Heterodyne detection with mismatch correction based on array detector

Abstract

Based on an array detector, a new heterodyne detection system, which can correct the mismatches of amplitude and phase between signal and local oscillation (LO) beams, is presented in this paper. In the light of the fact that, for a heterodyne signal, there is a certain phase difference between the adjacent two samples of analog-to-digital converter (ADC), we propose to correct the spatial phase mismatch by use of the time-domain phase difference. The corrections can be realized by shifting the output sequences acquired from the detector elements in the array, and the steps of the shifting depend on the quantity of spatial phase mismatch. Numerical calculations of heterodyne efficiency are conducted to confirm the excellent performance of our system. Being different from previous works, our system needs not extra optical devices, so it provides probably an effective means to ease the problem resulted from the mismatches.

1. Introduction

It is well known that a maximum signal-to-noise ratio (SNR) of heterodyne system will appear when signal field matches with LO field. That means that there are not any distribution mismatches of amplitude and phase on the photosensitive surface of a detector. However, such conditions are difficult to meet. Many influence factors, such as, misalignment angle [1,2], turbulence [3,4], aberration of optical system [5], will give rise to these mismatches. In view of the difficulty of avoiding them, the study about how to correct these mismatches becomes an important issue. In heterodyne detection, as far as phase mismatch concerned, to the best of our knowledge, there are two available methods to correct it. One is utilizing adaptive optical technique [6]. The fundamental is to detect firstly the wave-front distortion with Hartman wave-front sensor [7] or shearing interferometer [8], and then compensate for it with a system composed of electro-optic, acousto-optic devices and a deformable mirror. The technique needs a large-scale computer control system and servo device. No matter from the view of cost and project, this technique is difficult to realize. So it is only used in few astronomical observatories. Another method [9,10] is based upon phase conjugate mirror, which can generate a conjugate wave of the distorted wave. Exploiting the physical phenomena of self-pumped and mutually pumped phase conjugation, one can obtain a matching wave-front distribution. However, these phenomena appear only when the incident position and angle of the beams meet certain conditions, which give rise to the trouble of adjusting properly relevant components. Even if these conditions are met, the conjugate reflectivity is still very low. Hence, further works are still demanded if one wants to apply this technique in practice.

David Fink [11] pointed out that it is possible to improve the SNR of heterodyne detection by replacing the single detector with an array detector. Provided that the output from every detector element in the array is amplified with a certain coefficient and attached with a certain phase shift, the heterodyne efficiency will get improvement after adding up the outputs of all elements. But, how to determine the quantities of these mismatches, especially, how to correct the phase mismatch with an array detector, were not provided in the paper. Since then, more studies about the topic have not appeared. In this paper, we propose to set on–off controllers on the optical path. By use of the controllers, the field distributions of the signal and LO beams are gained in the form of digital data, respectively. The mismatch quantities can be calculated based on these data. Heterodyne signal can be seen as a cosine signal, hence, in time domain, there is a certain phase difference between adjacent two samples of ADC. By utilizing such characteristic, we propose to correct the spatial phase mismatch of every element by shifting its data sequence with corresponding steps. Namely, we may compensate the spatial phase mismatch by using the time-domain phase. The numerical analyzes show the good correction with our method. Finally, the calculations of heterodyne efficiency also are conducted to show the excellent performance of our system. In our method, one can correct these mismatches only by some simple calculations and without the need of extra optical devices, which greatly simplifies the system.

