Compressive coherent anti-Stokes Raman scattering holography

Alexander Cocking; Nikhil Mehta; Kebin Shi; Zhiwen Liu

doi:10.1364/OE.23.024991

1. Introduction

Holography can encode both the amplitude and the phase information of an incident optical field onto a recording medium and has found numerous applications [1–4]. With the inception of digital holography [5–8], the ability to digitally record the optical field has enabled digital algorithms based holographic reconstruction. Using the diffraction theory, direct digital propagation can estimate the complex field at different axial depths along the propagation direction (i.e., digital focusing). D. Brady et al. has recently proposed and demonstrated that compressive sensing based algorithms can estimate a sparse volumetric sample distribution from a single hologram [9] thereby reducing the number of measurements required as compared to diffraction tomography [10]. However one of the limitations of conventional holographic imaging is that it lacks chemical selectivity. Coherent anti-Stokes Raman scattering (CARS) holography [11–14], on the other hand, captures the complex anti-Stokes field produced in a sample upon excitation with a pump/probe and a Stokes beam and uses the intrinsic molecular vibrational response as a contrast mechanism. Thus it presents an attractive label-free chemically selective imaging modality based on vibrational resonance enhanced third-order optical nonlinear susceptibility $χ^{(3)}$ . Traditional digital propagation based reconstruction of CARS holograms has been previously investigated, whose quality of image reconstruction suffers due to presence of out-of-focus background. It is shown that the compressive sensing technique can improve the reconstruction quality of inline CARS holograms [11], in particular, the suppression of the twin image noise. Since the off-axis holography system [15] allows for efficiently filtering the twin image, here we present compressive off-axis CARS holography to enable inference of volumetric source distribution of $χ^{(3)}$ from measured complex anti-Stokes field. In particular, using compressive CARS holography, we show the significant reduction of out-of-focus background noise while estimating the sparse volumetric distribution from a single CARS hologram. Complex valued numerical model of the holographic reconstruction method, including the complex valued measurement as in our case, improves the performance of compressive sensing based inversion algorithms by allowing the system matrix to be well-conditioned, as has been noted before [9].

The discussions below are organized as follows. First, we discuss the formulation of CARS holography including the model of CARS signal generation and the formalism for compressive sensing based inversion. Next, we present the retrieval results from experimentally recorded CARS holograms of microspheres and biological cells (HeLa). Comparison with digital propagation based reconstruction is also discussed.

2. Formulation

2.1 Model of signal generation

The model of compressive CARS holography can be understood as the linearization of the equation governing the formation of a CARS hologram from a given $χ^{(3)} (x, y, z)$ source distribution. The pump field $E_{p} = A_{p} e^{i {\vec{k}}_{p} \cdot \vec{r}}$ and Stokes field $E_{s} = A_{s} e^{i {\vec{k}}_{s} \cdot \vec{r}}$ excite molecular vibrational modes in a sample of thickness L. The probe field is assumed to be the same as the pump. The resultant anti-Stokes field is given by $E_{a s} = A_{a s} (x, y, z) e^{i {\vec{k}}_{a s} \cdot z \hat{z}}$ ( $\hat{z}$ is the unit vector of ${\vec{k}}_{a s}$ ). The amplitude of the anti-Stokes field in the frequency domain is given by [12]:

{\tilde{A}}_{a s} (u, v, L) \propto A_{p}^{2} A_{s}^{*} \int_{0}^{L} d z e^{i Δ k z} ({\tilde{χ}}^{(3)} (u, v, z) e^{- i (L - z) (k_{x}^{2} + k_{y}^{2}) / (2 k_{a s})})

where the tilde represents the two dimensional Fourier transform with respect to x and y, and

Δ k z = (2 {\vec{k}}_{p} - {\vec{k}}_{s} - {\vec{k}}_{a s}) \cdot z \hat{z}

is the phase mismatch. Equation (1) can also be written in the spatial domain, i.e.,

A_{a s} (x, y, L) \propto \int_{0}^{L} d z e^{i Δ k z} (χ^{(3)} (x, y, z) \otimes e^{\frac{i π}{λ_{a s} (L - z)} (x^{2} + y^{2})} / i λ_{a s} (L - z))

, where

λ_{a s}

is the anti-Stokes wavelength. In other words, the output field is simply the summation of the diffracted anti-Stokes field generated at each slice in the volume distribution. This observation motivates the application of digital back-propagation for retrieving the unknown volume source distribution, provided that the complex anti-Stokes field at the exit plane can be digitally captured. To accomplish this goal, an off-axis digital CARS hologram can be recorded using a reference beam at the same frequency and incident at a small angle

θ

with respect to the anti-Stokes field. The recorded hologram can be written as

I = {| A_{a s} + A_{r} e^{i k_{a s} s i n (θ) x} |}^{2} = | A_{a s} |^{2} + {| A_{r} |}^{2} + A_{r}^{*} A_{a s} e^{- i k_{a s} s i n (θ) x} + A_{r} A_{a s}^{*} e^{i k_{a s} s i n (θ) x}

, where

A_{r} e^{i k_{a s} \sin (θ) x}

denotes the reference field at the appropriate anti-Stokes frequency. The sideband of interest

A_{r}^{*} A_{a s} e^{- i k_{a s} s i n (θ) x}

can be digitally separated from the other three terms and re-centered (see Fig. 1) to obtain the desired carrier-free complex anti-Stokes field.

