Accurate Shot-Noise-Limited Calibration of a Time-Domain Balanced Homodyne Detector for Continuous-Variable Quantum Key Distribution

The time-domain balanced homodyne detector (TBHD) is critical component in continuous-variable quantum key distribution (CVQKD). The shot-noise-limited attribute of a TBHD depends on the whole frequency spectrum of the detector, in contrast to a frequency-domain balanced homodyne detector that focuses only on part of the frequency spectrum. We propose a method for accurate calibration of the shot-noise-limited attribute of the TBHD by overlapping the gain lines of the TBHD and an integral photodetector which is realized by blocking one photodiode of the TBHD. The overlapping of two lines indicates that the detector is indeed shot noise limited, which is crucial to the security of CVQKD. Experimentally, we use this method to obtain a calibration error of $\pm 1$‰. Theoretical analysis showed that the CVQKD security is very sensitive to calibration errors of shot noise, and proper countermeasure was proposed to eliminate this issue. The presented methods are expected to contribute to CVQKD security and TBHD standardization.

Quantum key distribution is a key technology in quantum communication owing to its unconditional security in information theory [40], [41], [42], [43], [44], [45]. Depending on the measurement method, it can be divided into discreteand continuous-variable domains. In 1984, the BB84 protocol, which uses a single-photon detector, was proposed for the discrete-variable domain [46]. Up to now, a maximum transmission distance of 830 km in fiber has been achieved [47], and a satellite-ground platform has been demonstrated in this domain [48]. In 2002, the GG02 protocol using a TBHD was proposed for the continuous-variable domain [29], and demonstration experiment was performed in 2003 [31]. The GG02 protocol utilizes only the coherent state and does not require the squeezed or entangled states, so it has good compatibility with classical coherent communication technology. It has achieved a maximum transmission distance of 200 km [37], and is expected to be employed for quantum private networks in metropolitan areas.
In CVQKD, the quadratures of the signal field measured by the receiver are normalized to the shot noise, which can be determined by the power of the local oscillator (LO). To this end, a linear relationship between the shot noise variance and LO's power is calibrated before operation of a CVQKD system. Inaccurate calibration of the slope of the line will lead to security 0733-8724 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. loopholes. Calibration of the shot noise is the main difference between CVQKD and a coherent optical communication system, which utilizes an FBHD to measure the quadrature of electromagnetic modes to transmit information [49]. An FBHD only considers part of the frequency spectrum, and the shotnoise-limited attribute can be verified by using a standard radio frequency (RF) spectrometer [50]. This method is not suitable for TBHD which is used to detect pulsed beam. For a TBHD, the entire spectral range should be verified to be shot-noise-limited, especially at low frequencies. Due to limited CMRR, the low frequency intensity noise of LO pulse is difficult to be eliminated totally. The intensity noises of the LO pulse mainly come from the laser intensity noise and modulation process. To achieve a high CMRR and minimize the low-frequency remaining intensity noise for a TBHD, several techniques are utilized. For example, two matched photodiodes with adjustable bias voltages are employed and the output optical paths of the 50/50 coupler are set to be identical [39], [51], [52], [53]. Low noise pulsed modulation technology is also needed to calibrate a TBHD [54]. However, the variance of the output quadrature may still be mixed with the remaining intensity noise because of the limited CMRR. Thus, a linear relationship between the quadrature variance and LO power is not enough to make sure the TBHD is shot-noise-limited.
In this study, an accurate model for a TBHD was established, and the shot noise calibration process was analyzed in detail. A method is proposed for verifying the shot-noise-limited attribute by overlapping the fitting gain lines of the TBHD and the integral photodetector which is realized by blocking one photodiode of the TBHD. When the two gain lines overlap, the remaining intensity noise of output quadrature variance has been eliminated and the TBHD can be considered to be shot-noise-limited. Experiments were performed to validate the model and calibration performance, and a theoretical analysis is presented on the effects of inaccurate calibration on a CVQKD system. Finally, the countermeasures of decreasing the safe line of the TBHD is recommended to avoid security loophole due to inevitable calibration errors.

