Improved phase retrieval in holographic data storage based on a designed iterative embedded data

Abstract

Embedded data are used to retrieve phases quicker with high accuracy in phase-modulated holographic data storage (HDS). We propose a method to design an embedded data distribution using iterations to enhance the intensity of the high-frequency signal in the Fourier spectrum. The proposed method increases the anti-noise performance and signal-to-noise ratio (SNR) of the Fourier spectrum distribution, realizing a more efficient phase retrieval. Experiments indicate that the bit error rate (BER) of this method can be reduced by a factor of one after 10 iterations.

Graphical abstract

Keywords

holographic data storage (HDS) / phase retrieval / embedded data / high frequency

Introduction

Vertical-cavity surface-emitting lasers (VCSELs) have been considered as key devices for optical communications, optical interconnection and optical signal processing because of their prominent advantages over traditional edge-emitting lasers, such as low threshold current, single-longitudinal mode, large modulation bandwidth, low cost and wafer-scale integration capability for large arrays configuration [1-6]. Nowadays, mutually coupled VCSELs have attracted considerable interests due to their important application in bidirectional chaos secure communication fields. Since the first demonstration of chaos synchronization between mutually coupled VCSELs [3], nonlinear dynamics, polarization properties and chaos synchronization of mutually coupled scheme have been studied extensively [4-10]. Relevant reports have shown that the chaos synchronization characteristics between two mutually coupled semiconductor lasers (SLs) behave in an unstable way due to stochastic noise, and stochastic leader-laggard synchronization is usually observed in a time dependent manner [11-14]. In order to realize stable chaos synchronization in mutually coupled VCSELs, a method that operation parameters of two VCSELs are asymmetrical has been proposed. For instance, through adjusting frequency detuning between two mutually coupled VCSELs, stable chaos synchronization can be realized [4,15-18]. Additionally, through adopting asymmetrical injection level, stable chaos synchronization can also be obtained and the VCSEL subject to weaker injection rate plays a leader role [4]. However, to our knowledge, all the works mentioned above are focused on the influence of external parameters mismatch on the leader-laggard synchronization between two mutually coupled VCSEL. In practice, there always exist mismatched intrinsic parameters between two VCSELs. Therefore, it is essential to investigate the influences of mismatched intrinsic parameters on the leader-laggard relationship between two mutually coupled VCSEL under asymmetrical external operation parameters. In this paper, based on the spin-flip model (SFM), the impacts of mismatched intrinsic parameter on leader-laggard chaos synchronization induced by frequency detuning or injection rate detuning in mutually coupled VCSELs have been investigated numerically, and part results are different from that obtained under the case of two VCSLEs with identical intrinsic parameter.

System model

The schematic diagram of two mutually coupled VCSELs system is shown in Fig. 1. The output of VCSEL1 passes through microscopic lens (ML1), and then injects into VCSEL2 after passing through a beam splitter (BS1), an optical isolator (ISO1) and a neutral density filter (NDF1). The output of VCSEL2 experiences a similar process to form mutually delay-coupled structure. The NDF is used to control injected strength. In order to independently control the injection strength of each VCSEL, two separated light paths have been designed in this system. The ISO is used to ensure unidirectional transmission of light.

Fig.1 Schematic diagram of two mutually coupled VCSELs (M: mirror)

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Theory

Based on SFM [1], after considering the effect of external injected field, rate equations for the mutually coupled VCSELs can be described as [6,8]

(1)$$\frac{\text{d}{E}_{1,2}^x}{\text{d}t}=k\left(1+\text{i}\alpha \right)\left(N_{1,2}{E}_{1,2}^x-E_{1,2}^x+\text{i}{n}_{1,2}{E}_{1,2}^y\right)-\left({\gamma }_{\text{a}}+\text{i}{\gamma }_{\text{p}}\right)E_{1,2}^x+{\eta }_{21,12}{E}_{2,1}^x\left(t-\tau \right)\exp \left(-\text{i}{\omega }_0\tau \right)+{\left(-1\right)}^{1,2}\text{i}\Delta \omega E_{1,2}^x+F_{1,2}^x,$$