Fig. 1 (a) Schematic diagram of CARS holography (b) Central part (256x256 pixels) of a recorded CARS hologram; (c) DC filtered two dimensional Fourier transform (amplitude) of the hologram in (b); (d) A digitally filtered and re-centered sideband; (e) Reconstructed CARS field (amplitude) at the recording plane.

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2.2 Model of inversion algorithm based on compressive sensing

We define $Χ^{(3)} (x, y, z) \propto A_{p}^{2} A_{s}^{*} χ^{(3)} (x, y, z) \overset{F . T .}{⇄} {\tilde{X}}^{(3)} (u, v, z)$ , so that

\begin{array}{l} {\tilde{A}}_{a s} (u, v) = \int_{0}^{L} d z H (u, v, z) {\tilde{X}}^{(3)} (u, v, z) or, \\ {\tilde{A}}_{a s} (u, v) = \sum_{l} [\sum_{n, m} X^{(3)} (n Δ x, m Δ y, l Δ z) e^{- i 2 π (u n Δ x + v m Δ y)}] e^{i Δ k l Δ z} e^{- i π λ_{a s} (L - l Δ z) (u^{2} + v^{2})} \end{array}

where

H (u, v, z) = e^{i Δ k z} e^{- i π λ_{a s} (L - z) (u^{2} + v^{2})}

is the transfer function of the medium multiplied by the phase mismatch factor and “F.T.” refers to the two-dimensional Fourier transform. Let us denote by

K (u, v) [\cdot] ≜ Σ_{0}^{L} H (u, v, z) \cdot F . T . [\cdot]

the discretized sensing matrix operator, (which includes the 2D Fourier transform and the diffraction transfer function) which operates upon each slice of the volume distribution

Χ^{(3)} (x, y, z_{0})

throughout the depth

L

and generates the recorded complex CARS field

{\tilde{A}}_{a s} (u, v)

. In this notation, (2) is essentially analogous to a linear system of the form

{\tilde{A}}_{a s} = K Χ^{(3)}

. With a priori knowledge that the

χ^{(3)}

distribution is sparse, this under-constrained linear problem of estimating the 3D distribution from its 2D hologram can be solved using the method of compressive sensing. The theory of compressive sensing ensures retrieval of a sparse signal (or a sparse representation of the signal in a suitable basis) even from significantly incomplete measurement set, provided that the system matrix satisfies the so-called “restricted isometry property” [16, 17]. It is also shown that the sparsest approximation to

Χ^{(3)} (x, y, z)

can be obtained by solving for

Χ_{e}^{(3)} = \underset{Χ^{(3)}}{\arg \min} | | Χ^{(3)} | |_{1} such that {\tilde{A}}_{a s} = K Χ_{e}^{(3)}

, where

| | \cdot | |_{1} = Σ | \cdot |

is the

ℓ_{1}

norm if the condition

M \geq C \cdot μ^{2} (K, I) \cdot S \cdot \log N

is met, where M is the number of measurements, C is some constant, K is the sensing matrix, I is the identity matrix to represent the sparse

Χ^{(3)}

distribution in its canonical basis, μ is a coherence measure between K and I, N is the image size and the signal is assumed to be 'S-sparse' (i.e. it has only S significant components) [9,16,17]. In particular, we use the two-step iterative shrinkage thresholding (TwIST) algorithm [18] to minimize the objective function

f (Χ^{(3)}) = 0.5 | | {\tilde{A}}_{a s} - K Χ^{(3)} | |_{2}^{2} + λ | | Χ^{(3)} | |_{1}

where

λ

is the regularization parameter. Furthermore, in light of the complex valued measurement, the

ℓ_{1}

norm constrained objective function can be minimized using a complex thresholding operator to iteratively update the next estimate of

Χ^{(3)}

using the TwIST algorithm (see remark 2.5 in [19]). The results of the volume distribution retrieval using the single shot recorded CARS hologram are discussed in the next section.

