One-shot and aberration-tolerable homodyne detection for holographic storage readout through double-frequency grating-based lateral shearing interferometry

Yeh-Wei Yu; Shuai Xiao; Chih-Yuan Cheng; Ching-Cherng Sun

doi:10.1364/OE.24.010412

1. Introduction

Holographic data storage (simplified HDS) is a promising breakthrough for advanced optical storage, it has been studied for more than half a century because of its high storage capacity and its compatibility with various methods of holographic multiplexing [1]. The problems associated with commercializing HDS include the difficulty in making the system compact and robust to the real environment and in obtaining a high-quality readout signal. In amplitude-encoded holograms, the nonuniformity caused by the DC component consumes at least 2–10 times the M/# more than that required [2]. By contrast, the phase-only encoded hologram inherently removes the DC component and makes the M/# consumption effective [3, 4]. In addition, the phase-only encoded hologram has higher energy efficiency in the recording process because no energy is wasted by black signals. Moreover, it has potential for realizing high visibility in the readout process [5, 6]. However, the phase distortion of the interfering reference and the readout signal is a major concern because a constant unwanted phase distortion is present in the optical system [6]. Figure 1 as well as Visualization 1 depicts the simulated interferogram of homodyne detection obtained through inline interferometry after inducing a wavefront aberration of 0.25 λ rms, 10% pseudorandom coherent noise, and unstable dc phase shift. The phase distortion makes it impossible to readout the all signals through one-shot in-line interferometry. For an off-axis HDS system, use of a 2-step quadrature homodyne detection or 2-step phase-shifting interferometry is proposed to avoid the phase error [7, 8]. However, the 2-step scheme has lower reading speed, and control of the quadrature phase shift requires highly precise control. In a collinear algorithm, 2-step schemes have been proposed to encode the reading light with a plane-wave in the inner zone along with the original reference in the outer zone [9, 10]. The plane-wave carried by the reading light passes through the recording area in the holographic disc and interferes with the diffracted signal. The interferogram is used for retrieving the original phase in the signal. The main drawback in this scheme is the planewave can exhibit phase distortion when passing through the holographic disc, thus causing a phase error. An alternative approach is to use a slanted plane-wave to bypass the recording area at the holographic disc and interfere with the diffracted signal through off-axis interferometry [11]. In this approach, the phase distortion can be controlled, but off-axis interference can tighten the fringes such that the space bandwidth of the image detector is 4 times larger than necessary [12]. Therefore, the readout speed decreases, making the holographic storage system less efficient.

Fig. 1 The simulated interferogram of homodyne detection obtained through inline interferometry after inducing a wavefront aberration of 0.25 λ rms, 10% pseudorandom coherent noise, and unstable dc phase shift (see Visualization 1).

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Here, we propose a one-shot and aberration-tolerable scheme to make the readout signal form a shearing interference based on a double-frequency grating (simplified DFGSI) [13]. The theoretical discussion of the grating property is presented, and the experiment for the phase decoding is demonstrated.

2. Principle of grating-based homodyne detection

A schema of the DFGSI is depicted in Fig. 2. The spatial light modulator (simplified SLM) is illuminated by a plane wave. The signal modulated by SLM is used to simulate the readout signal of the volume holographic storage system. The modulated signal is imaged to the CMOS image sensor by using an optical system in which lens1 focuses the signal onto a double-frequency grating (simplified DFG) and the second lens collimates the light diffracted from the DFG to the CMOS image sensor. The DFG is used for diffracting two images at marginally different first-order diffraction angles to create shearing interferometry. If the grating is placed at the focus of lens₁ and lens₂, lens₂ can form the diffracted waves after DFG collimation, and the two diffracted waves will be sheared laterally, following which they interfere on the plane of the CMOS image sensor. Figure 3(a) presents the encoded binary phase distribution of the signal obtained through one-pixel shearing interference [Fig. 3(b)], and the bright and dark pixels in the interferogram correspond to constructive interference and destructive interference between two neighboring pixels, respectively, as illustrated in Fig. 3(c). Because the encoded phase of the first column is known, the encoded phase of the signal can be decoded by analyzing the interference pattern depicted in Fig. 3(c).

Fig. 2 A schematic diagram of the shearing interferometry with the DFG.

