[ {{p_0} + 2\sum\limits_{n = 1}^{2N} {{p_n}\cos ({2\pi n{f_r}t} )} } \right]\{{1 + {\gamma^2} + 2\gamma \cos [{m({1 + {a_e}\cos 2\pi {f_e}t} )\cos 2\pi {f_s}t + {\varphi_b}} ]} \}\\ = R\left\{ {2\gamma \sum\limits_{n = 0}^{{2N}} {{p_n}} \sum\limits_{p ={-} \infty }^{ + \infty } {\sum\limits_{q ={-} \infty }^{ + \infty } {\sum\limits_{k ={-} \infty }^{ + \infty } {{J_p}(m ){J_q}\left( {\frac{{{a_e}m}}{2}} \right){J_k}\left( {\frac{{{a_e}m}}{2}} \right)\cos [{2\pi ({n{f_r} \pm p{f_s} \pm q{f_s} \pm q{f_e} \pm k{f_s} \mp k{f_e}} )t \pm {\varphi_b}} ]+ \ldots } } } } \right\} \end{array}$$

where R is the responsivity of the PD. Through using Jacobi-Anger expansion, the amplitudes of the desired frequency components can be quantified by

(6a)$$A({n{f_r} \pm {f_e}} )= 8\gamma {p_n}R({n{f_r} \pm {f_e}} ){J_1}(m ){J_1}\left( {\frac{{{a_e}m}}{2}} \right){J_0}\left( {\frac{{{a_e}m}}{2}} \right)\cos {\varphi _b}$$

(6b)$$A[{n{f_r} \pm ({2{f_s} - {f_e}} )} ]= 4\gamma {p_n}R[{n{f_r} \pm ({2{f_s} - {f_e}} )} ]{J_1}(m ){J_1}\left( {\frac{{{a_e}m}}{2}} \right){J_0}\left( {\frac{{{a_e}m}}{2}} \right)\cos {\varphi _b}$$

It can be seen from Eqs. (6a) and (6b) that desired frequency components have a common factor of 4γJ₁(m)J₁(a_em/2)J₀(a_em/2)cosφ_b after up-and-down conversion sampling. The frequency component at f_e is taken as the reference in the case of n=0, and the frequency component at nf_r+2f_s-f_e is taken as the probe signal. Therefore, the frequency response of the PD at any frequency of nf_r+2f_s-f_e with respect to the fixed low-frequency of f_e can be calculated by

(7)$$\frac{{R({n{f_r} + 2{f_s} - {f_e}} )}}{{R({{f_e}} )}} = \frac{{2{p_0}}}{{{p_n}}} \cdot \frac{{A({n{f_r} + 2{f_s} - {f_e}} )}}{{A({{f_e}} )}},({n = 0,1,2,\ldots ,2N} )$$

It can be seen from Eq. (7) that the modulation response of the MZM is fully cancelled out. However, the obtained relative frequency response involves the uneven comb intensity p₀/p_n of the optical pulse train.

In order to measure the coefficient p₀/p_n, the microwave subcarrier is set at the specific joint-frequency of f_s’≈f_r/4 to guarantee that if_r+2f_s’-f_e≈(i+1)f_r-2f_s’+f_e (i=0, 1, 2, …, n-1) is maintained. In such a case, the responsivities of the PD at if_r+2f_s’-f_e and (i+1)f_r-2f_s’+f_e can be considered to be equal to each other, i.e., R(if_r+2f_s’-f_e)≈R[(i+1)f_r-2f_s’+f_e]. Thus, the uneven comb intensity p₀/p_n of the optical pulse train can be obtained by

(8)$$\frac{{{p_0}}}{{{p_n}}} = \prod\limits_{i = 0}^{n - 1} {\frac{{{p_i}}}{{{p_{i + 1}}}}} = \prod\limits_{i = 0}^{n - 1} {\frac{{A({i{f_r} + 2{f_s}^{\prime} - {f_e}} )}}{{A[{({i + 1} ){f_r} - 2{f_s}^{\prime} + {f_e}} ]}}}$$

After calibrating the uneven comb intensity p₀/p_n induced by the MLLD, the relative frequency response of the PD at any frequency of nf_r+2f_s-f_e can be extracted through subtly varying f_s from 0 to f_r/2. The most prominent advantage of this method is that the maximum measurement range is extended to (2N+1)f_r by using a single microwave source with a tuning range of only f_r/2.