(2)$$\frac{\text{d}{E}_{1,2}^y}{\text{d}t}=k\left(1+\text{i}\alpha \right)\left(N_{1,2}{E}_{1,2}^y-E_{1,2}^y-\text{i}{n}_{1,2}{E}_{1,2}^x\right)+\left({\gamma }_{\text{a}}+\text{i}{\gamma }_{\text{p}}\right)E_{1,2}^y+{\eta }_{21,12}{E}_{2,1}^y\left(t-\tau \right)\exp (-i{\omega }_0\tau )+{(-1)}^{1,2}\text{i}\Delta \omega E_{1,2}^y+F_{1,2}^y,$$

(3)$${\displaystyle \frac{d{N}_{1,2}}{dt}}=-{\gamma }_e{N}_{1,2}\left(1+{\left|E_{1,2}^x\right|}^2+{\left|E_{1,2}^y\right|}^2\right)+{\gamma }_e\mu -i{\gamma }_e{n}_{1,2}\left(E_{1,2}^y{E}_{1,2}^x{}^{*}-E_{1,2}^x{E}_{1,2}^y{}^{*}\right),$$

(4)$${\displaystyle \frac{d{n}_{1,2}}{dt}}=-{\gamma }_s{n}_{1,2}-{\gamma }_e{n}_{1,2}\left({\left|E_{1,2}^x\right|}^2+{\left|E_{1,2}^y\right|}^2\right)-i{\gamma }_e{N}_{1,2}\left(E_{1,2}^y{E}_{1,2}^x{}^{*}-E_{1,2}^x{E}_{1,2}^y{}^{*}\right),$$

where subscripts 1 and 2 stand for VCSEL1 and VCSEL2, respectively, and superscripts x and y stand for x and y linear polarization (LP) modes, respectively. E is the slowly varied complex amplitude of the field, N is the total carrier inversion between conduction and valence bands, n accounts for the difference between carrier inversions for spin-up and spin-down radiation channels, k is the decay rate of field, α is line-width enhancement factor, γ_e is the decay rate of total carrier population, γ_s is spin-flip rate, γ_a and γ_p are linear anisotropies representing dichroism and birefringence, respectively, μ is normalized injection current (μ takes the value 1 at threshold), τ is the propagation delay time between the VCSEL1 and VCSEL2, η₁₂ (η₂₁) is the injection rate of VCSEL1 (VCSEL2) to VCSEL2 (VCSEL1). In the symmetric reference frame, averaged angle frequency ω₀ and detuning ∆ω are defined as follows:

(5)$${\omega }_{\mathrm{0}}={\displaystyle \frac{{\omega }_{\mathrm{1}}+{\omega }_{\mathrm{2}}}{\mathrm{2}}},$$

(6)$$\Delta \omega ={\displaystyle \frac{{\omega }_{\mathrm{2}}-{\omega }_{\mathrm{1}}}{\mathrm{2}}},$$

where ω is free-running optical frequency, and the last terms in Eqs. (1) and (2) represent the spontaneous emission noises which are modeled by Langevin sources [2]:

(7)$$F_{\mathrm{1},\mathrm{2}}^x=\sqrt{{\beta }_{sp}/\mathrm{2}}(\sqrt{N_{\mathrm{1},\mathrm{2}}+n_{\mathrm{1},\mathrm{2}}}{\xi }_{\mathrm{1},\mathrm{2}}^{\mathrm{1}}+\sqrt{N_{\mathrm{1},\mathrm{2}}-n_{\mathrm{1},\mathrm{2}}}{\xi }_{\mathrm{1},\mathrm{2}}^{\mathrm{2}}),$$