2. Heterodyne system with mismatch correction

The signal and LO fields are given by

$$E_S=E_S(\mathrm{r})\text{exp}(i{\omega }_{S}t),E_L=E_L(\mathrm{r})\text{exp}(i{\omega }_{L}t),$$ (1)

where E_S (r) and E_L (r) are the complex amplitudes of signal and LO fields, respectively, ω_S and ω_L angular frequency, r the space vector, and the two fields are assumed to have the same polarization. Heterodyne efficiency is an effective measurement to analyze the SNR of heterodyne detection. For a system composed of single detector, it may be expressed as [12]

$$\xi =\frac{{\left\{{\int }_A\mid E_S\mid \mid E_L\mid cos[\varphi (x,y)]dA\right\}}^2+{\left\{{\int }_A\mid E_S\mid \mid E_L\mid sin[\varphi (x,y)]dA\right\}}^2}{\left({\int }_A{\mid E_S\mid }^{2}dA{\int }_A{\mid E_L\mid }^{2}dA\right)}.$$ (2)

where A is the area of the detector, ϕ(x, y) is the phase difference between the two beams and it is related to the spatial position. From above equation, it can be found that if ϕ(x, y) becomes constant, namely, it is independent of spatial position, and |E_S| = |E_L|, one can obtain ξ = 1. Otherwise, we may declare that there are the mismatches of amplitude and phase, and define ϕ(x, y) as the phase mismatch. For reducing these mismatches, some rigorous requirements must be met, such as, exact mechanical and optical adjustment, excellent aberration correction of optical components, good phase front alignment. However, in practice, it is difficult to do these works adequately.

For a system composed of an array detector, if the output of the ith element in the array is enlarged to β_i times and attached a phase shift ψ_i. After adding up all outputs as the total output of the system, ξ can be rewritten by [11]

$$\xi =\frac{{\left\{{\int }_A\mid E_S\mid \mid E_L\mid {\beta }_{i}cos[\varphi (x,y)-{\psi }_i]dA\right\}}^2+{\left\{{\int }_A\mid E_S\mid \mid E_L\mid {\beta }_{i}sin[\varphi (x,y)-{\psi }_i]dA\right\}}^2}{\left({\int }_A{\mid E_S\mid }^{2}dA{\int }_A{\mid E_L\mid }^2{\beta }_i^{2}dA\right)}.$$ (3)

Setting

$${\beta }_i\approx \mid E_S\mid /\mid E_L\mid ,{\psi }_i\approx \phi (x,y)+\text{constant},$$ (4)

and substituting Eq. (4) into Eq. (3), we have ξ = 1. Like this, the mismatches of amplitude and phase achieve the corrections. However, the problems appear here, i.e. how to determine β_i and ψ_i, especially, what approach should be used to correct the phase distortion without a exact optical device.

Our system composed of an array detector is illustrated in Fig. 1(a). An acousto-optic modulator (AOM) is set in the optical path and it will produce a frequency shifting. The signal beams reflected from the target interferes with LO beams on the surface of detector. Then, the heterodyne current signal will be got from the detector. Two on–off controllers, which can shut the optical path by mechanical means, are set in the system. When controller 1 is on and controller 2 is off, the array detector can achieve |E_S (x, y)|², the intensity profile of signal beams. Neglecting the constant irrelevant to subsequent analyzes, we may describe the current output from the mth row and the nth column in the array as following

Fig. 1.

Scheme of heterodyne detection system. (a) The configuration of system based on an array detector and (b) the processing module of the array detector.

$$i_S(m,n)={\iint }_{A_{m,n}}{\mid E_S(x,y)\mid }^2\mathit{dxdy},$$ (5)

where A_m,n is the area of photosensitive surface of the detector element. On the contrary, when controller 1 is off and controller 2 is on, |E_L (x, y)|², the intensity profile of LO beams, can also be got. The output from the element can be described by

$$i_L(m,n)={\iint }_{A_{m,n}}{\mid E_L(x,y)\mid }^2\mathit{dxdy}.$$ (6)

Referring to Eq. (4), the amplification coefficient of the mth row and nth column element can be gained by

$$\beta (m,n)={\left[\frac{i_S(m,n)}{i_L(m,n)}\right]}^{1/2}.$$ (7)