3. Experimental demonstration

In the proof-of-concept experiment, an optical pulse train with a repetition rate of 9.954 GHz (f_r) is generated by a semiconductor MLLD (u2t TMLL 1550), whose optical spectrum is shown in Fig. 2(a). Then, the optical pulse train enters a push-pull LiN_bO₃ MZM (EOSPACE AX-0MSS-10) biased at its minimum transmission point to sample an envelope-modulated microwave subcarrier (f_e=100 kHz) generated by a microwave source (MS, R&S SMB100A). Figure 2(b) presents the output optical spectrum from the MZM when the center frequency of the microwave subcarrier is set to be 3 GHz (i.e., f_s=3 GHz). It can be seen from Fig. 2(b) that the comb teeth are suppressed, and the first-order sidebands are retained. Each modulation sideband should involve three frequency components corresponding to the envelope-modulated microwave subcarrier as shown in the inset of Fig. 2(a). Nevertheless, due to the limited resolution of the employed optical spectrum analyzer (YOKOGAWA AQ6370C, 0.02 nm, ∼2.5 GHz at 1550 nm), the three frequency components in each modulation sideband cannot be distinguished in Fig. 2(b). Finally, the sampled pulse train is detected by the PD under test (DSC 10H), and the output signal is analyzed by an ESA (R&S FSU50). The optical pulse train acts as an ultra-wideband optical stimulus to measure the frequency response of the PD up to 49.77 GHz [=(n+1)f_r, n=0, 1, 2, 3, 4].

Fig. 2. Measured optical spectra of the optical pulse train (a) before and (b) after photonic sampling, where the inset shows the electrical spectrum of the envelope-modulated microwave subcarrier.

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In order to characterize the uneven comb intensity p₀/p_n of the optical pulse train, the envelope-modulated microwave subcarrier is set at the joint-frequency of f_s’=f_r/4-0.95 MHz=2.48755 GHz. The desired frequency components if_r+2f_s’-f_e and (i+1)f_r-2f_s’+f_e (i=0, 1, 2, 3) keep a frequency difference of 4 MHz to guarantee that R(if_r+2f_s’-f_e)≈R[(i+1)f_r-2f_s’+f_e] is maintained. The electrical spectra are shown in Fig. 3 with the emphasis on the desired frequency components. For example, in the case of i=0, the desired two frequency components are measured to be -57.07 dBm at 4.975 GHz (2f_s’-f_e) and -58.37 dBm at 4.979 GHz (f_r-2f_s’+f_e), indicating the coefficient p₀/p₁=1.16 (=10^1.3/20). In the case of i=1, the frequency components are measured to be -58.95 dBm at 14.929 GHz (f_r+2f_s’-f_e) and -60.92 dBm at 14.933 GHz (2f_r-2f_s’+f_e), indicating the coefficient p₁/p₂=1.25 (=10^1.97/20). Therefore, the coefficient p₀/p₂ is calculated as 1.46 [=10^{(1.3+1.97)/20}] according to Eq. (8). Similarly, other uneven comb intensity p₀/p_n of the optical pulse train can also be obtained by multiplying these coefficients p_i/p_i₊₁ (i=0, 1, 2, 3).

Fig. 3. Measured electrical spectra emphasized on the desired frequency components at if_r+2f_s’-f_e and (i+1)f_r-2f_s’+f_e (i=0, 1, 2, 3).

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After calibrating the uneven comb intensity induced by the optical pulse train, the frequency response of the PD can be obtained at any frequency of nf_r+2f_s-f_e with respect to that at the frequency of f_e. Figure 4 illustrates typical electrical spectra emphasized on the desired frequency components at nf_r+2f_s-f_e and f_e. For example, when the relative frequency response of the PD at 20.56 GHz is under test, the microwave subcarrier is set at the frequency of f_s=0.32605 GHz (n=2) based on Eq. (7), and the frequency components are measured to be -57.13 dBm at 20.56 GHz (2f_r+2f_s-f_e) and -47.55 dBm at 100 kHz (f_e). Therefore, the frequency response of the PD is calculated to be -0.29 dB (=-57.13 + 47.55 + 3.27 + 20Log₁₀2) at 20.56 GHz (2f_r+2f_s-f_e) with respect to 100 kHz (f_e) according to Eq. (7). Similarly, the measurement can be easily implemented at other frequency by simply changing the frequency f_s (0<f_s≤f_r/2) of microwave subcarrier. Therefore, the relative frequency response of the PD under test can be extracted within about 50 GHz frequency range, as is displayed in Fig. 5. For the frequency response at nf_r, the assumption of R(nf_r)≈R(nf_r+f_e) is employed, and the frequency response of the PD at nf_r+f_e with respect to that at f_e can be obtained based on Eq. (6a). The measurement result with the proposed method is also compared to those with the method in [23] as well as the manufacturer data, as shown in Fig. 5. The consistency among these results verifies that the method is applicable for self-calibrating extraction of the PD frequency response. Moreover, the frequency measurement range of 49.77 GHz is achieved by using an MLLD with a repetition rate of 9.954 GHz (f_r) and an electro-optic frequency sweeping up to 4.977 GHz (f_s≤f_r/2).