(8)$$F_{\mathrm{1},\mathrm{2}}^y=-i\sqrt{{\beta }_{sp}/\mathrm{2}}(\sqrt{N_{\mathrm{1},\mathrm{2}}+n_{\mathrm{1},\mathrm{2}}}{\xi }_{\mathrm{1},\mathrm{2}}^{\mathrm{1}}-\sqrt{N_{\mathrm{1},\mathrm{2}}-n_{\mathrm{1},\mathrm{2}}}{\xi }_{\mathrm{1},\mathrm{2}}^{\mathrm{2}})$$

where ξ₁ and ξ₂ indicate independent Gaussian white noise with zero mean and unitary variance, and β_sp is spontaneous-emission rate.

Results and discussion

To specifically describe synchronization quality between the two lasers, the quality of chaos synchronization and its time shift can be quantified by calculating shifted correlation coefficient C(Δt):

(9)$$C(\Delta t)={\displaystyle \frac{\langle [I_{\mathrm{1}}(t)-\langle I_{\mathrm{1}}\rangle ][I_{\mathrm{2}}(t+\Delta t)-\langle I_{\mathrm{2}}\rangle ]\rangle }{{\{\langle {[I_{\mathrm{1}}(t)-\langle I_{\mathrm{1}}\rangle ]}^{\mathrm{2}}\rangle \langle {[I_{\mathrm{2}}(t)-\langle I_{\mathrm{2}}\rangle ]}^{\mathrm{2}}\rangle \}}^{\mathrm{1}/\mathrm{2}}}}$$

where ∆t is the time shift between two VCSELs output, the brackets $\langle \rangle$ denote temporal average, I = |E|² is the output intensity of the laser. A large value of C indicates that good synchronization has been achieved. For perfect chaotic synchronization, C equals to 1.

The rate equations (1)-(4) can be used to solve numerically with the fourth-order Runge-Kutta algorithm. During the calculations, typical parameters are used as follows [2]: k = 300 ns^-1, α = 3, γ_a = 0.1 ns^–1, γ_p = 10 ns^–1, γ_e = 1 ns^–1, γ_s = 50 ns^–1, ω₁ = 2.2176 × 10¹⁵ rad/s (the corresponding central wavelength is 850 nm), μ = 2.5, τ = 3 ns, β_sp = 10^–6 ns^–1, and η₂₁ fixed at a moderate strength of 30 GHz.

The chaotic time series of each LP of the two VCSELs are given in Figs. 2(a) and 2(b) under the condition of symmetric coupling. Figures 2(c1) and 2(c2) give cross correlation coefficients corresponding to Figs. 2(a1), 2(b1) and Figs. 2(a2), 2(b2) as a function of time shift. In Figs. 2(c1) and 2(c2), there are two main peaks, respectively, which corresponds to the cross-correlation values shifted by the one-way coupling delay time of ∆t = + τ(called C₊) and ∆t = -τ (called C_-) [16]. If C_- is smaller than C₊, the chaotic time series of VCSEL1 anticipate that of VCSEL2 by τ. On the contrary, if C_- is larger than C₊, the chaotic time series of VCSEL1 lag behind that of VCSEL2 by τ.

Fig.2 (a) Time series of x LP, y LP of VCSEL1; (b) time series of x LP, y LP of VCSEL2; (c) corresponding cross-correlation coefficient of x LP, y LP between VCSEL1 and VCSEL2

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Effects of frequency detuning or asymmetrical injection level on leader-laggard relationship of two mutually coupled VCSELs with identical intrinsic parameter

The cross-correlation coefficients C₊ and C_- between the two VCSELs are numerically calculated as a function of the frequency detuning under ∆η = η₂₁–η₁₂ = 0 GHz in Fig. 3 and the asymmetrical injections under ∆ν = 0 GHz in Fig. 4. An exchange of the leader-laggard role depends on the sign of the frequency detuning or asymmetrical injection rate, and the switching point of leader-laggard synchronization is exactly at ∆ν = 0 GHz or ∆η= 0 GHz for both x and y LP modes. Therefore, the VCSEL oscillated at higher frequency or subject to weaker injection plays a leader role, which is coincided with the results of Refs. [4,18].