If both controllers are on, a normal heterodyne detection will be in progress. In this case, the output of the element can be expressed by

$$i_I(m,n,t)={\iint }_{A_{m,n}}\left\{{\mid E_L(x,y)\mid }^2+{\mid E_L(x,y)\mid }^2+2\mid E_L(x,y)E_S(x,y)\ast \mid \times cos\left[{\omega }_{h}t+\phi (x,y)+\mathrm{\Delta }{\phi }_0\right]\right\}\mathit{dxdy}$$ (8)

From Eqs. (5), (6) and (8), we can see that, if the area of detector element is small enough, the digital data from the array can reflect nearly these field distributions. Let f_h denote heterodyne frequency and assume that the sampling frequency of ADC is c times than f_h. We can get the sampling time interval Δt = 1/cf_h. For facilitating the subsequent analysis, t is rewritten by

$$\begin{gathered} t=k\mathrm{\Delta }t+t_0 \\ =k/c{f}_h+t_0, \end{gathered}$$ (9)

where t₀ is the initial time and k is the sequence number of sampling data from ADC chip and k = 0, 1, 2, ..... Then, the kth sampling data of i_I (m, n, t) can be rewritten by

$$i_I(m,n,k)={\iint }_{A_{m,n}}\left\{{\mid E_L(x,y)\mid }^2+{\mid E_L(x,y)\mid }^2+2\mid E_L(x,y)E_S(x,y)\ast \mid \times cos\left[{\omega }_h(k/c{f}_h+t_0)+\phi (x,y)+\mathrm{\Delta }{\phi }_0\right]\right\}\mathit{dxdy}$$ (10)

The processing module of the array detector is shown in Fig. 1(b). The outputs from detector elements are acquired synchronously by ADCs. Then the data are sent into the micro controller unit (MCU) for further processing. After the actions of β_i and ψ_i, the outputs from all elements are summed as the final output of the system.

3. Correction of phase mismatch

As mentioned in above part, the spatial phase mismatch ϕ(x, y) in the cos(.) term of Eq. (8) will severely degrade the heterodyne efficiency. In this section, we will elaborate our method of phase correction. Prior to all works, the quantity of phase mismatch on every element surface must be determined.

3.1. Quantity of phase mismatch

For the mth cow and the nth column element, by use of i_S (m, n), i_L (m, n) and i_I (m, n, k), we may compute Ω(m, n) as follows:

$$\begin{gathered} \mathit{\Omega }(m,n)=\frac{i_I(m,n,k_a)-i_S(m,n)-i_L(m,n)}{2i_S(m,n)i_L(m,n)} \\ =\frac{2{\iint }_{A_{m,n}}{E}_S(x,y)E_L(x,y)\left\{cos\left[{\omega }_h{t}_a+\varphi (x,y)+\mathrm{\Delta }{\phi }_0\right]\right\}\mathit{dxdy}}{2{\iint }_{A_{m,n}}{E}_S(x,y)\mathit{dxdy}{\iint }_{A_{m,n}}{E}_L(x,y)\mathit{dxdy}} \end{gathered}$$ (11)

where k_a is an arbitrary sequence number, this is, i_I(m, n, k_a) is selected arbitrarily from the sequence i_I (m, n, k). If the area A_m,n is small enough, we have

$$\mathit{\Omega }(m,n)\approx cos\left[{\omega }_h{t}_a+\phi (x,y)+\mathrm{\Delta }{\phi }_0\right].$$ (12)

Define ψ (m, n) as follows:

$$\begin{gathered} \psi (m,n)=\text{acos}[\mathit{\Omega }(m,n)]. \\ \approx {\psi }_h{t}_a+\phi (x,y)+\mathrm{\Delta }{\phi }_0. \end{gathered}$$ (13)

When A_m,n is small enough, ψ (m, n) can reflect nearly the value of ϕ(x, y) on the detector element. One can find that there are two excess terms ω_h t_a and Δφ₀ in ψ (m, n). If ψ (m, n) is seen as the real phase mismatch ϕ(x, y), we may define ϕ′(x, y) as the phase mismatch after correction and have,

$$\begin{gathered} {\phi }^{\prime }(x,y)=\phi (x,y)-\psi (m,n) \\ =-({\omega }_h{t}_a+\mathrm{\Delta }{\phi }_0). \end{gathered}$$ (14)

Being irrelevant to spatial position, ω_ht_a + Δφ₀ will not influence the heterodyne efficiency according to Eq. (3). So, it is reasonable to define ψ (m, n) as the quantity of phase mismatch, and it can be calculated by Eqs. (11) and (13).