II. MODEL OF THE TBHD
An accurate model of the TBHD was established to clarify the calibration process as shown in Fig. 1. The signal pulseâ i and LO pulseâ l interfere at an ideal 50/50 coupler, and the output beamsl 1 andl 2 are guided into two ideal photodiodes PD 1 and PD 2 , both have quantum efficiencies of 100%. The paths from the coupler to PD 1 and PD 2 are denoted as paths 1 and 2, respectively. Beam splitters (BS) are used to model the losses of realistic couplers. PD 1 and PD 2 convert the input photons into photoelectrons, which are subtracted from each other. The subtracted photoelectrons are amplified by the charge amplifier and shaping amplifier, which consist of a high-pass filter, low noise-operation amplifier (OPA), and low-pass filter. R i is the input resistor, C i is the input capacitor, and C f is the feedback capacitor. The output impedance of the TBHD is 50Ω. This interference of the signal beamâ i and LO beamâ l can be described as follows l 1 where the beamsl 1 andl 2 have photon numbersn 1 =l † 1l 1 and n 2 =l † 2l 2 , respectively. In the idea case, the ideal photodiodes transformn 1 andn 2 photons inton e1 andn e2 photoelectrons, respectively, with perfect 100% efficiency. The subtracted pho-toelectronsN r are given bŷ Here we have treated the strong LO beam as classical field β = |β| · e iθ i , where |β| represents the amplitude of the LO beam and θ i is the phase of the LO beam,X θ i is the quadrature. Then, the subtracted photoelectronsN r are amplified by the following amplifiers with gain G. The peak value of the output voltage is given byÛ When the signal beam is in vacuum, the variance of the output peak value is Here, the variance Δ 2X θ i of the vacuum field is set to 1. The variance V peak has linearity dependence with the photon number per LO pulse |β| 2 , and the slope is G 2 . In the above derivation, N r represents the total subtracted photoelectrons number per LO pulse. The instantaneous subtracted photoelectrons current is defined asî r (t). The charge amplifiers integrateî r (t) over the duration time T of the pulsed photoelectrons current, which can be expressed bŷ The duration time T should be much shorter than the time constant of the charge amplifier. (5) can be rewritten aŝ When one photodiode of the TBHD is blocked, it is can be considered to be an integral photodetector. All of the photoelec-tronsN i of one photodiode are integrated and the output peak voltage iŝ The gain G can also be measured by using the integral photodetector. Note that an FBHD can also be used to detect the pulsed light. However, there is no charge amplifier. Normally, a transimpedance amplifier with a feedback resistor R is used to amplify the subtracted current signal directly. When the duration time of the optical pulse is much greater than the response time of the FBHD, the output voltage is given bŷ To obtain the total subtracted number of photoelectrons with the gain R, which is given byN r · R. A number of points should be sampled in a single signal pulse and then numerical integration is required to calculate the pulse area. The above analysis shows that both the TBHD and FBHD can be used to detect the quadrature of a pulsed signal. The TBHD only needs to acquire the peak point, but the FBHD needs to acquire many samples and then integrating them, which increases the amount of data processing complexity.
A realistic 50/50 coupler may have a small deviation from the ideal coupler ratio of 1/2, and there may be losses when the signal beamâ i and LO beamâ l transmit through the 50/50 coupler. The four losses from beamsâ i ,â l to beamsl 1 ,l 2 have slight discrepancies. The quantum efficiencies η PD1 , η PD2 of the two photodiodes also differ in general. To analyze various losses, a model was constructed under realistic conditions (see Appendix A). We find that the measurement of the shot noise depends on the detected photoelectrons number. To simplify the calibration process, the model in Fig. 1 can be transformed to have two output paths with the transmission efficiencies η 1 and η 2 . When the two output paths are tuned to a balance, the two models are equivalent for the calibration of the shot-noiselimited attribute.