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Fig. 3 (a) The 2-D encoded binary phase distribution, where the phase difference between the yellow and green pixels is π. (b) Schema of one-pixel shearing through the DFG. (c) The interference result by the original image and the one-pixel sheared image with destructive interference. The first and last columns blocked by red dash lines are out of the interference region.

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3. Optical model of the DFGSI

To apply the DFGSI in a holographic storage system, the alignment of the DFG is crucial. Figure 4(a) depicts the schema of shearing interferometry of the signal incorporated with the DFG for simulating the holographic readout, and Fig. 4(b) depicts the schema for simulating the DFG shifts ∆z along the z axis. The angular spectrum is split by DFG with angle θ₁ and θ₂ to the z axis. When DFG shifts longitudinally, the wave diffracted from the first and second gratings propagates similar to the spectrum shifting to positions P₁ and P₂, respectively. The position of P₁ and P₂ is (x₁, z₁) and (x₂, z₂), respectively, and can be expressed as

x_{1} = Δ z \sin θ_{1},

z_{1} = Δ z (1 - \cos θ_{1}),

x_{2} = Δ z \sin θ_{2},

z_{2} = Δ z (1 - \cos θ_{2}) .

(x’, z’) coordinate is attached to the optical axis of lens₂. The position of P₁, P₂ in (x’, z’) coordinate is (x₁’, z₁’) and (x₂’, z₂’), respectively, and is calculated by coordinate transform:

x_{1}^{'} = Δ z \sin (θ_{1} - θ) + Δ z \sin θ,

z_{1}^{'} = - Δ z \cos (θ_{1} - θ) + Δ z \cos θ,

x_{2}^{'} = Δ z \sin (θ_{2} - θ) + Δ z \sin θ,

z_{2}^{'} = - Δ z \cos (θ_{2} - θ) + Δ z \cos θ,

where, z axis is the optical axis of lens₁; z’ axis is the optical axis of lens₂; θ is the angle from z axis to z’ axis; θ₁ and θ₂ are the angles to the z axis of the first diffracted light and the second diffracted light, respectively. Under paraxial approximation, the optical distribution in the front-focal plane of lens₂ is expressed as convolution of the tilting spectrum in the back-focal plane of lens₁, a shifting term and a propagation term. The wave diffracted from the first grating is expressed as

\begin{array}{l} u_{1} (x^{'}, y^{'}) =, \\ {ℑ [U (- u, - v)] |_{\begin{matrix} u \to \frac{x^{'}}{λ f_{1}} \\ v \to \frac{y^{'}}{λ f_{1}} \end{matrix}} \exp [i k \sin (θ_{1} - θ) x^{'}]} \otimes δ (\frac{x^{'}}{λ f_{2}} - \frac{x_{1}^{'}}{λ f_{2}}) \otimes \exp [\frac{- i k ({x^{'}}^{2} + {y^{'}}^{2})}{2 z_{1}^{'}}] \end{array}

where, U (u, v) is the readout image from the volume holographic storage system with wave-front aberration; λ is the optical wavelength; f₁ and f₂ are the focal lengths of the lens₁ and lens₂, respectively; ⊗ denotes the convolution operator. ℑ [U (-u,-v)] is the spectrum of the readout image; exp [i k sin (θ₁ - θ) x’] denotes the tilting phase;δfunction is the shifting term which shifts the spectrum to x’ = x₁’;

\exp [- i k ({x^{'}}^{2} + {y^{'}}^{2}) / 2 z_{1}^{'}]

is the propagation term which propagates the spectrum from z₁’ to the front-focal plane of lens₂. Similarly, the wave diffracted from the second granting is expressed as

Fig. 4 (a)The geometry of the shearing interferometry with the DFG. (b) The schema when the DFG shifts ∆z along z axis.

