Effects of spatiotemporal coherence on interferometric microscopy

Seungwoo Shin; Kyoohyun Kim; KyeoReh Lee; SangYun Lee; YongKeun Park

doi:10.1364/OE.25.008085

Effects of spatiotemporal coherence on interferometric microscopy

Seungwoo Shin, Kyoohyun Kim, KyeoReh Lee, SangYun Lee, and YongKeun Park

Optics Express
Vol. 25,
Issue 7,
pp. 8085-8097
(2017)
•https://doi.org/10.1364/OE.25.008085

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Seungwoo Shin, Kyoohyun Kim, KyeoReh Lee, SangYun Lee, and YongKeun Park, "Effects of spatiotemporal coherence on interferometric microscopy," Opt. Express 25, 8085-8097 (2017)

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Related Topics
Table of Contents Category
- Coherence and Statistical Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Coherence (030.1640)
- Interferometric imaging (110.3175)
- Phase measurement (120.5050)
- Interference microscopy (180.3170)

About this Article
History
- Original Manuscript: January 27, 2017
- Revised Manuscript: March 18, 2017
- Manuscript Accepted: March 19, 2017
- Published: March 29, 2017
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 12, Iss. 6

Accessible

Open Access

Abstract

Illumination coherence plays a major role in various imaging systems, from microscopy, metrology, digital holography, optical coherence tomography, to ultrasound imaging. Here, we present a systematic study on the effects of degrees of spatiotemporal coherence of an illumination (DSTCI) on imaging quality of interferometric microscopy. An optical field with arbitrary DSTCI was decomposed into wavelets with constituent spatiotemporal frequencies, and the effects on image quality were quantitatively investigated. The results show the synergistic effects on reduction of speckle noise when DSTCI is decreased. This study presents a method to systematically control DSTCI, and the result provides an essential reference on the effects of DSTCI on the imaging quality. We believe that the presented methods and results can be implemented in various imaging systems for characterizing and improving the imaging quality.

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References

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X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
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J. A. Rodrigo and T. Alieva, “Illumination coherence engineering and quantitative phase imaging,” Opt. Lett. 39(19), 5634–5637 (2014).
[Crossref] [PubMed]
S. Shin, Y. Kim, K. Lee, K. Kim, Y.-J. Kim, H. Park, and Y. Park, “Common-path diffraction optical tomography with a low-coherence illumination for reducing speckle noise,” Proc. SPIE 9336, 933629 (2015).
[Crossref]
C. Edwards, B. Bhaduri, T. Nguyen, B. G. Griffin, H. Pham, T. Kim, G. Popescu, and L. L. Goddard, “Effects of spatial coherence in diffraction phase microscopy,” Opt. Express 22(5), 5133–5146 (2014).
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[Crossref] [PubMed]
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J. Jung, K. Kim, J. Yoon, and Y. Park, “Hyperspectral optical diffraction tomography,” Opt. Express 24(3), 2006–2012 (2016).
[Crossref] [PubMed]
G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[Crossref] [PubMed]
Y. Kim, H. Shim, K. Kim, H. Park, J. H. Heo, J. Yoon, C. Choi, S. Jang, and Y. Park, “Common-path diffraction optical tomography for investigation of three-dimensional structures and dynamics of biological cells,” Opt. Express 22(9), 10398–10407 (2014).
[Crossref] [PubMed]
J. Jung and Y. Park, “Spectro-angular light scattering measurements of individual microscopic objects,” Opt. Express 22(4), 4108–4114 (2014).
[Crossref] [PubMed]
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[Crossref] [PubMed]
J. Jung and Y. Park, “Experimental observations of spectral changes produced by individual microscopic spheres,” Opt. Lett. 40(6), 1093–1096 (2015).
[Crossref] [PubMed]
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[Crossref]
S. K. Debnath and Y. Park, “Real-time quantitative phase imaging with a spatial phase-shifting algorithm,” Opt. Lett. 36(23), 4677–4679 (2011).
[Crossref] [PubMed]

2016 (3)

K. Kim, J. Yoon, S. Shin, S. Lee, S.-A. Yang, and Y. Park, “Optical diffraction tomography techniques for the study of cell pathophysiology,” J. Biomed. Photonics Eng. 2, 020201 (2016).