As shown in Fig. 1, BS 1 and BS 2 , which model the transmission efficiencies η 1 and η 2 , are placed on output paths 1 and 2, respectively. For path 1, beaml 1 interferes with the signal beam in vacuum fieldv 1 . The output beams areô 1 andû 1 . The incident photon number in PD 1 is given bŷ For path 2, the photon number received by PD 2 is given bŷ In experiment, a variable optical attenuator (VOA) based on bending the fiber can be inserted into one path with a higher transmission efficiency to balance the two paths [53]. For convenience, the transmission efficiencies of the two paths are assumed to be identical, or η 1 = η 2 = η. Then, the subtracted photoelectronsN r are given bŷ The photon number per LO pulse is usually about 10 6 ∼10 8 , and the photon number per signal pulse is usually about 10 0 ∼ 10 1 . The value of the terms that do not includeâ l is very small compared to the terms that includeâ l . By neglecting the small terms withoutâ l , (11) is rewritten aŝ whereX v1 andX v2 are the quadratures of vacuum fieldsv 1 andv 2 , respectively. Combine (6) and (12), the variance of the output peak values of the TBHD is given by In reality, the output ratio of the 50/50 coupler may change because of the temperature fluctuations. This is equivalent to that BS 1 and BS 2 having different transmittances, which can be represented by η+ε and η−ε, respectively (see Appendix A). Then, the photon number received by PD 1 and PD 2 are The number of subtracted photoelectronsN r is given bŷ The term εâ † lâ l can be transformed by usingâ l = β+δâ l The term ε has a value range from 10 −3 to 10 −4 , and |β| 2 has a value range from 10 6 to 10 8 . Thus, the term ε · δâ † l δâ l can be neglected. The term can be simplified as where η is 0.6 to 0.8 for a fiber-based TBHD. Thus, the term can be neglected. The number of subtracted photoelectronsN r can be simplified tô At this stage, we have Eq. (22) shows that an imbalance between the two beams will introduce the remaining intensity noise G 2 · (ε|β|) 2 · (δX l ) 2 . The term (δX l ) 2 represents the intensity noise of the LO pulse and usually has a value greater than 1 at low frequencies. In experiments, the displacement term ε|β| 2 does not vary with the phase θ. Thus, it can be measured and used to balance the two paths with auto-balancing technique [53]. For a TBHD, the balance or CMRR of the two paths not only depends on the intensity of the two paths but also affected by the response characteristics of the two photodiodes. In a CVQKD system, the transmission efficiencies of signal and LO beams from the input port of the receiver to the input port of the TBHD are η s and η o respectively, which are modeled by BS s and BS o in Fig. 1. The transmission efficiency of the quantum channel is η ch (modeled by BS ch ). This model is suitable for a CVQKD system with or without a LO beam transmitting together with the signal beam [55]. Similar to the derivation method described above, the subtracted photoelectrons N r can be derived as follows whereX s is the quadrature of the signal beamâ s .X ch andX vs are the quadratures of the vacuum fieldsv ch andv s . |γ| is the amplitude of the LO beam at the input port of the receiver. When the signal beamâ s is in vacuum, the variance V peak becomes (24) shows that V peak has a linear relationship with the photoelectron number per pulse η|β| 2 and that it has no relationship with the losses of the signal beam.

III. CALIBRATION OF THE SHOT NOISE
When a TBHD is shot-noise-limited, it should have a linear gain G and its output quadrature values are independent [39]. The traditional method of measuring the linear gain G of TBHD is based on (24). However, this method may be affected by the intensity noise of the LO and the limited CMRR as shown in Section II. To accurately calibrate the TBHD, three gain measurement methods are presented in this section. By introducing high pass filter and comparing the fitting gain lines, the shot noise limited attribute can be accurately calibrated. Fig. 2(a) shows a fiber-based TBHD. The signal and LO beams were injected into a 2 × 2 polarization-maintaining 50/50 fiber coupler, whose two output ports were connected to two photodiodes. A VOA based on bending the tail fiber of the coupler was utilized to balance the TBHD [53]. The subtracted photoelectrons from the two photodiodes are amplified by a low noise charge amplifier. Then, a shaping amplifier was used to amplify and reshape the output pulses to a Gaussian shape.