3.2. Principle of correcting spatial phase mismatch ϕ(x, y) by using time-domain phase

At an arbitrary time t_a, Φ₁, Φ₂ and Φ₃, the phase terms of different spatial positions in Eq. (10), can be expressed by

$${\mathit{\Phi }}_i={\omega }_h{t}_a+\phi (x_i,y_i)+\mathrm{\Delta }{\phi }_0,i=1,2,3$$ (15)

In terms of the two methods of phase correction mentioned in Section 1, their principles are to construct a spatial phase distribution by using deformable mirror and conjugate mirror, and the distribution can compensate or match the phase mismatch. Being different from the two methods, here, we propose to correct phase mismatch by use of time-domain phase. Provide that Φ₁ is considered as the reference phase, in order to correct Φ₂ and Φ₃, two time phase terms can be attached as following

$${\mathit{\Phi }}_2=\left[{\omega }_h{t}_a+\phi (x_2,y_2)+\mathrm{\Delta }{\phi }_0\right]+{\omega }_h{t}_2,{\mathit{\Phi }}_3=\left[{\omega }_h{t}_a+\phi (x_3,y_3)+\mathrm{\Delta }{\phi }_0\right]+{\omega }_h{t}_3,$$ (16)

where

$${\omega }_h{t}_2=\varphi (x_1{y}_1)-\varphi (x_2{y}_2),{\omega }_h{t}_3=\varphi (x_1{y}_1)-\varphi (x_3{y}_3),$$ (17)

Thus, one can obtain Φ₁ = Φ₂ = Φ₃. Then, it may be declared that the spatial phase mismatch in the three positions is corrected.

The heterodyne signal is actually a cosine signal in the form of A cos(ω_h t + φ). In the time domain, there is a certain phase difference between the adjacent two samples of ADC when ω_h is certain. So, it is feasible that one can correct the spatial phase mismatch by shifting the sequence of sampling data, which is illustrated in Fig. 2. Examining Eq. (10), we can find that the sequence shift will not affect the current output of the direct-current term |E_L (x, y)|² + |E_S (x, y)|², because it is not correlative with time.

Fig. 2.

Illustration of correction of phase mismatch.

3.3. Realization of correction

Let Δθ denote the phase difference between adjacent two samples of ADC. For a heterodyne signal, we have

$$\begin{gathered} \mathrm{\Delta }\theta ={\omega }_h\mathrm{\Delta }t \\ ={\omega }_h/c{f}_h \end{gathered}$$ (18)

So, with regard to the heterodyne signal in Eq. (10), there is a phase difference between i_I (k) and i_I (k + 1). Hence, if there is a phase advance or lag θ_g between i_I (k) and i_I(k_g), the sequence number k_g may be determined according to k_g = k ± p, where p = [θ_g/Δθ]_int, and symbol [.]_int denotes rounding to the nearest integer.

In line with above method, we select i_I (m′, n′, k) as the reference sequence, where i_I (m′, n′, k) is the output sequence of an arbitrary element in the array. Then, with the known ψ (m, n), the finial output of the heterodyne system is denoted by i_h (k) and it may be got by

$$i_h(k)=\sum _{m=1}^N\sum _{n=1}^N{i}_I(m,n,k_{m,n})$$ (19)

where N is the number of elements and

$$k_{m,n}=k-l_{m,n},$$ (20)

where

$$\begin{gathered} l_{m,n}={\left[\frac{\psi (m^{\prime },n^{\prime })-\psi (m,n)}{\mathrm{\Delta }\theta }\right]}_{\mathit{int}} \\ ={\left[\frac{\psi (m^{\prime },n^{\prime })-\psi (m,n)}{2\pi }\times c\right]}_{\mathit{int}} \end{gathered}$$ (21)

