As we all know, for IM/DD system directly incident light field on the photodetector only signal light E_S(t), so the output of the detector should be:

S P D = i P D 2 R L = β 2 P S 2 R L . (1)

In Equation (1), β is the responsiveness of the photodetector.

Suppose the signal light of heterodyne detection is E S ( t ) = A s cos ( ω S ( t ) + ϕ s ) , The local oscillator light is E L ( t ) = A L cos ( ω L ( t ) + ϕ L ) , Then the output of ordinary heterodyne detection is [6]:

i P D = β [ E S ( t ) + E L ( t ) ] 2 ¯ = β { A S 2 cos 2 ( w s t + ϕ s ) ¯ + A L 2 cos 2 ( w L t + ϕ L ) ¯ + A s A L cos [ ( w s + w L ) t + ( ϕ s + ϕ L ) ] ¯ + A s A L cos [ ( w s − w L ) t + ( ϕ s − ϕ L ) ] ¯ } . (2)

Due to the limitation of intermediate frequency bandwidth, the photodetector can only respond to the difference frequency term, then the final detected current signal is [7]:

i out = β A s A L cos [ ( w s − w L ) t + ( ϕ s − ϕ L ) ] . (3)

The principle of balance detection is shown in Figure 1. After the signal light E_s and the local oscillator light E_L are input to the beam splitter B through the optical fiber, two signals lights (E_s1 and E_s2) and two local oscillator lights (E_L1 and E_L2) are generated respectively. Where E_S1 and E_L1 are received by photodetector 1 and output photocurrent i1; E_s2 and E_L2 are received by photodetector 2 and the output current is i2. The phase of the photocurrent output by the two detectors is opposite, and the final IF signal is obtained after the difference frequency.

It is known that the signal light of heterodyne detection is E S ( t ) = E S cos ( ω S ( t ) + ϕ S ) , and the local oscillator light is

E L ( t ) = E L cos ( ω L ( t ) + ϕ L )

where: E S , ω S , ϕ s and E L , ω L , ϕ L are the amplitude, frequency and phase of the signal light and the local oscillator light respectively. Assuming that the fraction ratio of the beam splitter is ε, the quantum efficiency of the two detectors is η₁ and η₂, and the bandwidth ∆f is equal, the energy of the signal light and the local oscillator light after passing through the beam splitter can be calculated as [8] [9]:

E s 1 = 1 − ε E s cos ( ω s ( t ) + ϕ s ) E s 2 = ε E s cos ( ω S ( t ) + ϕ S ) E L 2 = 1 − ε E L cos ( ω L ( t ) + ϕ L ) E L 1 = ε E L cos ( ω L ( t ) + ϕ L ) (3)

The output of the four-way signal after mixing with the photodetector is:

i 1 = η 1 e h ω { ( 1 − ε ) E s 2 + ε E L 2 − 2 1 − ε ε × E s E L cos [ ( ω s − ω L ) t + ϕ s − ϕ L ] } + n 1 i 2 = η 2 e h ω { ε E s 2 + ( 1 − ε ) E L 2 + 2 1 − ε ε × E s E L cos [ ( ω s − ω L ) t + ϕ s − ϕ L ] } + n 2 (4)

In the Equation (4), n₁(t) and n₂(t) are noise currents. The response current signal of the final balanced detection system is as follows:

i = i 2 − i 1 = e h ω { [ ε η 2 − ( 1 − ε ) η 1 ] E s 2 + [ ( 1 − ε ) η 2 − ε η 1 ] E L 2 } + e h ω { 2 ( η 1 + η 2 ) 1 − ε ε E s E L × sin [ ( ω s − ω L ) t + ϕ s − ϕ L ] } + { n 2 ( t ) − n 1 ( t ) } . (5)

The double balance detection system is based on the structure of the balance heterodyne detection system, which uses four almost identical photoelectric detectors to form two groups of balance heterodyne detectors at the receiving end [10] [11] [12]. The photocurrent signal received by the detector is processed by differential processing to obtain the difference frequency signal. Its principle block diagram is shown in Figure 2.