The linear gain G is determined by the charge amplifier and shaping amplifier, which can be measured directly using the electronic method as shown in Fig. 2(b). A pulse generator generated electronic square-pulsed voltage signals with pulse width of 1 μs and pulse repetition frequency of 500 kHz. The electronic signals were attenuated by a precision attenuator (50DR-119). Then the attenuated voltage signals with amplitude U i were guided into electronic amplifiers. The input resistor R i had a resistance of 5 MΩ, and the input capacitor C i had a capacitance of about 100 pF. The input electron number per electronic pulse N ele could be calculated by where Q i is the quantity of electric charge per electronic pulse, and e is the electric charge of one electron. By varying the input voltage U i (N ele ), the output peak voltage value U e varied accordingly as shown in Fig. 3(a). Do a linear fitting to the experimental data, a fitting black solid line GE was obtained. Its slope is the measured gain. To discriminate the real gain G and the measured gain, the measured gain is noted G e . In experiment, the measured gain was G e = 7.3 × 10 −6 V/electron. The sum of residual squares for the linear fitting was 6.10 × 10 −29 V 2 , which indicates that the amplifiers had good linearity. The relationship between U e and N ele can be expressed by When one photodiode of the TBHD is blocked, it becomes an integral photodetector as shown in Fig. 2(c). A 1550.4 nm pulsed laser with a repetition rate of 500 kHz and pulse width of 100 ns, followed by a fiber VOA, was directed into the photodiode PD 2 on path 2. The photon number per optical pulse N i was tuned from 3 × 10 4 to 2 × 10 5 . Fig. 3(b) shows relationship between the measured output peak voltage value U p versus the photoelectron number per pulse N phe , which is the effective photon number per optical pulse η P D2 N i . The quantum efficiency of the photodiode was calibrated to η P D2 = 80.2%. From the slope of the fitting line GP, the gain is determined to G p = 7.21 × 10 −6 V/photoelectron. The sum of residual squares for the fitting was 4.55 × 10 −4 V 2 . The integral photodetector exhibited good linearity It should be noted that the measured peak values U e and U p of output voltage pulses contain the electronic noise. They are measured by averaging many peak values, so various fluctuations of noise will be eliminated. The gains G e and G p showed only slight differences, which can mainly be attributed to a ±5% deviation in the capacitor C i . Thus the measured gain G p is more accurate than G e , and fitting line GP can be seen as the safe line.
For the TBHD, only LO port was injected by 1550.4 nm pulsed laser. A CMRR of 76 dB was achieved by selecting matched photodiodes. The output pulsed voltage had a repetition rate of 500 kHz and full width of about 400 ns. Fig.  4(a) shows the time traces of the output pulsed voltage. The photoelectron number per pulse is 10 7 , and the signal to noise ratio is SNR = 10 log 10 (V m /V ele − 1) = 11.5 dB. Here V m is the variance of the measured peak voltage value U m , and V ele is the electronics variance. Fig. 4(b) shows the various noise variances versus photon number per LO pulse |β| 2 . The black squares represents the variance V m . The red circles represent the variance V ele . The green triangles represent the measured variance V m peak , which can be obtained by Finding the square roots of both sides of (28) results in Fig. 3(c) plots the squared data of the green triangle points in Fig. 4(b). The title of horizontal coordinate of Fig. 3(c) is square root of photoelectron number per LO pulse η · |β| 2 , and the title of longitudinal coordinate is standard variance of V m peak or V m peak . From the slope of the fitting line GM, the gain is determined to G m = 7.16 × 10 −6 V/particle. The sum of residual squares for the fitting was 7.63 × 10 −8 V 2 . Fig. 3(d) shows that the fitting gain lines GM and GP nearly overlapped with an error about ±1‰ in the experiment, a stable low-loss fiber connector and suitable high-pass filter were used to minimize the measurement error. Fig. 4(c) shows the whole spectra of TBHD when a continuous 1550.4 nm laser beam was injected into the LO port. Fig. 4(d) and (e) plot the spectra at low frequency when one and three orders high-pass filter were used respectively. Due to the low frequency intensity noise of the pulsed LO and limited CMRR, the gain G m is usually higher than the gain G p . When the low frequency intensity noise decreases, the gain G m will approach the gain G p . In the calibration process, the gain G m of the TBHD using one-order high-pass filter at low frequency is 5% -6% higher than the gain G p . By using the three-order high-pass filter the fitting line GM is well overlapped with fitting line GP with error about 1‰.
