Three-dimensional wide-field pump-probe structured illumination microscopy: erratum

Yang-Hyo Kim; Peter T. C. So

doi:10.1364/OE.25.031423

Optics Express
Vol. 25,
Issue 25,
pp. 31423-31430
(2017)
•https://doi.org/10.1364/OE.25.031423

Three-dimensional wide-field pump-probe structured illumination microscopy: erratum

Yang-Hyo Kim and Peter T. C. So

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Yang-Hyo Kim and Peter T. C. So, "Three-dimensional wide-field pump-probe structured illumination microscopy: erratum," Opt. Express 25, 31423-31430 (2017)

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About this Article
History
- Original Manuscript: November 15, 2017
- Published: December 1, 2017
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 13, Iss. 2

Abstract

We found errors in Eqs. (5), (6), (7), (8), (23), (25), Figs. 2, 3, 4, 5, 10, and Discussion of our article “Three-dimensional wide-field pump-probe structured illumination microscopy.” Here we publish the revised equations, figures, and discussion. In general, the corrections do not affect the essential conclusion and image reconstruction quality improves with these corrections.

1. Configuration of a wide-field pump-probe structured illumination microscope (ppSIM)

The 3rd plane wave component, ${\vec{e}}_{3} (y, z)$ of Eq. (5) in [1] has an error for the positive/negative signs in front of the unit vectors in y and z directions. The revised equation is as the following.

\begin{array}{l} {\vec{e}}_{1} (x, z) = \hat{y} \exp [i (k \sin θ x + k \cos θ z + ϕ_{1})] \\ {\vec{e}}_{2} (x, z) = \hat{y} \exp [i (- k \sin θ x + k \cos θ z + ϕ_{2})] \\ {\vec{e}}_{3} (y, z) = \hat{y} \cos θ \exp [i (k \sin θ y + k \cos θ z + ϕ_{3})] \\ - \hat{z} \sin θ \exp [i (k \sin θ y + k \cos θ z + ϕ_{3})] \\ {\vec{e}}_{4} (y, z) = \hat{y} \cos θ \exp [i (- k \sin θ y + k \cos θ z + ϕ_{4})] \\ + \hat{z} \sin θ \exp [i (- k \sin θ y + k \cos θ z + ϕ_{4})] \\ {\vec{e}}_{5} (z) = \hat{y} \exp [i (k z + ϕ_{5})] \end{array}

The above-mentioned revision affects the plus/minus signs in the following equations and the revised equations for Eq. (6), (7), and (8) in [1] are shown below. The structured pump beam intensity for S-pol interference pattern (Eq. 6) in [1]) can be calculated as

\begin{array}{l} e (x, y, z) = {| {\vec{E}}_{1} + {\vec{E}}_{2} + {\vec{E}}_{3} + {\vec{E}}_{4} + {\vec{E}}_{5} |}^{2} = [5 \\ + 2 \cos (2 k \sin θ \cdot x + Δ ϕ_{12}) \\ + 2 \cos θ \cos (k \sin θ \cdot x - k \sin θ \cdot y + Δ ϕ_{13}) \\ + 2 \cos θ \cos (k \sin θ \cdot x + k \sin θ \cdot y + Δ ϕ_{14}) \\ + 2 \cos (k \sin θ \cdot x + k \cos θ \cdot z - k z + Δ ϕ_{15}) \\ + 2 \cos θ \cos (- k \sin θ \cdot x - k \sin θ \cdot y + Δ ϕ_{23}) \\ + 2 \cos θ \cos (- k \sin θ \cdot x + k \sin θ \cdot y + Δ ϕ_{24}) \\ + 2 \cos (- k \sin θ \cdot x + k \cos θ \cdot z - k z + Δ ϕ_{25}) \\ + 2 \cos 2 θ \cos (2 k \sin θ \cdot y + Δ ϕ_{34}) \\ + 2 \cos θ \cos (k \sin θ \cdot y + k \cos θ \cdot z - k z + Δ ϕ_{35}) \\ + 2 \cos θ \cos (- k \sin θ \cdot y + k \cos θ \cdot z - k z + Δ ϕ_{45})] \end{array}

where

Δ ϕ_{i j} \equiv ϕ_{i} - ϕ_{j}

By Fourier transforming this pump beam intensity, $F {e (x, y, z)}$ , we obtain the pump beam spectrum in the spatial-frequency domain (Eq. 7) in [1]) as