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\begin{array}{l} u_{2} (x^{'}, y^{'}) =, \\ {ℑ [U (- u, - v)] |_{\begin{matrix} u \to \frac{x^{'}}{λ f_{1}} \\ v \to \frac{y^{'}}{λ f_{1}} \end{matrix}} \exp [i k \sin (θ_{2} - θ) x^{'}]} \otimes δ (\frac{x^{'}}{λ f_{2}} - \frac{x_{2}^{'}}{λ f_{2}}) \otimes \exp [\frac{- i k ({x^{'}}^{2} + {y^{'}}^{2})}{2 z_{2}^{'}}] \end{array}

The optical field of the diffracted waves propagating to the CMOS sensor are Fourier Transform of Eq. (5) and Eq. (6), and are expressed as

U_{1} (ξ, η) = \exp [\frac{j π {z^{'}}_{1} (ξ^{2} + η^{2})}{λ f_{2}^{2}}] \exp (\frac{- j 2 π {x^{'}}_{1} ξ}{λ f_{2}}) U {\frac{f_{1}}{f_{2}} [ξ - f_{2} \sin (θ_{1} - θ)], \frac{f_{1} η}{f_{2}}},

and

U_{2} (ξ, η) = \exp [\frac{j π {z^{'}}_{2} (ξ^{2} + η^{2})}{λ f_{2}^{2}}] \exp (\frac{- j 2 π {x^{'}}_{2} ξ}{λ f_{2}}) U {\frac{f_{1}}{f_{2}} [ξ - f_{2} \sin (θ_{2} - θ)], \frac{f_{1} η}{f_{2}}},

Using Eqs. (5)–(8), Eq. (11), and Eq. (12), the intensity distribution of the interferogram can be written as

I (ξ, η) = {| \begin{array}{l} \exp [i Δ ϕ_{0} + \frac{i π Δ z \cos (θ - θ_{1}) (ξ^{2} + η^{2})}{λ f_{2}^{2}} + \frac{i 2 π Δ z \sin (θ - θ_{1}) ξ}{λ f_{2}}] \\ U {\frac{f_{1}}{f_{2}} [ξ - f_{2} \sin (θ_{1} - θ)], \frac{f_{1} η}{f_{2}}} \\ + \exp [i Δ ϕ_{x} + \frac{i π Δ z \cos (θ - θ_{2}) (ξ^{2} + η^{2})}{λ f_{2}^{2}} + \frac{i 2 π Δ z \sin (θ - θ_{2}) ξ}{λ f_{2}}] \\ U {\frac{f_{1}}{f_{2}} [ξ - f_{2} \sin (θ_{2} - θ)], \frac{f_{1} η}{f_{2}}} \end{array} |}^{2},

∆Φ₀ is the phase difference between the two gratings. ∆Φ_x is the phase difference induced by lateral displacement of the DFG and can be written as

Δ ϕ_{x} = \frac{2 π}{λ} (\sin θ_{1} - \sin θ_{2}) Δ x .

Equation (13) and Eq. (14) show that the amount of lateral shear is a function of f₂, θ₁ and θ₂. The dc phase difference between the two interference waves is controlled by lateral shift (Δx) of DFG. In this paper, it is controlled as πto eliminate the wavefront aberration. Otherwise, the wavefront aberration will degrade the image quality. On the other hand, Δz inducing unwanted phase error should be nearly zero.

4. Experimental verification

For the experiment, first, the DFG was fabricated by recording the fringes generated from two separate exposures of two planewaves. In these exposures, one of the planewaves was normally incident, and the other was incident on the recording plate at an angle of 10° in the first exposure and 10.5° in the second exposure. A 532-nm Verdi laser was used as the light source. The DFG was placed in the back-focal plane of lens₁, as depicted in Fig. 2. The DFG must be placed at a specific location for realizing destructive interference between the two diffracted images. Two experiments were designed to observe the lateral and longitudinal shifting of the DFG.

The first experiment was conducted to analyze the alignment in the longitudinal direction. The focal lengths of lens₁ and lens₂ were 50 and 105 mm, respectively. The interferogram is changed by adjusting the DFG longitudinally, i.e., shift along z axis. When the DFG is located exactly at the back-focal plane of lens₁, the interferogram is uniform. Figure 5(a) as well as Visualization 2 shows the interferogram obtained when the DFG was shifted from 0 to 1000 μm and with no phase coding in the input signal. The corresponding simulation is depicted in Fig. 5(b) as well as Visualization 3, where the phase difference between the two gratings is set at 0.2π. Furthermore, tilting the axis by 3° with respect to the z axis was necessary, meaning that an error in the shifting axis may have been induced in the experiment. Figure 5 shows fringe appears when ∆z>0, and it causes errors in the readout signal. In this paper, the fringes is applied to alignment of DFG along z axis.