J. Dohet-Eraly, C. Yourassowsky, A. E. Mallahi, and F. Dubois, “Quantitative assessment of noise reduction with partial spatial coherence illumination in digital holographic microscopy,” Opt. Lett. 41(1), 111–114 (2016).
[Crossref] [PubMed]

J. Jung, K. Kim, J. Yoon, and Y. Park, “Hyperspectral optical diffraction tomography,” Opt. Express 24(3), 2006–2012 (2016).
[Crossref] [PubMed]

2015 (2)

S. Shin, Y. Kim, K. Lee, K. Kim, Y.-J. Kim, H. Park, and Y. Park, “Common-path diffraction optical tomography with a low-coherence illumination for reducing speckle noise,” Proc. SPIE 9336, 933629 (2015).
[Crossref]

J. Jung and Y. Park, “Experimental observations of spectral changes produced by individual microscopic spheres,” Opt. Lett. 40(6), 1093–1096 (2015).
[Crossref] [PubMed]

2014 (4)

J. A. Rodrigo and T. Alieva, “Illumination coherence engineering and quantitative phase imaging,” Opt. Lett. 39(19), 5634–5637 (2014).
[Crossref] [PubMed]

C. Edwards, B. Bhaduri, T. Nguyen, B. G. Griffin, H. Pham, T. Kim, G. Popescu, and L. L. Goddard, “Effects of spatial coherence in diffraction phase microscopy,” Opt. Express 22(5), 5133–5146 (2014).
[Crossref] [PubMed]

Y. Kim, H. Shim, K. Kim, H. Park, J. H. Heo, J. Yoon, C. Choi, S. Jang, and Y. Park, “Common-path diffraction optical tomography for investigation of three-dimensional structures and dynamics of biological cells,” Opt. Express 22(9), 10398–10407 (2014).
[Crossref] [PubMed]

J. Jung and Y. Park, “Spectro-angular light scattering measurements of individual microscopic objects,” Opt. Express 22(4), 4108–4114 (2014).
[Crossref] [PubMed]

2013 (4)

J. H. Jung, J. Jang, and Y. Park, “Spectro-refractometry of individual microscopic objects using swept-source quantitative phase imaging,” Anal. Chem. 85(21), 10519–10525 (2013).
[Crossref] [PubMed]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
[Crossref] [PubMed]

K. Lee, K. Kim, J. Jung, J. Heo, S. Cho, S. Lee, G. Chang, Y. Jo, H. Park, and Y. Park, “Quantitative phase imaging techniques for the study of cell pathophysiology: from principles to applications,” Sensors (Basel) 13(4), 4170–4191 (2013).
[Crossref] [PubMed]

K. Lee, H. D. Kim, K. Kim, Y. Kim, T. R. Hillman, B. Min, and Y. Park, “Synthetic Fourier transform light scattering,” Opt. Express 21(19), 22453–22463 (2013).
[Crossref] [PubMed]

2012 (2)

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012).
[Crossref] [PubMed]

2011 (3)

S. K. Debnath and Y. Park, “Real-time quantitative phase imaging with a spatial phase-shifting algorithm,” Opt. Lett. 36(23), 4677–4679 (2011).
[Crossref] [PubMed]

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19(2), 1016–1026 (2011).
[Crossref] [PubMed]

Y. Choi, T. D. Yang, K. J. Lee, and W. Choi, “Full-field and single-shot quantitative phase microscopy using dynamic speckle illumination,” Opt. Lett. 36(13), 2465–2467 (2011).
[Crossref] [PubMed]

2010 (2)

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
[Crossref] [PubMed]

2009 (2)

Y. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. S. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17(15), 12285–12292 (2009).
[Crossref] [PubMed]

S. K. Narayanan and R. Wahidabanu, “A view on despeckling in ultrasound imaging,” IJSIP 2, 85–98 (2009).

1999 (1)

J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Speckle in optical coherence tomography,” J. Biomed. Opt. 4(1), 95–105 (1999).
[Crossref] [PubMed]

1987 (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326(6111), 363–365 (1987).
[Crossref]

1982 (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
[Crossref]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
[Crossref]

Alieva, T.