The following mathematical model is used to illustrate this process.

Let the mathematical expression of the local oscillator light and the signal light be as follows:

E L = [ k 1 k 2 ] exp ( i ϕ ) E S = [ k 3 k 4 ] exp ( i ω i f t ) (6)

where, k₁, k₂, k₃ and k₄ represent the respective components of the local oscillator light and the signal light in two directions respectively.

Let E L ′ be the circularly polarized light formed after the local oscillator light passes through the 1/4 wave plate, then:

E L ′ = [ k 1 exp ( i π / 2 ) k 2 ] exp [ i ( ϕ − π / 4 ) ] . (7)

After the two beams of light pass through the beam splitting prism, they are divided into two beams of light in the horizontal and vertical directions, and the beam splitting prism transmits and reflects the matrix.

T = [ t ∥ exp ( i τ 1 ) 0 0 0 ] , R = [ 0 0 0 r ⊥ exp ( i ρ ⊥ ) ] . (8)

It can be obtained from the above equation.

E 1 = [ t I exp ( i τ ∥ ) k 1 exp [ i ( ϕ + π / 4 ) ] r ⊥ exp ( i ρ ⊥ ) k 4 exp ( i ω i f t ) ] E 2 = [ t 1 exp ( i τ ∥ ) k 3 exp ( i ω i f t ) r ⊥ exp ( i ρ ⊥ ) k 2 exp [ i ( ϕ + π / 4 ) ] ] (9)

The transmission matrix of 1/2 wave plate in the system is

Λ 1 / 2 = 2 2 [ 1 1 1 − 1 ] . (10)

In Equation (10), the angle between the wave plate and the X-axis is 22.5˚.

Two beams of light through the beam splitter are divided into four different directions of the light signals, the photodetector after detection of the four channels phase difference π/2 photocurrent signal.

I 0 = 1 2 { k 1 2 + k 4 2 + 2 k 1 k 4 cos [ ω i f t + ρ ⊥ − τ ∥ − ( ϕ + π / 4 ) ] } . (11)

Similarly, the expressions of I90, I180 and I270 can be derived, which will not be expanded in detail here. Four channels of signal for differential processing, obtained two groups of phase difference π/2 differential signals, namely orthogonal component I and vertical component Q.

I : I 0 − I 180 = 2 k 1 k 4 cos [ ω i f − ( ϕ + π / 4 ) ] Q : I 90 − I 270 = 2 k 2 k 3 sin [ ω i f − ( ϕ + π / 4 ) ] (12)

Compared with the direct detection method (Equation (1)), the ordinary heterodyne detection (Equation (3)) has the advantage of realizing the detection of the amplitude, frequency and phase of the signal light and improving the detection accuracy. It's good for detecting weak signals. According to the formula of photocurrent, it can be seen that the size of the photocurrent is proportional to the square root of the product of the power of the signal light and the local oscillator light. Therefore, when detecting weak signal, even if the signal light is very weak, as long as the local oscillator light power is large enough, a better signal can still be obtained eventually.

From Equation (5), it can be seen that balanced detection has advantages over single-source detection: on the one hand, it eliminates the DC component in the signal, which is convenient for signal processing, on the other hand, the amplitude of the AC component is double that of the output current of the single-source detector. In addition, due to the differential amplification used in the balanced detection, the common mode noise signals in the two detectors are greatly suppressed, and the signal-to-noise ratio of the output signals is improved. Dual balance detection not only has the advantage of balance detection, but also has a variety of modulation and demodulation methods, so its application prospect is very broad.

There are three types of noise in microwave photonic link: thermal noise, excess intensity noise and shot noise. The characteristics of each noise are analyzed in detail in the following sections [13].