Fig. 5 The interferogram is changed by adjusting the DFG longitudinally. When the DFG locates exactly at the back-focal plane of lens₁, the interferogram is uniform. (a) Experimental result for shifting DFG longitudinally (see Visualization 2), (b) the corresponding theoretical simulation (see Visualization 3).

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The second experiment was conducted to analyze the alignment in the lateral direction, i.e., shift along x axis. Lateral shifting of the DFG is crucial because the phase difference between the interference waves depends on the amount of lateral shifting of the DFG. For the phase difference to be π for realizing wavefront aberration elimination, the DFG must be ideally located. To observe the dc phase difference as a function of the lateral displacement (Δx) of the DFG, Δz was adjusted to 1000 μm such that the interference pattern was a running 1-D sinusoidal wave corresponding to phase difference. It makes the dc phase difference easily observed when the grating was laterally shifted. Figure 6(a) (Visualization 4) and Fig. 6(b) (Visualization 5) depicts the experimental observation with 10μm shifting step and the corresponding simulation, respectively.

Fig. 6 The phase change by adjusting the DFG laterally can be observed as fringes shifting. (a) Experimental result for shifting DFG laterally (see Visualization 4), (b) the corresponding theoretical simulation (see Visualization 5).

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After careful alignment of the DFG for realizing destructive interference of the two diffracted waves, the SLM was encoded with a 2-D binary phase code. The lateral shearing amount was controlled by adjusting the difference in the diffraction angles of the two diffracted waves and the focal length of lens 2. In the experiment, the focal lengths of lens 1 and lens 2 were 50 and 105 mm, respectively, and the lateral shearing amount was 417 μm. The phase encoding was achieved by a phase-only SLM by using a Jasper Display (JD8554). Figure 7(a) depicts the diffracted wave with binary phase encoding, and Fig. 7(b) depicts the interferogram obtained when the phase-encoded wave passed through the shearing interference system, where the effective pixel size of the phase encoding was set to 417 μm.

Fig. 7 (a) The phase encoding without passing the lateral shearing interferometer. (b) The interferogram by the lateral shearing interferometer, where the effective pixel size is 417 μm.

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Decoding is applicable when the lateral shearing amount is as large as several pixels. Figure 8(a) illustrates the interferogram obtained for an effective pixel size of 140 μm, which is nearly one-third the amount of lateral shearing. The quality of the interferogram depends on the precision level of the phase difference between the two diffracted waves. Figure 8(b) depicts the destructive interferogram obtained for constant phase encoding in the SLM; the figure confirms that destructive interference was realized for the whole image.

Fig. 8 (a) The interferogram by the lateral shearing interferometer, where the effective pixel size is 417 μm. (b) The corresponding destructive interference between two diffracted waves in the case of constant phase encoding.

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5. Wavefront aberration eliminated by the DFGSI

To analyze the ability of wavefront aberration elimination, the interferograms of the DFGSI with linear wavefront aberration varying from 0 to 10 λ rms were simulated according to Eq. (13). In the simulation, function U is a matrix of signal distribution multiplied by wavefront aberration; the wavelength used was 532 nm; the pixel size of the SLM and the effective pixel size of the signal were both 6.4 µm; the image size were 8.8 mm × 14.7 mm, and f₁ and f₂ were 50 mm and 105 mm, respectively; θ₁ and θ₂ were 10.2° and 10.26°, respectively. A lateral shearing of 51 µm was recorded. When the induced wavefront aberration is linear distribution,

Figure 9(a) as well as Visualization 6 showed simulation result of the inline interferometry. And serious noise were induced. Figure 9(b) as well as Visualization 7 simulated the inteferogram with the DFGSI. It revealed that a linear wavefront aberration as large as 10 λ rms was removed. When the induced wavefront aberration was quadratic distribution, Fig. 10(a) as well as Visualization 8 showed the simulated interferogram of the inline interferometry. Figure 10(b) as well as Visualization 9 showed with aid of the DFGSI, quadratic wavefront aberration was eliminated until it was getting larger. When the induced wavefront aberration was biquadratic distribution, Fig. 11(a) as well as Visualization 10 showed the simulated interferogram of the inline interferometry. Figure 11(b) as well as Visualization 11 showed with aid of the DFGSI, biquadratic wavefront aberration was eliminated until it was getting larger.