\begin{array}{l} E (m, n, s) = F {e (x, y, z)} = 5 δ (m, n, s) \\ + \exp (- i Δ ϕ_{12}) δ (m + 2 k_{m n}, n, s) \\ + \exp (i Δ ϕ_{12}) δ (m - 2 k_{m n}, n, s) \\ + \cos θ {\exp (- i Δ ϕ_{13}) + \exp (i Δ ϕ_{24})} δ (m + k_{m n}, n - k_{m n}, s) \\ + \cos θ {\exp (i Δ ϕ_{13}) + \exp (- i Δ ϕ_{24})} δ (m - k_{m n}, n + k_{m n}, s) \\ + \cos θ {\exp (- i Δ ϕ_{14}) + \exp (i Δ ϕ_{23})} δ (m + k_{m n}, n + k_{m n}, s) \\ + \cos θ {\exp (i Δ ϕ_{14}) + \exp (- i Δ ϕ_{23})} δ (m - k_{m n}, n - k_{m n}, s) \\ + {\exp (- i Δ ϕ_{15}) δ (m + k_{m n}, n, s + k_{s}) + \exp (i Δ ϕ_{15}) δ (m - k_{m n}, n, s - k_{s})} \\ + {\exp (- i Δ ϕ_{25}) δ (m - k_{m n}, n, s + k_{s}) + \exp (i Δ ϕ_{25}) δ (m + k_{m n}, n, s - k_{s})} \\ + \cos 2 θ {\exp (- i Δ ϕ_{34}) δ (m, n + 2 k_{m n}, s) + \exp (i Δ ϕ_{34}) δ (m, n - 2 k_{m n}, s)} \\ + \cos θ {\exp (- i Δ ϕ_{35}) δ (m, n + k_{m n}, s + k_{s}) + \exp (i Δ ϕ_{35}) δ (m, n - k_{m n}, s - k_{s})} \\ + \cos θ {\exp (- i Δ ϕ_{45}) δ (m, n - k_{m n}, s + k_{s}) + \exp (i Δ ϕ_{45}) δ (m, n + k_{m n}, s - k_{s})} \end{array}

where

k_{m n} \equiv \frac{k \sin θ}{2 π} = \frac{\sin θ}{λ}

and

k_{s} \equiv \frac{k \cos θ - k}{2 π} = \frac{\cos θ - 1}{λ}

Substituting the previous result into Eq. (4) in [1], and mathematically expanding, we get the corresponding image field in the Fourier space (Eq. 8) in [1]) as

\begin{array}{l} U (m, n, s) = [5 O (m, n, s) \\ + \exp (- i Δ ϕ_{12}) O (m + 2 k_{m n}, n, s) \\ + \exp (i Δ ϕ_{12}) O (m - 2 k_{m n}, n, s) \\ + \cos θ {\exp (- i Δ ϕ_{13}) + \exp (i Δ ϕ_{24})} O (m + k_{m n}, n - k_{m n}, s) \\ + \cos θ {\exp (i Δ ϕ_{13}) + \exp (- i Δ ϕ_{24})} O (m - k_{m n}, n + k_{m n}, s) \\ + \cos θ {\exp (- i Δ ϕ_{14}) + \exp (i Δ ϕ_{23})} O (m + k_{m n}, n + k_{m n}, s) \\ + \cos θ {\exp (i Δ ϕ_{14}) + \exp (- i Δ ϕ_{23})} O (m - k_{m n}, n - k_{m n}, s) \\ + \exp (- i Δ ϕ_{15}) O (m + k_{m n}, n, s + k_{s}) \\ + \exp (i Δ ϕ_{15}) O (m - k_{m n}, n, s - k_{s}) \\ + \exp (- i Δ ϕ_{25}) O (m - k_{m n}, n, s + k_{s}) \\ + \exp (i Δ ϕ_{25}) O (m + k_{m n}, n, s - k_{s}) \\ + \cos 2 θ \exp (- i Δ ϕ_{34}) O (m, n + 2 k_{m n}, s) \\ + \cos 2 θ \exp (i Δ ϕ_{34}) O (m, n - 2 k_{m n}, s) \\ + \cos θ \exp (- i Δ ϕ_{35}) O (m, n + k_{m n}, s + k_{s}) \\ + \cos θ \exp (i Δ ϕ_{35}) O (m, n - k_{m n}, s - k_{s}) \\ + \cos θ \exp (- i Δ ϕ_{45}) O (m, n - k_{m n}, s + k_{s}) \\ + \cos θ \exp (i Δ ϕ_{45}) O (m, n + k_{m n}, s - k_{s})] H (m, n, s) \end{array}

2 Appendix A

The light field in the image plane with a shifted object (Eq. (23) in [1]) was represented incorrectly according to the standard convolution definition. The right representation is as the following.