The experiment results of Fig. 3 demonstrate that the improved TBHD has a good linear gain G. To further ensure that the  TBHD is indeed shot noise limited, the output peak voltage values should be independent. To ensure the measured peak voltage values or the corresponding quadratures are independent [39], we investigated the correlation coefficients between adjacent values. A sequence of measured peak voltage values denoted by X i (i = 1, 2, . . . , n) were used. The correlation coefficient CC(m) can be calculated as where E(X) represents the expectation value of X [51]. The value of n is 10 5 , and m takes the value between −5 and 5. The calculation results are shown in Fig. 4(f). The black star points represent the correlation coefficients of the measured peak voltage values of pulsed shot noise at SNR = 11.5 dB, and the red square points represent the correlation coefficients of electronic noise. We can see clearly that a complete correlation occurs when m = 0, whereas the correlation vanishes for m = 0. Then we can conclude that measured peak voltage values (corresponding to the quadratures of vacuum field) are independent and the TBHD is surely shot noise limited.

IV. THE SECURITY ANALYSIS OF CVQKD SYSTEM DUE TO INACCURATE SHOT NOISE CALIBRATION
In the CVQKD systems, various types of variances are normalized to the shot noise variance. If the shot noise cannot be calibrated accurately, security loopholes may be generated.
In CVQKD experiments, the measured variance V m peak can be obtained using (28).
where V m is the total variance measured at the receiver side and V ele is the electronic noise. V B and V e are the corresponding normalized values. V 0 represents the shot noise variance with the real gain G. The variance of the signal beam is given by where V A is the normalized modulation variance and the value of 1 represents the shot noise variance. They are both the normalized values. Based on (23), (31) can be rewritten as where ξ is the excess noise of the CVQKD system. We use the calibrated gain G p and the corresponding shot noise V p 0 to normalize the measure variance. In this case, (33) becomes V B and V e are calibrated as V p B = V B V 0 /V p 0 and V p e = V e V 0 /V p 0 respectively, and the transmission efficiency is calculated as η p ch = η ch V 0 /V p 0 . The estimated excess noise will be Then, (34) can be rewritten as When G p > G or V p 0 > V 0 , the estimated excess noise ξ p is less than the real excess noise ξ. The values of the other parameters used to calculate the secret key rate also change. The secret key rate under collective attack is given by [32] where ΔI AB is the secret key rate, I AB is the Shannon mutual information between Alice and Bob, χ BE is the von Neumann information eavesdropped by Eve, and β is the reconciliaiton efficiency. Fig. 5 shows the estimated excess noise and secret key rate versus the transmission distance at different ratios R = V 0 /V p 0 under collective attack. The green, blue, and red dotted lines represent the estimated excess noise ξ p at ratios of 0.9999, 0.999, and 0.99 respectively. When the calibrated value G p or V p 0 is larger, the ratio is smaller, and the estimated excess noise ξ p decreases more rapidly with the distance. The decreasing of estimated excess noise ξ p will lead to overestimation of the secret key rate, as indicated by the green, blue, and red solid lines. The black solid line represents the secret key rate when the shot noise is calibrated accurately (R = 1). When the estimated excess noises ξ p decrease to zero or less than zero, it is set to zero. This will lead to the changing of the secret key rate trend. The black dot line represents the real excess noise ξ = 0.05 of the CVQKD system. By comparing the red, blue and green solid lines with the black solid line, we can see that when G p > G, the calculated secret key rate will be overestimated. It means that loophole is generated.