Thermal noise is the noise caused by the Brownian motion of free electrons in the resistor. It has randomness and can only be increased but not eliminated. Therefore, in the balanced detection system, the thermal noise is the superposition of the thermal noise of the two detectors. If the resistance of the two detectors is equal, the thermal noise power is:

p th = 4 k T Δ f . (13)

where k is Boltzmann constant, T is the temperature, ∆f is the bandwidth of the receiver.

Excess intensity noise is caused by fluctuations in local oscillator light output intensity and is usually described as relative intensity noise (RIN):

R I N = 10 log 〈 p N 2 〉 〈 p 〉 2 Δ f . (14)

where 〈 p N 2 〉 is the mean square value of noise power, 〈 p 〉 2 is the average output power of the laser, ∆f is the bandwidth of the receiver. The unit of relative intensity noise is dBc/Hz. It can be seen that the mean square value of laser noise power is proportional to the relative intensity noise, and the value range of relative intensity noise is −160 - −120 dBc/Hz [14].

p rin = 2 e γ I 2 dc Δ f . (15)

2 e γ = 10 R I N 10 . (16)

where, γ represents the variation of noise power of the local oscillator with the square of the average power of the local oscillator. The unit is A⁻¹, and its corresponding value range is 10² - 10⁶ A⁻¹.

Assuming that the bandwidth of the two detectors is the same, since the excess intensity noise incident to the two detectors respectively is related, the square value of the excess intensity noise current of the balanced detector is [15]:

p rin = 2 e γ [ η 1 ε E 1 2 − η 2 ( 1 − ε ) E 1 2 ] Δ f . (17)

Shot noise is caused by the random fluctuation of photoelectric current in the photodetector. The shot noise current is a random process of Poisson distribution. The power spectral density of the shot noise is flat, so for the balanced detection system, the shot noise of the two signals after differential processing can be expressed as:

p sh = 2 e I D C Δ f = 2 e ( I D C 1 + I D C 2 ) Δ f . (18)

Therefore, the SNR of the balanced detection method is:

S N R balance = i signal 2 p s h + p r i n + p t h = 2 e 2 E S 2 E L 2 ( η 1 + η 2 ) 2 ( 1 − ε ) ε / h 2 ω 2 2 e 2 h ω E L 2 [ η 1 ε + η 2 ( 1 − ε ) ] Δ f + 2 e 3 h 2 ω 2 γ E L 4 [ η 1 ε − η 2 ( 1 − ε ) ] 2 Δ f + 4 k T Δ f . (19)

Compared with shot noise, thermal noise can often be ignored:

S N R balance = ( η 1 + η 2 ) 2 ( 1 − ε ) ε E S 2 / h ω ( η 1 ε + η 2 ( 1 − ε ) ) Δ f + ( e h ω ) γ E 1 2 [ η 1 ε − η 2 ( 1 − ε ) ] 2 Δ f . (20)

Assume that the photosensitive area of one of the balanced detectors is A₁, and let A₁ approach to infinity or 0, then the signal-to-noise ratio formula of detector 1, namely ordinary heterodyne detection, is:

S N R 1 = ( A 1 η 1 + η 2 ) 2 ( 1 − ε ) ε E S 2 / h ω ( A 1 η 1 ε + η 2 ( 1 − ε ) ) Δ f + ( e h ω ) γ E 1 2 [ A 1 η 1 ε − η 2 ( 1 − ε ) ] 2 Δ f = η 1 ( 1 − ε ) E S 2 / h ω B W + ( e h ω ) γ E L 2 η 1 ε Δ f . (21)

According to the SNR formula of Equation (20) for balanced coherent detection and the signal-to-noise ratio formula of Equation (21) for single source coherent detection, MATLAB software is used to carry out numerical simulation respectively. Some parameters in the SNR formula are as follows: the selected laser frequency is 150 THz, the photoelectric probe adopts InGaAs high-sensitivity PIN photoelectric diode, the band width BW = 1.0 × 10⁶ Hz, and the quantum efficiency is η = 1 A/W. It is assumed that the local oscillator power output by the fiber laser is 1 mW, and the signal light power is 0.1% of the local oscillator power, i.e., 1 μW. The relative intensity noise coefficient γ of the local oscillator light is 10³ A⁻¹. Beam splitting ratio is variable, and the relation diagram of system noise and beam splitting ratio can be obtained.