\begin{array}{l} U_{4} (x_{4}, y_{4}, z_{5}) = \frac{\exp [- 2 i k (f_{1} + f_{2})]}{λ^{2} f_{1} f_{2}} \\ \iint \int_{- \infty}^{\infty} o (x_{1}, y_{1}, z_{1} - z_{5}) \exp (- i k z_{1}) h (x_{1} + M x_{4}, y_{1} + M y_{4}, z_{1}) d x_{1} d y_{1} d z_{1} \\ = \frac{\exp [- 2 i k (f_{1} + f_{2})]}{λ^{2} f_{1} f_{2}} \iint \int_{- \infty}^{\infty} o (x_{1}, y_{1}, z_{1}) h^{'} (- x_{1} - M x_{4}, - y_{1} - M y_{4}, - z_{1} - z_{5}) d x_{1} d y_{1} d z_{1} \end{array}

with a change of a integration variable in z direction and a new function

h^{'} (x, y, z) = \exp (i k z) h (- x, - y, - z)

. Thus the acquired image field is the 3D convolution of the object function with the point spread function

h^{'} (x, y, z)

. Here the negative sign before the coordinate variable (

x_{4}, y_{4}, and z_{5}

) inside the point spread function

h^{'}

implies that the image is inverted.

With the above mentioned change, the final equation for 3D CTF (Eq. (25) in [1]) changes as

c (l, s) = {(λ f_{1})}^{2} p (l) δ (s - \frac{1}{λ} + \frac{λ l^{2}}{2})

where

p (l) = {\begin{cases} 1, l \leq N A / λ \\ 0, l > N A / λ \end{cases}

(NA: numerical aperture of the optical system) and

δ ()

is a delta function.

This 3D CTF is an axially shifted cap of a paraboloid of revolution about the s axis in Fig. 1(a) (Fig. 10) in [1]). However, this axial shift of the 3D CTF comes from the wave vector of the incident light and the 3D CTF in Fig. 1(a) is a 3D Fourier transform of the diffracted light over the object. Therefore, the effective 3D CTF to measure the pass/block of the object information passes the origin without the axial shift in Fig. 1(b) [2].

Fig. 1 Schematic diagram of the 3D coherent transfer function for coherent imaging with a circular lens in 4f system. (a) 3D Fourier transform of the diffracted light. (b) Effective 3D CTF of the object. The object diffracts the incident wave and the resulting diffracted wave is captured by the imaging system. The vector $K$ represents a characteristic wave vector of the incident wave, the object as a grating, and the diffracted wave.

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3. Theoretical framework

The Fig. 2 corresponds to the Fig. 2 in [1]. In Fig. 2(d)–2(f), we can see the change of 3D CTF. With the revised equations and the effective 3D CTF (without the axial shift), we re-simulated all the results of [1] in the following.

Fig. 2 Pump beam configuration and probe beam extended transfer function for pump-probe structured illumination microscopy (ppSIM). (a) The five wave vectors corresponding to the each pump beam direction. All five wave vectors have the same magnitude $k = 2 π / λ$ . (b, c) The resulting spatial frequency components of the illumination intensity for the ppSIM with (b) a single grating period and (c) grating period scanning. (d-e) The transfer function for (d) the conventional wide-field microscopy, (e) the single grating period ppSIM, and (f) grating period scanning ppSIM in (1) 3D, (2) mn plane, and (3) ms plane. The color in 3D transfer function represents the position in s axis, not a weighting.

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4. Pump-probe structured illumination imaging of a planar target: calibration chart

The Fig. 3 corresponds to the Fig. 3 in [1]. In Fig. 3(a4), 3(b4), and 3(c4), we can see the change of 3D CTF. In both the lateral and axial cases, excellent agreement with the theoretical expectations is observed in Fig. 3(e) and 3(g). The field amplitude modulation of lateral case is close to one because of the dense spatial frequency support in the lateral direction, while that of axial case is smaller than one because of relatively coarse spatial frequency support in the axial direction.

Fig. 3 Numerical simulation results of USAF 1951 test chart: a whole image data for (a) conventional wide-field microscope, (b) ppSIM with a single grating period, and (c) ppSIM with grating period scanning in (1) xy (2) xz, (3) mn, and (4) ms planes. The image spectra (logarithmic scale) are displayed for amplitude with their axes normalized with the lateral cut-off frequency of the conventional microscope. An isolated three bar pattern simulation results comparing the conventional microscope and ppSIM with grating period scanning in (d, e) lateral and (f, g) axial dimensions: (d, f) cross-sectional profiles in space and (e, g) field amplitude modulations according to the spatial frequency. For the conventional wide-field imaging, the three bar patterns inside the dashed red rectangle in (a1) appear blurred and they are barely resolvable where the period of element 1 (the most coarse set) is close to the coherent Abbe diffraction limit λ/NA (1.177 µm in this study). For the ppSIM imaging, on the other hand, three bars inside the dashed rectangles in (b1, c1) are clearly distinguished with more sharply defined edges.