Fig.3 Dependence of C+ and C- on frequency detuning (∆ν = ν2–ν1). (a) x LP; (b) y LP

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Fig.4 Dependence of C+ and C- on injection detuning (∆η = η21–η12). (a) x LP; (b) y LP

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Effects of mismatched intrinsic parameters on leader-laggard synchronization induced by frequency detuning between two VCSELs

The external parameters such as oscillated frequency and injection strength can be controlled easily, but intrinsic parameters are difficult to be accurately controlled. Therefore, it is essential to investigate the influence of the mismatched parameters on the leader-laggard relationship. For convenience, we fix intrinsic parameters of VCSEL1, and only change the intrinsic parameters of VCSEL2. The relative mismatched intrinsic parameters are defined as

(10)$$\mathrm{\Delta }k=\left(k_2-k_1\right)/k_1,\mathrm{\Delta }\alpha =\left({\alpha }_2-{\alpha }_1\right)/{\alpha }_1,\mathrm{\Delta }{\gamma }_{\mathrm{a}}=\left({\gamma }_{a2}-{\gamma }_{a1}\right)/{\gamma }_{a1},\mathrm{\Delta }{\gamma }_{\mathrm{e}}=\left({\gamma }_{\mathrm{e}2}-{\gamma }_{\mathrm{e}1}\right)/{\gamma }_{\mathrm{e}1},\mathrm{\Delta }{\gamma }_{\mathrm{s}}=\left({\gamma }_{s2}-{\gamma }_{\mathrm{s}1}\right)/{\gamma }_{\mathrm{s}1}.$$

Figure 5 shows the dependence of C₊ and C_- on the ∆ν when each relative mismatched intrinsic parameter is set at 10%. For the case of ∆ν = 0 GHz, the leader-laggard relationship is induced only by the mismatched intrinsic parameter. In this case, the VCSEL with smaller k or γ_e plays a leader role, while the VCSEL with smaller α, γ_a or γ_s plays a laggard role. Therefore, after considering the influences of mismatched intrinsic parameters, the location of switching point ∆ν_sw is not at ∆ν = 0 GHz but shifted to negative or positive frequency detuning for different mismatched intrinsic parameter. The location of switching point ∆ν_sw shifted to the left (as shown in Figs. 5(a) and 5(d)). That is to say, for ∆ν is varied from ∆ν_sw to 0 GHz, the VCSEL with lower-frequency will takes the leader role because the influences of mismatched intrinsic parameter plays a dominant role, which is opposite to the result obtained under two VCSELs with matched intrinsic parameter. On the contrary, if the switching point shifted from zero to positive frequency detuning (as shown in Figs. 5(b), 5(c) and 5(e)), for ∆ν is varied in the range from zero to ∆ν_sw, the opposite result can also be obtained after taking the mismatched α, γ_e or γ_s into account. Figure 6 further shows variation of C₊ and C_- of each LP mode with different internal mismatched parameters (Figs. 6(a) k, 6(b) α, 6(c) γ_a, 6(d) γ_e, and 6(e) γ_s) on condition of ∆ν = 1 GHz. From this diagram, it can be observed that the curve C₊ and C_- can also appear the cross point. Fortunately, for ∆ν = 1 GHz, if the mismatched intrinsic parameter is not larger than 10%, the conclusion that the higher-frequency VCSEL takes a leader role is always true. Additionally, it should be pointed out that the mismatched intrinsic parameters may not be only one at most of the time, so the influence of intrinsic parameter mismatch on leader-laggard relationship could be even more complex.