When G p < G or V p 0 < V 0 , the estimated excess noise ξ p is greater than the real excess noise ξ. Fig. 6 presents the estimated excess noise and secret key rate versus the transmission distance at different ratios R = V 0 /V p 0 under collective attack. The green, blue, and red dotted lines represent the estimated excess noise ξ p at ratios of 1.0001, 1.001, and 1.01, respectively. The black dotted line represents the real excess noise of the CVQKD system. When the calibrated value G p or V p 0 is smaller, the ratio is larger, and the estimated excess noise ξ p increases more rapidly with the distance. Correspondingly, the secret key rate decreases more rapidly, as shown by the green, blue, and red solid lines. By comparing the red, blue and green solid lines with the black solid line in Fig. 6, we can see that when G p < G, the calculated secret key rate will be underestimated, which degrade the performance of the CVQKD.
In the CVQKD systems, the laser and modulators used to generate LO pulses may be disturbed by the electromagnetic interference or other factors, and the LO pulses in the transmission channel may be attacked by Eavesdropper. Then, the intensity noise of the LO pulses will become larger, and it may exist in all the frequency band of the TBHD. In this case, the noise V m 0 of the LO pulses will be greater than V 0 . Eq. (33) becomes When the gain G p = G is used to normalize the measured variance, (38) is transformed to Here, the the estimated excess noise ξ p and secret key rate versus the transmission distance are the same as the results shown by the dotted and solid lines in Fig. 6 at the ratios 1.0001, 1.001, and 1.01. However, the meaning is different. When the gain G p = G, intensity noise of the LO pulse will degrades the performance of CVQKD. However, no loophole is generated. In paper [56], there is no safe line GP or V p 0 to normalize the variance, the intensity noise of LO pulse will generate loopholes.
Notice that, measurement errors during calibration of the TBHD are usually inevitable. To avoid security loopholes caused by a high gain G p , we can decrease the slope of the safe line GP or gain G p . The decreased gain is noted as G d , and the corresponding shot noise is represented by V d 0 . Eq. (33) becomes In this situation, the estimated excess noise ξ p and secret key rate versus the transmission distance are the same as the results shown by the dotted and solid lines in Fig. 6 Then the estimated excess noise ξ p and secret key rate versus the transmission distance are represented by the dot-dash and dash lines in Fig. 6 at the ratios R = V m 0 /V d 0 of 1.0002, 1.002, and 1.02. It means at the situation of LO pulse with higher intensity noise V m 0 and decreased shot noise V d 0 , the secret key rate will be underestimated. It should be noted that no loophole is generated. We can see that the method of decreasing the slope of safe line can be used to ensure that noise V m 0 will not generate loopholes but only reduce the performance of the CVQKD system.
The above results show that the secret key rate is very sensitive to the shot noise calibration. Calibrating the shot-noise-limited attribute of TBHD with high accuracy is critical for ensuring the safety of CVQKD systems. By decreasing the calibrated gain of the TBHD, we can effectively avoid the security loopholes due to the measurement error during the shot-noise calibration.