Thus, ϕ(x, y) is corrected in the output of the system. For example, for 8 × 8 array, let the sampling frequency be seven times than heterodyne frequency, namely, c = 7. Assume that there are ψ (1, 1) = π,…, ψ (3, 7) = − π/2.3,…, ψ (5, 2) = π/7,…, ψ (8, 8) = 3π/4 according to Eq. (13). Select U_I (5, 2, k) as the reference sequence. Referring to Eq. (21), we have l_1,1 = 3,…, l_3,7 = − 2,…, l_5,2 = 0, …,l_8,8 = 2. Then, i_h (k) should be calculated in the following form

$$i_h(k)=i_I(1,1,k+3)+\dots +i_I(3,7,k-2)+\dots +i_I(5,2,k)+\dots +i_I(8,8,k+2),$$ (22)

as illustrated in Fig. 3

Fig. 3.

Example of data sequence process.

The realization of our method is based on the acquiring of i_S (m, n), i_L (m, n) and i_I (m, n, k), i.e. the digital forms of |E_S (x, y)|², |E_L (x, y)|² and I (x, y, t), respectively. During the course of detection, if some influence factors give rise to the changes of these field distributions, the corrections will perhaps become invalid. For this situation, one may set a threshold value of SNR for the detection system, where SNR denotes the ratio value of the time-averaged signal power to the noise power. When these changes cause the SNR to be below the threshold value, the system may repeat the detection process automatically or artificially, as shown in Fig. 4. Like this, the new field distributions are obtained for the corrections. One may also find from Fig. 4 that, the process is not complicated and the corrections can be performed only by some simple calculations, which are easy to conduct with high speed MCU.

Fig. 4.

Process diagram of detection with corrections.

4. Numerical simulation of phase correction

Many factors can result in the phase mismatch, such as, misalignment angle, turbulence, aberration of optical system. Among them, the misalignment angle is a common factor which will severely influent the heterodyne detection due to the tilt wave-front. In the light of pervious study [2], even several milliradians will cause the heterodyne efficiency to decrease to 50%. So, by using Gaussian field mode and taking misalignment angle as an example, we will analyze numerically the correction performance with the method proposed in this paper. We define a Cartesian coordinate system oxyz and locate the origin of the system on the central of photosensitive surface. The detector array is illuminated by a convergent signal beams with angle γ and α, together with a LO beams whose beam waist coincides with the surface, as shown in Fig. 5(a).

Fig. 5.

Relative coordinate system and configuration of array detector.

Rotating oxyz with an angle α along z axis, we can get a new coordinate system ox′y′z′ as shown in Fig. 5(b), and then ox″y″z″, the principal axis system of signal beams, may be got by rotating ox′y′z′ with an angle γ along y′ axis. Consequently, the field of signal beams can be written by

$$E_S(x^{″},y^{″})=\frac{E_{0S}}{w_S(z^{″})}exp\left[-\frac{{x^{″}}^2+{y^{″}}^2}{w_S^2(z^{″})}\right]exp\left\{i{k}_S\left[z^{″}+\frac{{x^{″}}^2+{y^{″}}^2}{2R_S(z^{″})}\right]-iarctan\frac{z^{″}}{f_S}+i{\phi }_S\right\},$$ (23)

where

$$\{\begin{array}{l}{x}^{″}=(xcos\alpha +ysin\alpha )-zsin\gamma \hfill \\ y^{″}=ycos\alpha -xsin\alpha \hfill \\ z^{″}=zcos\gamma +(xcos\alpha +ysin\alpha )sin\gamma \hfill \end{array},$$ (24)

and z = 0,w_S(z″) = ω_0S[1 + (z″)/f_S)²]^1/2, $f_S=\pi {\omega }_{0S}^2/{\lambda }_S$, R_S(z″) = z″ + f_S²/z″.−ω_0S is the beams waist radius of signal beams, E_0S is constant, φ_S the initial phase of signal beams. That of LO beams on the surface can be expressed as