Fig. 9 The simulated interferogram when linear wavefront aberration from 0 to 10λ rms was induced, (a) The interferogram of the inline interferometry (see Visualization 6), (b) The interferogram of DFGSI (see Visualization 7).

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Fig. 10 The simulated inteferogram when quadratic wavefront aberration varying from 0 to 10λ rms was induced. (a) The interferogram of the inline interferometry (see Visualization 8), (b) The interferogram of DFGSI (see Visualization 9).

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Fig. 11 The simulated inteferogram when biquadratic wavefront aberration varying from 0 to 10λ rms was induced. (a) The interferogram of the inline interferometry (see Visualization 10), (b) The interferogram of DFGSI (see Visualization 11).

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Because the shearing image eliminated the wave aberration by the neighboring pixels, a higher aberration tolerance was realized. For quantitatively analysis, bit error rate (BER) was induced. The readout value of a pixel was distinguished on the basis of the threshold value. The value distribution of signal 0 (G₀(v)) and signal 1 (G₁(v)) were both fitted by Gaussian distribution.

G_{0} (v) = \frac{1}{δ_{0} \sqrt{2 π}} e^{- \frac{{(v - μ_{0})}^{2}}{2 σ_{0}^{2}}},

G_{1} (v) = \frac{1}{δ_{1} \sqrt{2 π}} e^{- \frac{{(v - μ_{1})}^{2}}{2 σ_{1}^{2}}},

Where υ is the signal value; μ₀ and μ₁ are the mean values of the readout signals 0 and 1, respectively; δ₀ and δ₁ are the standard deviations of the readout signals 0 and 1, respectively. Accordingly, the threshold value V_th was calculated from the crossing point of G₀ and G₁.

V_{t h} = \frac{- (\frac{μ_{0}}{σ_{0}^{2}} - \frac{μ_{1}}{σ_{1}^{2}}) \pm \sqrt{{(\frac{μ_{0}}{σ_{0}^{2}} - \frac{μ_{1}}{σ_{1}^{2}})}^{2} - 4 (\frac{1}{2 σ_{1}^{2}} - \frac{1}{2 σ_{0}^{2}}) [\frac{μ_{1}^{2}}{2 σ_{1}^{2}} - \frac{μ_{0}^{2}}{2 σ_{0}^{2}} - \ln (\frac{δ_{0}}{δ_{1}})]}}{2 (\frac{1}{2 σ_{1}^{2}} - \frac{1}{2 σ_{0}^{2}})},

The error bits was detected by comparing the readout signal to the input signal, and BER was calculated as the number of error bits divided by the total number of pixels.

Figure 12 shows that the BER depends on linear wavefront aberration for lateral shearing. Generally, BER <0.5% is adequate for error-free recovery [14]. Accordingly, 8 SLM pixels (51 μm) of lateral shearing could eliminate 10 λ rms of linear wavefront aberration. As the lateral shearing became smaller, aberration elimination increased. Figure 13 shows that the BER depends on quadratic wavefront aberration for lateral shearing. For 10 λ rms of quadratic wavefront aberration, a lateral shearing of 2 SLM pixel (13 μm) was required. Figure 14 illustrates that BER depended on biquadratic wavefront aberration for lateral shearing; 1 SLM pixel (6.4 μm) of lateral shearing could eliminate 10 λ rms of biquadratic wavefront aberration. In other words, the DFGSI could eliminate 10 λ rms of wavefront aberration for all three types of aberrations as long as the lateral shearing was ≤1 SLM pixel.

Fig. 12 BER depended on linear wave-front aberration for different lateral shearing.

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Fig. 13 BER depended on quadratic wave-front aberration for different lateral shearing.

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Fig. 14 BER depended on biquadratic wave-front aberration for different lateral shearing.