Fig. 4. Measured electrical spectra emphasized on the desired frequency components at nf_r+2f_s-f_e and f_e.

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Fig. 5. Measured frequency response of PD with the proposed method and the method in [23] as well as the manufacturer data.

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Figure 6 shows the measured frequency response of the PD in the vicinity of 15.17 GHz by using the proposed method and the one in [23]. Hyperfine measurement with a frequency step of 300 kHz is achieved in the frequency range of 300 MHz via the proposed method, where further decreasing frequency step is mainly limited by the tuning resolution of the MS, the resolution bandwidth (RBW) of the ESA and the coherent status of the optical comb from the MLLD. As shown in Fig. 3, the electrical spectra feature with extremely narrow spectrum lines due to the inherent coherence of the optical pulse train from the MLLD. Meanwhile, the minimum RBW of the ESA is 10 Hz, and the tuning resolution of the MS is 1 Hz. Thus, the measurement step in the experiment is expected to reach 10 Hz level by further reducing the sweeping frequency step of the MS. In Fig. 6, the hyperfine results are also demonstrated in the case of the measurement frequency step of 50 kHz and 10 Hz, respectively, which proves the accuracy and the high resolution of the proposed method. It should be pointed out that only four measured data spaced by 96.9 MHz are obtained within the frequency range of 300 MHz by using the low-repetition-rate photonic sampling method [23], which is limited by the repetition rate of the MLLD. We also make a comparison between the proposed method, the method in [23], and other conventional methods, as illustrated in Table 1.

Fig. 6. Measured frequency response of the PD in the vicinity of 15.17 GHz with the proposed method and the method in [23].

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Table 1. Comparison between our method, method in [23], and other conventional methods.^a

View Table

4. Measurement uncertainty

For the measurement accuracy, the uncertainty of the frequency response of PD can be derived by the total derivative of Eq. (7)

(9)$$\frac{{\delta R({n{f_r} + 2{f_s} - {f_e}} )}}{{R({n{f_r} + 2{f_s} - {f_e}} )}} = \frac{{\delta ({{{{p_0}} / {{p_n}}}} )}}{{{{{p_0}} / {{p_n}}}}} + \frac{{\delta A({n{f_r} + 2{f_s} - {f_e}} )}}{{A({n{f_r} + 2{f_s} - {f_e}} )}}\textrm{ + }\frac{{\delta A({{f_e}} )}}{{A({{f_e}} )}}$$

In the proposed method, the uncertainty of p₀/p_n mainly depends on the response fluctuation of PD since the method is based on the approximation of R(if_r+2f_s’-f_e)≈R[(i+1)f_r-2f_s’+f_e]. The two frequencies at if_r+2f_s’-f_e and (i+1)f_r-2f_s’+f_e are set with only 4 MHz interval, corresponding to an uncertainty of less than 0.05 dB according to the specification of the PD under test. Therefore, the uneven comb intensity p₀/p_n of the MLLD would have an uncertainty of no more than 0.05×n dB. Meanwhile, since this method is based on the analysis and measurement of the electrical spectra, there will be a maximum power uncertainty of 0.1 dB or an amplitude uncertainty of less than 0.05 dB according to the specification of the employed ESA. Finally, according to Eq. (9), the measured frequency response of PD at any frequency of nf_r+2f_s-f_e would have a total relative error of no more than {10.^[0.05×(n+2)/20]-1}×100% (n=0, 1, 2, …, 2N) in the worst case. It means that a maximum relative error is less than 3.51% (n=4) in the experiment.