$$E_L(x,y)=\frac{E_{0L}}{{\omega }_{0L}}\text{exp}\left(-\frac{x^2+y^2}{{\omega }_{0L}^2}\right)exp(i{\phi }_L),$$ (25)

where E_0L is constant, ω_0L the beams waist radius, φ_L the initial phase of LO beams. The configuration of the detector array is given in Fig. 5(c). Let a and d denote the side length of the array and detector element respectively. Symbol e is defined as the distance between elements.

By use of Eqs. (23) and (25), Fig. 6 shows the calculation results for γ = 0.6°. Fig. 6(a) shows cos[ϕ(x, y) ] on the photosensitive surface of the array detector. After determining the phase mismatch from Eq. (13), cos[ϕ′(x, y) ] may be obtained according to Eq. (14) and shown in Fig. 6(b). Although, there is still a distribution fluctuation on the surface of every detector element, cos[ϕ′(x, y) ] can be approximately seen as a plane throughout the whole surface of the array, as shown in Fig. 6(b). Compared with that in Fig. 6(a), the phase mismatch is corrected well.

Fig. 6.

Relative diagram about phase correction. λ = 10 μm, γ = 0.6°, α = 23°, N = 8, a = 2mm, ω_0L = a/2, ω_0S = a/2, c = 7, d = 150μm, e = 100μm.

When angle γ becomes larger, the fluctuation of the phase mismatch is more dramatic as shown in Fig. 7(a) where γ = 2.6°. Correspondingly, the correction shown in Fig. 7(b) becomes degraded, which can be explained that Eqs. (5), (6) and (8) cannot continue to describe accurately the distribution of these fields. So a smaller detector element is expected in this situation. Set N = 11 and N = 15 in Fig. 7(c) and (d), respectively. Because the size of the array is unchanged, the elements become smaller with the increasing of element number, which will improve the correction as shown in the two figures. Certainly, the cost of the improvement is to let the processing module of the array become more complicated. Hence, there should be an overall consideration on the problem in practice.

Fig. 7.

Relative diagram about phase correction when milignment angle increases. γ = 2.6°. In Fig. 7(a) and (b), N = 8, d = 150μm, e = 100μm. In Fig. 7(c), N = 11, d = 109μm, e = 73μm. In Fig. 7(d) N = 15, d = 80μm, e = 53μm. Other parameters are same as that of previous part.

5. Numerical analysis of heterodyne efficiency

Based on Gaussian beams, we will analyze numerically the heterodyne efficiency of the system proposed in this paper. For the purpose of comparison, that of a system composed of single detector will also be given. Fig. 8 gives the optical configuration we study, in which several typical influence factors exist. The signal beams is assumed to incident with a angle γ, and the central of beams waist is set on the photosensitive surface. In addition, there is a position offset x₀ along x axis. Replacing x with (x − x₀) in Eq. (24), one may use Eq. (23) to describe the signal field. Assume that the beams waist of LO field coincides with the photosensitive surface. One can make Eq. (25) denote LO field on the surface. Setting ρ = ω_0L/ω_0S and x₀ = κω_0S, we can find that there are three kinds of influence factors in Fig. 9, which will appear when ρ ≠ 1, κ; ≠ 0 and γ ≠ 0, respectively. In practice, these common factors are inevitable and difficult to eliminate entirely. The effects of ρ ≠ 1 and κ ≠ 0 on heterodyne efficiency may be explained as the amplitude mismatch between the two beams, while, that of γ ≠ 0 should be classified as a phase mismatch owing to the wave-front tilt of signal beams.

Fig. 8.

Sketch of different influence factors in system.

Fig. 9.

Variation of ξ with ρ,κ,γ and c. Other parameters are same as that in above content.