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6. Conclusion

To retrieve the phase of the stored signal in HDS, a one-shot and aberration-tolerant method is necessary to decode the encoded phase of the diffracted wave. Therefore, we proposed a new scheme for decoding the phase encoding by using a DFG formed by recording two gratings with marginally different spatial frequencies. The proposed scheme is a self-reference by diffraction from the DFG. The DFG was located at the focal plane of lens 1 and lens 2, which were used for collimating the two diffracted waves to realize lateral shearing interference. The lateral shearing amount was controlled by adjusting the frequency difference between the gratings and the focal length of lens 2. An optical model for the DFGSI was developed to analyze phase-error induction and phase-difference control by shifting the double-frequency grating longitudinally and laterally, respectively. The optical model of the DFGSI was demonstrated experimentally. Moreover, the lateral shearing interferogram was successfully demonstrated with the lateral shearing amount of 417 μm incorporated with phase encoding, with an effective pixel size of 140 and 417 μm, respectively. The visibility of the interferogram enabled phase retrieval with high fidelity and simplicity in the optical system. Finally, a simulation based on the optical model of the DFGSI was developed for observing its ability of aberration elimination. For an image of size 8.8 mm × 14.7 mm, lateral shearing of 51 μm could eliminate 10 λ rms of linear wavefront aberration and lateral shearing of 13 μm could eliminate 10 λ rms of quadratic wavefront aberration. When lateral shearing was as small as 6.4 μm, the DFGSI could eliminate 10 λ rms of wavefront aberration in all three types of aberrations.

Acknowledgments

This study was supported in part by the “Plan to Develop First-Class Universities and Top-Level Research Centers” of National Central University (Grants 995939 and 100G-903-2) and the Ministry of Science and Technology of Taiwan (Contract MOST 104-2221-E-008-078-MY3 and 103-2221-E-008-063-MY3).

References and links

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6. K. Curtis, L. Dhar, W. Wilson, A. Hill, and M. Ayres, Holographic Data Storage: From Theory to Practical Systems (Wiley, 2010).

7. M. R. Ayres, “Method for holographic data retrieval by quadrature homodyne detection,” United States Patent, US 8,233,205 B2 (2012).

8. M. He, L. Cao, Q. Tan, Q. He, and G. Jin, “Novel phase detection method for a holographic data storage system using two interferograms,” J. Opt. A. 11(6), 065705 (2009). [CrossRef]

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14. L. Dhar, K. Curtis, M. Tackitt, M. Schilling, S. Campbell, W. Wilson, A. Hill, C. Boyd, N. Levinos, and A. Harris, “Holographic storage of multiple high-capacity digital data pages in thick photopolymer systems,” Opt. Lett. 23(21), 1710–1712 (1998). [CrossRef] [PubMed]

Name	Description
Visualization 1: MP4 (7696 KB)	The simulated interferogram of homodyne detection obtained through inline interferometry after inducing a wavefront aberration of 0.25 ? rms and 10% pseudorandom coherent noise.
Visualization 2: MP4 (168 KB)	Experimental result of the phase change by adjusting the DFG longitudinally.
Visualization 3: MP4 (1521 KB)	Simulation result of the phase change by adjusting the DFG longitudinally.
Visualization 4: MP4 (147 KB)	Experimental result of the phase change by adjusting the DFG laterally.
Visualization 5: MP4 (1543 KB)	Simulation result of the phase change by adjusting the DFG laterally.
Visualization 6: MP4 (7253 KB)	The simulated interferogram of the inline interferometry when linear wavefront aberration varying from 0 to 10? rms was induced.
Visualization 7: MP4 (7292 KB)	The simulated interferogram of DFGSI when linear wavefront aberration varying from 0 to 10? rms was induced.
Visualization 8: MP4 (8264 KB)	The simulated interferogram of the inline interferometry when quadratic wavefront aberration varying from 0 to 10? rms was induced.
Visualization 9: MP4 (8117 KB)	The simulated interferogram of DFGSI when quadratic wavefront aberration varying from 0 to 10? rms was induced.
Visualization 10: MP4 (6771 KB)	The simulated interferogram of the inline interferometry when biquadratic wavefront aberration varying from 0 to 10? rms was induced.
Visualization 11: MP4 (6728 KB)	The simulated interferogram of DFGSI when biquadratic wavefront aberration varying from 0 to 10? rms was induced.

One-shot and aberration-tolerable homodyne detection for holographic storage readout through double-frequency grating-based lateral shearing interferometry

Abstract

1. Introduction

2. Principle of grating-based homodyne detection

3. Optical model of the DFGSI

4. Experimental verification

5. Wavefront aberration eliminated by the DFGSI

6. Conclusion

Acknowledgments

References and links

Supplementary Material (11)

Cited By

Figures (14)

Equations (17)

Optics Express