V. CONCLUSION
In this study, an accurate model and method for calibrating the shot-noise-limited attribute of a TBHD were established. The proposed method involves overlapping the gain lines of the integral photodetector and TBHD to calibrate the shot-noise-limited attribute. The gain line measured by the integral photodetector can eliminate the effect of various noises. To eliminate the influence of remaining intensity noise at low frequencies, a highorder high-pass filter is utilized. In experiments, a measurement error of about ±1‰ was achieved. The effect of measurement error on the security of the CVQKD system was analyzed, and a countermeasure was proposed to avoid security loopholes. The proposed methods can be applied to CVQKD systems with or without LO beam transmitting together with the signal beam, We can't sure whether the losses of realistic coupler are before or after the coupler, or both. Then, all the losses before and after the coupler are considered, and they are equivalent to BS 1 , BS 2 , BS 3 , and BS 4 with transmission efficiencies η 1 , η 2 , η 3 , and η 4 . We assume that PD 1 and PD 2 are ideal photodiodes with 100% efficiency. The nonideal efficiencies of the photodiodes are combined into the transmission efficiencies of BS 1 and BS 2 , respectively.
The signal beamâ i interferes with the LO beamâ l on the 50/50 coupler. The output beamsl 1 andl 2 interfere with the vacuum beamsv 1 andv 2 , and outputô 1 andô 2 , respectively. Above transformations are given as follows: The number of photoelectronsN 1 in PD 1 is given bŷ The number of photoelectronsN 2 in PD 2 is expressed bŷ The number of subtracted photoelectrons is given bŷ To derive (46), small terms have been neglected.X θ is the quadrature, and θ is the LO phase. To calibrate the shot-noiselimited attribute, the signal beam is a vacuum field. Then, the variance of the subtracted photoelectrons is given by Considering the losses of signal path and LO path, the relations ofâ 3 ,â 4 andâ i ,â l are given by Using (48), (46) is rewritten aŝ Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Assume that the signal beamâ 3 is a vacuum field and |γ| is the amplitude of the LO beamâ 4 , the variance of the subtracted photoelectrons is given by A comparison between (47) and (50) shows that the transmission efficiency η 3 of the signal path has no influence on the variance V r when the signal beamâ 3 is a vacuum field. The transmission efficiency η 4 affects the number of photoelectrons, with η 4 |γ| 2 = |β| 2 . To calibrate the shot-noise-limited attribute, a VOA is inserted into one of the output paths of the coupler to ensure that the two output paths are balanced In this case, the variance of the subtracted photoelectrons is given by where 2η i η 1 |β| 2 or 2(1 − η i )η 2 |β| 2 is the number of photoelectrons of the two photodiodes. The efficiency η i has a small deviation from 1/2 that can be written as η i = 1/2±δ. Before balancing the TBHD, suppose that η i η 1 = (1/2±δ)η 1 is greater than (1 − η i )η 2 = (1/2∓δ)η 2 . The transmission efficiency η 1 can be decreased by the a VOA to balance the two output paths. For the model in Fig. 1, when the two output paths are balanced, the transmission efficiency of either path 1 or path 2 is The two models have the same number of photoelectrons. We can see that all the losses in the model of Fig. 7 can be placed after the coupler in the model of Fig. 1. Thus, the two models are equivalent when the two output paths are balanced. The transmission efficiency of path 2 without the VOA can be used to calibrate the total transmission efficiency. The transmission efficiency η i may change due to the environmental temperature. If the transmission efficiency changes slightly, η i becomes η i = 1/2 ± δ ± Δ, and 1 − η i becomes 1 − η i = 1/2 ∓ δ ∓ Δ. For path 1, the transmission efficiency changes ±Δ · η 1 ; for path 2, the transmission efficiency changes ∓Δ · η 2 . Because Δ is a small value, the difference between η 1 and η 2 is very small, the approximation Δ · η 1 ≈ Δ · η 2 represented by ε can be used. When the VOA is used to rebalance the two paths, the efficiency of the calibration path is changed from (1/2∓δ)η 2 to (1/2 ∓ δ ∓ Δ)η 2 . This will introduce a measurement error during the calibration process.