3. Numerical results and discussion

CARS holograms were recorded using the experimental setup detailed in Ref [12], whereby a pulsed nanosecond laser supplying the pump/probe beam and a type II OPO was used to generate the Stokes and reference beams. The generated anti-Stokes signal was then combined with a reference beam at the identical wavelength and propagating at a small off-axis angle of approximately 1.7 degrees and the resultant hologram was digitally recorded on a CCD camera as shown in Fig. 1(a). The recorded hologram of multiple polystyrene microspheres used in this study is shown in Fig. 1(b). The two-dimensional Fourier transform shown in Fig. 1(c) reveals the expected sidebands for this off-axis recorded hologram, of which we digitally retain only one sideband Fig. 1(d). The inverse transform of the re-centered sideband is the complex anti-Stokes field recorded in the hologram Fig. 1(e).

Figure 2 shows typical images of the distribution of the microspheres in a series of ‘z’ slices, ranging from propagation distance of 5 μm to 14 μm, in steps of 1 μm. The out-of-focus background noise is clearly seen affecting the image quality in each slice. We then use the TwIST algorithm to iteratively minimize the $ℓ_{1}$ constrained objective function to retrieve the image in each slice. The regularization parameter $λ$ is chosen by trial and error, i.e., by running the algorithm as $λ$ varies from $10^{- 5}$ to $10^{- 4}$ . A typical reconstructed distribution of microspheres in each of the 10 ‘z’ slices is shown in Fig. 3. We notice that the image in each slice is visually seen to have reduced noise and sharper images.

Fig. 2 Image reconstruction using conventional digital propagation. The nine polystyrene microspheres are visible, but are surrounded by out-of-focus background contributions.

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Fig. 3 Compressive sensing based image reconstruction using the l₁ regularization. The diffraction noise is suppressed and the spheres signal can clearly be seen to be localized at different depth positions. Red arrow indicates spheres used for SNR analysis in Fig. 4.

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To help better appreciate the improvement of image quality and removal of out-of-focus noise, in Fig. 4 (a-b) we plot the signal-to-noise ratio (SNR) for two spheres (indicated by arrows in Fig. 3) as function of the z depth position. The ‘signal’ is computed as the mean value of a small 5x5 pixel region around the sphere center, and ‘noise’ is computed as mean value of the background in the immediate neighborhood of the sphere, within a 101x101 pixel region. The dotted curves are Gaussian fit to the SNR. It is clearly seen that as compared to the conventional digital propagation algorithm, our compressive CARS holography technique improves the localization of the spheres along z, because it minimizes the out-of-focus background noise. This is further corroborated by Fig. 4 (c-d), which shows an axial slice of the reconstruction of the lower sphere. The apparent improved “sectioning” occurs because the $ℓ_{1}$ constrained inversion favors the strong signal in the image plane and shrinks the weak signal (noise) based on an empirically chosen threshold constant. Since compressive sensing algorithms search for the sparsest image, the apparent optical “sectioning” is ultimately limited by the inherent sparsity of the sample.

Fig. 4 Axial localization using compressive CARS holography. (a) and (b) show SNR for two spheres (indicated by red arrows in Fig. 3 along the z depth position using conventional digital back-propagation algorithm and compressive reconstruction, respectively. (c) and (d) show the x-z cross section of the lower sphere using digital back-propagation and the compressive reconstruction, respectively.

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We further test our minimization algorithm on HeLa cells to demonstrate the applicability of compressive CARS holography to imaging biological specimens. The experimental details can be found in Ref [13]. The frequency difference between the pump/probe and the Stokes was tuned to 2913 cm⁻¹ to excite the C-H aliphatic vibrational mode, which was reported to be rich in mitochondria clusters or lipid vesicles [20, 21]. The manually focused multiple-shot CARS images and the digital propagation based reconstruction of a CARS hologram are compared in Ref [13], and the significant features are comparable to those obtained using compressive CARS holography seen in Fig. 5, where the out-of-focus background is suppressed.

Fig. 5 (a-e) Compressive CARS holographic reconstruction of HeLa cells at various depths.

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4. Conclusion

We have demonstrated the efficacy of using compressive sensing for 3D label-free chemically selective volumetric estimation through off-axis compressive CARS holography. Although off-axis CARS hologram helps remove the twin image noise associated with inline holograms, the traditional digital propagation based retrieval still suffers from out-of-focus background noise due to the limited axial resolution, which is inherent in the fact that the hologram only samples a very small subset in the ‘k-space’ describing the volume distribution of the sample. It is well known that diffraction tomography can help improve axial resolution through sampling a larger subset of the object ‘k-space’. If the volumetric distribution of interest has sparse features, compressive off-axis CARS holography can emulate the higher axial resolution using only a single shot CARS hologram compared to multiple field measurements and/or rotation of the sample required for diffraction tomography.

Acknowledgement

This work is partially supported by the National Science Foundation (CBET 1264750).

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Compressive coherent anti-Stokes Raman scattering holography

Abstract

1. Introduction

2. Formulation

2.1 Model of signal generation

2.2 Model of inversion algorithm based on compressive sensing

3. Numerical results and discussion

4. Conclusion

Acknowledgement

References and links

Cited By

Figures (5)

Equations (2)

Optics Express