Fig.5 C+ (circle) and C- (square) as a function of frequency detuning for a fixed mismatched (a) k, (b) α, (c) γa, (d) γe, and (e) γs of 10%. Left column is for x LP, and right column is for y LP

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Fig.6 C+ (circle) and C- (square) as a function of mismatched (a) k, (b) α, (c) γa, (d) γe, and (e) γs for a fixed frequency detuning of ∆ν = 1 GHz. Left column is for x LP, and right column is for y LP

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Effects of mismatched intrinsic parameters on leader-laggard relationship induced by asymmetrical injection strength between two mutually coupled VCSELs

Similar to Fig. 5, Fig. 7 gives the dependence of C₊ and C_- on ∆ηfor each of relative mismatched parameters is set at 10%. As seen in Figs. 7(a) and 7(d), the location of switching point ∆η_sw slightly shifted to positive value, that is to say, at the range of 0<∆η<∆η_sw, the VCSEL subject to stronger injection level takes the leader role under ∆k = 10% or ∆γ_e = 10%. Therefore, combining Fig. 4 and Figs. 7(a) and 7(d), it can be seen that the opposite result has been obtained for 0<∆η<∆η_sw after considering the mismatched k or γ_e. For mismatched α, γ_a and γ_s, the switching point ∆η_sw takes negative value (see Figs. 7(b), 7(c) and 7(e)). So, at the range of ∆η_sw<∆η<0, the VCSEL subject to stronger injection level takes the leader role at this case. Figure 8 shows the dependence of C₊ and C_- on the relative mismatched parameters (Figs. 8(a) k, 8(b) α, 8(c) γ_a, 8(d) γ_e, and 8(e) γ_s of 10%) for ∆η fixed at 1.5 GHz. From this diagram, it can be observed that the curve C₊ and C_- can also appear the cross point, but the cross point only appeared when relative mismatch is beyond 10%. It can be predicted that, with the increase of injection level detuning, the separated degree of the two curves C₊ and C_- will enlarge obviously, then the influences of mismatched intrinsic parameters on the relations of anticipating and laggard synchronization will become weak.

Fig.7 C+ (circle) and C- (square) as a function of the detuning of injection strength for a fixed mismatched (a) k, (b) α, (c) γa, (d) γe, and (e) γs of 10%. Left column is for x LP, and right column is for y LP

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Fig.8 C+ (circle) and C- (square) as a function of mismatched (a) k, (b) α, (c) γa, (d) γe, and (e) γs for a fixed detuning of injection strength of ∆η = 1.5 GHz. Left column is for x LP, and right column is for y LP

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Conclusions

Based on the framework of SFM, the influences of mismatched intrinsic parameter on leader-laggard synchronization induced by frequency detuning or asymmetrical injection between two VCSELs are investigated numerically. The results show that, for the case of frequency detuning, the switching point appeared at zero detuning when the intrinsic parameters of two VCSELs are identical. Therefore, the VCSEL oscillated at higher frequency takes the leader role. However, after taking the mismatched intrinsic parameter into account, the switching point will deviate from zero frequency detuning. Therefore, between the switch point and the zero frequency detuning, the result that VCSEL is leader is opposite to that obtained under identical intrinsic parameters between two VCSELs. For the case of asymmetrical injection, the transition point appeared at zero detuning of injection strength when internal parameters of two VCSELs are identical, and the VCSEL subject to lower-injection plays a leader role. However, the switching point slightly shifted to positive value for mismatched k or γ_e, while the switching point slightly shifted to negative value for mismatched α, γ_a or γ_s. In general, after considering the influences of mismatched intrinsic parameter, the switching point of leader-laggard synchronization will deviate from the zero frequency detuning or symmetric injection level as observed experimentally in Ref. [16]. Additionally, all of the results for x LP mode are as the same as those for y LP mode.

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Acknowledgements

This work was partially supported by the Open Project Program of Wuhan National Laboratory for Optoelectronics (No. 2019WNLOKF007) and the National Key R & D Program of China (No. 2018YFA0701800).

Disclosures

The authors declare no conflicts of interest.

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