5. Discussion and conclusion

The proposed measurement method is independent of the phase bias of MZM, because the phase bias has the same influence on the desired frequency components and can be completely cancelled out through the amplitude ratio. In practice, it is recommended to adjust the phase bias of the MZM to make |cosφ_b| as large as possible for better signal-to-noise ratio (SNR), where the MZM is biased at its minimum transmission point in the experiment. Meanwhile, the method can work with different types of MZMs, such as push-pull MZMs, single-drive MZMs, and dual-drive MZMs. Besides, in the experiment, a high-repetition-rate optical pulse train is used as the optical source to measure the frequency response of the PD under test. However, the method is also applicable for the low-repetition-rate optical pulse train, thereby further reducing the frequency sweep range of the microwave subcarrier (0<f_s≤f_r/2). However, in this case, it should be noted that more multiplication coefficients are required during the characterization of the optical pulse train, which might lead to larger measurement error. Therefore, there is a trade-off between the repetition rate of MLLD and the measurement accuracy of PD. In the method, the frequency response of the PD at n+1 frequency components can be extracted when the microwave subcarrier is set at a single frequency. Then, the frequency response in the frequency measurement range can be extracted by sweeping the frequency of microwave subcarrier. If it is only necessary to measure the frequency response of the PD at a specific frequency, the center frequency of the microwave subcarrier can be inversely calculated according to Eq. (7).

Compared to the all-optical methods, this method shows very narrow-linewidth comb spectrum due to the inherently mutual coherence of the optical pulse train from the MLLD. In contrast to the electro-optic frequency sweeping method and the carrier-suppressed electro-optic frequency sweeping method, the proposed method does not depend on any extra electro-optic calibration, and achieves self-calibration and bias-drift-free measurement. Different from the frequency-shifted heterodyne method, the method achieves ultra-wideband frequency response measurement of PDs. Superior to the low-repetition-rate photonic sampling method, the method enables extracting hyperfine frequency response of PD over ultrawide frequency range.

In summary, we have proposed and demonstrated a photonic sampling method for frequency response measurement of high-speed PDs by employing an ultrashort optical pulse train to sample an envelope-modulated microwave subcarrier. In this method, the uneven efficiency of electro-optic modulation is fully cancelled out, and the uneven comb intensity of the optical pulse train is corrected through choosing a specific microwave subcarrier. Moreover, the hyperfine frequency response of PD at any frequency can be extracted by subtly varying the frequency of the microwave subcarrier from 0 to f_r/2. More importantly, the maximum frequency measurement range of (2N+1)f_r can be realized based on the ultra-wideband comb-like spectrum of the ultrashort optical pulse train. In the experiment, the self-calibrated frequency response measurement of a high-speed PD is demonstrated by employing an optical pulse train with a repetition rate of 9.954 GHz and an electro-optic frequency sweeping up to 4.977 GHz. The frequency measurement range is achieved up to 49.77 GHz (n=0, 1, 2, 3, 4), and the frequency resolution reaches 300 kHz in the rough measurement and 10 Hz level in the fine measurement. The measurement results obtained with the proposed method are compared to those with the conventional method, in terms of accuracy and fineness, proving the self-calibrated, hyperfine and ultra-wideband measurement ability of the proposed method.

Funding

National Key R & D Program of China (2018YFE0201900); National Natural Science Foundation of China (NSCF) (61927821); The joint research fund of MOE (6141A02022436); Fundamental Research Funds for the Central Universities (ZYGX2019Z011).

Disclosures

The authors declare no conflicts of interest.

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	Method	Optical source	MFR/GHz	SFR/ GHz	FRSF	Resolution	Calibration	Bias drifting
[18]	EOSF	CW	12	12	1	MHz	Extra Cal.	Sensitive
[21]	CSEOSF	CW	50	25	2	MHz	Extra Cal.	Sensitive
[22]	FSH	CW	30	15	2	kHz	Self Cal.	Independent
[23]	LSOS	MLLD	50	2.5	20	Tens of MHz	Self Cal.	Independent
Ours	EMMSPS	MLLD	50	5	10	10 Hz	Self Cal.	Independent

Frequency response measurement of high-speed photodiodes based on a photonic sampling of an envelope-modulated microwave subcarrier

Abstract

1. Introduction

2. Operation principle

3. Experimental demonstration

4. Measurement uncertainty

5. Discussion and conclusion

Funding

Disclosures

References

Cited By

Figures (6)

Tables (1)

Equations (11)

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