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5. Pump-probe structured illumination imaging of a non-planar target: a 3D MIT logo

The re-simulated Fig. 4 in [1] shows almost the same as before except for the better contrast in Fig. 4(c) because near DC spatial information is allowed to pass through the system’s transfer function.

Fig. 4 3D MIT logo image simulation result: (a) original 3D object and (b) the image from the conventional wide-field microscope and (c) the image from ppSIM with grating period scanning in (1) xy plane and (2) xz plane. xy plane passes through the middle of the letter ‘i' (z = 0 µm) and xz plane cuts only the legs of each letter (y = 0 µm).

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6. Pump-probe structured illumination imaging of biomolecules in HeLa cells

The Fig. 5 corresponds to the Fig. 5 in [1]. In Fig. 5(f3), 3(g3), and 3(h3), we can see the change of 3D CTF. For high NA pump-probe SIM simulation, we changed the NA from 0.9 to 1.0 to match the sectioning capability similar to the previous result (grating period is also changed from 0.84 µm ~2.77 μm to 1.11 µm ~3.66 µm). For all the HEC1 protein and the DNA, both the low NA pump-probe SIM and the high NA pump-probe SIM successfully reconstruct original sample information in both lateral and axial dimensions in Fig. 5(c), 5(d), 5(g), and 5(h). Comparing the images in Fig. 5, we can observe the 3D imaging capability of our grating period scanning pump-probe SIM is consistent regardless of the characteristics of the sample itself and NA of the objective lens.

Fig. 5 The HEC1 (a–c) and DNA (e–h) in HeLa cells. (a, e) Original 3D data, (b, f) conventional wide-field microscope with 0.68 NA objective, (c, g) grating period scanning ppSIM with 0.68 NA objective, and (d, h) grating scanning ppSIM with 1.0 NA objective in (1) xy plane, (2) xz plane, and (3) ms plane. The image spectra (logarithmic scale) are displayed for amplitude with their axes normalized with the lateral cut-off frequency of the conventional wide-field microscope with 0.68 NA objective. m = 0 and n = 0 axes lines are added to help analyzing the CTF support change. ppSIM with 1.0 NA shows the axial cut-off frequency of ~2.21 (which gives ~0.53 µm sectioning capability).

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7. Discussion

We found an error in counting the number of images for the intensity SIM framework to reconstruct a single 3D resolved image ( $M_{F}$ ). The revised numbers are $M_{F} = 11851$ images for $N_{g} = 10$ and $M_{F} = 31717$ images for $N_{g} = 16$ ). On the other hand, the field SIM framework that we suggested requires 5 pump beams (17 Fourier copies), 4 measurements for optical phase and amplitude, and $N_{g}$ grating period scanning which results in total $4 \times 17 N_{g} = 68 N_{g}$ images (680 images for $N_{g} = 10$ and 1088 images for $N_{g} = 16$ ). In conclusion, our optical field based pump-probe SIM framework requires 30~50 times less number of pump beams and 17~29 times less number of images to reconstruct a single 3D image with the same synthetic CTF than the intensity SIM framework.

Funding

National Institutes of Health (9P41EB015871-26A1, 5R01NS051320, 4R44EB012415, and 1R01HL121386-01A1); National Science Foundation (CBET-0939511); Hamamatsu Corporation; Singapore–Massachusetts Institute of Technology Alliance for Research and Technology (SMART) Center, BioSystems and Micromechanics (BioSyM); Samsung Scholarship.

References and links

1. Y.-H. Kim and P. T. C. So, “Three-dimensional wide-field pump-probe structured illumination microscopy,” Opt. Express 25(7), 7369–7391 (2017). [PubMed]

2. S. S. Kou and C. J. R. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express 15(21), 13640–13648 (2007). [PubMed]

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K

represents a characteristic wave vector of the incident wave, the object as a grating, and the diffracted wave.

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k = 2 π / λ

. (b, c) The resulting spatial frequency components of the illumination intensity for the ppSIM with (b) a single grating period and (c) grating period scanning. (d-e) The transfer function for (d) the conventional wide-field microscopy, (e) the single grating period ppSIM, and (f) grating period scanning ppSIM in (1) 3D, (2) mn plane, and (3) ms plane. The color in 3D transfer function represents the position in s axis, not a weighting.

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