It can be seen from Figure 3(a) that the maximum SNR of the balanced coherent detection system is 3.5 dB higher than that of the ordinary coherent detection. When the beam splitting ratio is 0.13 or 0.87, the SNR of the balanced coherent detection is exactly equal to that of the ordinary coherent detection. When the beam splitting ratio is less than 0.13 or more than 0.87, the SNR of the balanced coherent detection system is less than the SNR of the single source coherent detection.

If the above conditions remain unchanged and the signal optical power is reduced from 0 dBm (1 mW) to −60 dBm (10⁻⁶ mW), the SNR of the balanced coherent detection system and the SNR of the single source coherent detection are shown in the Figure 3(b).

It can be seen from the Figure 3(b) that the SNR of balanced detection is about 3 dB higher than that of ordinary heterodyne detection, under the condition that the signal optical power is taken as the variable.

The balance detection system model and the ordinary heterodyne detection system model are set as shown in Figure 4(a) and Figure 4(b). The local oscillator light power is set as 1mW, and the signal light carrier frequency is 100 MHz. The polarization direction of the signal light is strictly the same as the local oscillator light. The photoelectric probe adopts PIN photoelectric diode with band width BW = 1.0 × 10⁷ Hz, and the value of quantum efficiency is η = 1 A/W. The signal light power is 0.1% of the local oscillator light power, that is, 1uW, and the relative intensity noise coefficient γ of the local oscillator light is 10³ A⁻¹. THE beam splitting ratio is variable, and its simulation diagram is shown in Figure 5(a).

It can be seen from Figure 5(a) that the maximum SNR of the balanced coherent detection system is 5 dB higher than that of the ordinary coherent detection. When the beam splitting ratio is 0.10 or 0.90, the SNR of the balanced coherent detection is exactly equal to that of the ordinary coherent detection. When the beam splitting ratio is less than 0.10 or more than 0.90, the SNR of the balanced coherent detection system is less than the SNR of the single source coherent detection. The simulation results are basically consistent with the SNR formula derived by us, which verifies the correctness of the derivation.

As shown in Figure 5(b), the SNR of balance detection is about 4 dB higher than that of ordinary heterodyne detection when the above conditions remain unchanged and the signal optical power is taken as variable, which is consistent with the theoretical results.

The laser source is a continuous laser signal source, and the frequency is set to 150 THz. In order to make the difference frequency between the signal light and the local oscillator light exist, the frequency of the signal light is shifted by 1 G relative to the local oscillator light, that is, the difference frequency of the two signals is 1 G. The local oscillator light power is 10 mW, and the echo signal light power is 1 uW, both of which are linearly polarized light. Four PIN type photodetectors are selected, the response rate is 0.9 A/W, the dark current is 10 nA, and the noise characteristics of the detectors are the default of the system. The simulation diagram is shown in Figure 6.

Figure 7(a) and Figure 7(b) is the time domain simulation waveform diagram of the signals of system I and Q. It can be seen from the simulation results that the phase difference of the components of system I and Q is π/2.

Rotate the 1/4 wave plate and 1/2 wave plate respectively, and the simulation results are shown in Figure 8 and Figure 9. When the 1/4 glass plate changes

from a 45˚ polarization angle to a 20˚ polarization angle, the phase difference of signals in channel I and Q is not equal to π/2.

After the 1/2 glass slide was changed from 22.5˚ to 30˚, the SNR is reduced by 3 dB.

Figure 7(c) is the frequency spectrum of the difference frequency signal. As can be seen from the figure, the difference frequency signal of 1 GHz can be demodulated after differential processing, and the detected signal also has a better power value.