In calculation, let a′ denote the side length of single detector and set ω_0S = a/4 in the array-detector system and ω_0S = a′/4 in single-detector system. According to Eq. (3), Fig. 9(a) and (b) show the variation of heterodyne efficiency ξ with ρ and κ, respectively. For the single-detector system, the heterodyne efficiency ξ decreases remarkably with the increasing ρ and κ. On the contrary, it remains high in the array-detector system, which reflects a good correction of amplitude mismatch. When ρ increases to a certain extent, the amplitude distribution of LO field is near to a plane on the photosensitive surface. Like this, no more change of ξ will occur if ρ keeps increasing, which can also be seen in Fig. 9(a). In addition, we can see that the curves for a′ = 2mm and a′ = 100μm coincident in Fig. 9(a) and (b), which is reasonable due to the assumption that the ratios of ω_0S to a′ are same in the two situations.

The variation of ξ with γ is given in Fig. 9(c). Being different from the situations in Fig. 9(a) and (b), the single-detector system with a smaller detector displays a better performance in the effect of misalignment angle. The reason of that may be explained here. When γ is certain, for a smaller detector, the fluctuation of phase mismatch is also smaller during the range of photosensitive surface, thus, the influence of misalignment angle will be decreased. So, it seems that a small enough detector will overcome the effect of misalignment angle. However, the cost is to limit the view angle of a system. For example, for a′ = 100μm and the focal length of lens f = 10 cm, according to atan(a′/f), the view angle is only 0.0573°. Thus, the performance for a′ = 100μm in Fig. 9(c) has not application significance. Compared with the system of single detector with the same size, the heterodyne efficiency of array-detector system is still high enough. For example, for γ = 1°, the efficiency of array detector is around 75%, while, that of single detector is near to zero. According to Eq. (21), the sampling frequency of ADC will affect the correction of phase mismatch. Fig. 9(d) gives the heterodyne efficiency with different sampling frequencies. One can see from the figure that, there is not obvious improvement when the sampling frequency is over seven times than heterodyne frequency, which should be considered in the practical application.

When the three factors exist at the same time, which is more close to the realistic situation, the heterodyne efficiency is shown in Fig. 10. In the figures, the advantage of our system is more obvious. Despite, the system of single detector for ά= 100μm, exhibits good adaption when being affected by misalignment angle, as shown in Fig. 9(c), its performance will also be degraded severely when various factors exist. For example, there are κ = 0.4 and ρ = 1.7 in Fig. 10(a). For and ω_0S = a′/4, we have x₀ = 10μm. Such an extremely small offset will also result in an overall degradation of the efficiency curve. However, our system continue to show the excellent performance in the two figures. The heterodyne efficiencies with different N are also given in Fig. 10. We can find that, in general, there is a improvement of the efficiency when N increases, which benefits from the better correction of phase mismatch as shown in Fig. 7.

Fig. 10.

Variation of ξ when the three factors exist. For N = 8, d = 80μm and e = 53μm. For N = 10, d = 120μmand e = 80μm. For N = 15, d = 80μm and e = 53μm. Other parameters are same as that of above content.

6. Conclusion

For achieving a good detection performance, some rigorous requirements must be met in heterodyne system, as mentioned in above part. Being aimed at the characteristic of heterodyne signal, we propose to compensate the spatial phase mismatch by using time-domain phase difference. The nature of our method is to turn these rigorous requirements into the difficulty of realizing the processing module of the array detector. Obviously, the latter is easier to handle. However, in brief, one can find that the algorithm of our method is not complicated and these electronic components in the system are conventional. Compared with the techniques based on phase conjugate mirror and deformable mirror, our system is easier to realize. Although the heterodyne efficiency of our system is analyzed based on numerical calculations, these results provide a proof that the system has the ability to improve notably the performance of heterodyne detection. Our method is valuable for decreasing the effect of the mismatching. But, because that it is based on the acquirement of the field distribution of LO and signal beams, there will be a limitation in the application of our method especially when the signal optical power too weak to obtain the accurate field distribution. In the case, more researches are needed in future.