J. A. Rodrigo and T. Alieva, “Illumination coherence engineering and quantitative phase imaging,” Opt. Lett. 39(19), 5634–5637 (2014).
[Crossref] [PubMed]

Badizadegan, K.

Y. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. S. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17(15), 12285–12292 (2009).
[Crossref] [PubMed]

Barbastathis, G.

L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
[Crossref] [PubMed]

Bhaduri, B.

B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012).
[Crossref] [PubMed]

Bishara, W.

Callens, N.

Cao, H.

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

Chang, G.

Cho, S.

Choi, C.

Choi, W.

Y. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. S. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17(15), 12285–12292 (2009).
[Crossref] [PubMed]

Choi, Y.

Choma, M. A.

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

Dasari, R.

Y. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. S. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17(15), 12285–12292 (2009).
[Crossref] [PubMed]

Dasari, R. R.

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[Crossref] [PubMed]

Debnath, S. K.

S. K. Debnath and Y. Park, “Real-time quantitative phase imaging with a spatial phase-shifting algorithm,” Opt. Lett. 36(23), 4677–4679 (2011).
[Crossref] [PubMed]

Ding, H.

Dohet-Eraly, J.

Dubois, F.

Edwards, C.

Feld, M. S.

Y. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. S. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17(15), 12285–12292 (2009).
[Crossref] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[Crossref] [PubMed]

Gillette, M. U.

Goddard, L. L.

Griffin, B. G.

Heo, J.

Heo, J. H.

Hillman, T. R.

K. Lee, H. D. Kim, K. Kim, Y. Kim, T. R. Hillman, B. Min, and Y. Park, “Synthetic Fourier transform light scattering,” Opt. Express 21(19), 22453–22463 (2013).
[Crossref] [PubMed]

Horstmeyer, R.

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
[Crossref] [PubMed]

Hoyos, M.

Ikeda, T.

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[Crossref] [PubMed]

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
[Crossref]

Isikman, S. O.

Istasse, E.

Iwai, H.

Jang, J.

Jang, S.

Jo, Y.

Jung, J.

J. Jung, K. Kim, J. Yoon, and Y. Park, “Hyperspectral optical diffraction tomography,” Opt. Express 24(3), 2006–2012 (2016).
[Crossref] [PubMed]

J. Jung and Y. Park, “Experimental observations of spectral changes produced by individual microscopic spheres,” Opt. Lett. 40(6), 1093–1096 (2015).
[Crossref] [PubMed]

J. Jung and Y. Park, “Spectro-angular light scattering measurements of individual microscopic objects,” Opt. Express 22(4), 4108–4114 (2014).
[Crossref] [PubMed]

Jung, J. H.

Khademhosseini, B.

Kim, H. D.

K. Lee, H. D. Kim, K. Kim, Y. Kim, T. R. Hillman, B. Min, and Y. Park, “Synthetic Fourier transform light scattering,” Opt. Express 21(19), 22453–22463 (2013).
[Crossref] [PubMed]

Kim, K.

K. Kim, J. Yoon, S. Shin, S. Lee, S.-A. Yang, and Y. Park, “Optical diffraction tomography techniques for the study of cell pathophysiology,” J. Biomed. Photonics Eng. 2, 020201 (2016).

J. Jung, K. Kim, J. Yoon, and Y. Park, “Hyperspectral optical diffraction tomography,” Opt. Express 24(3), 2006–2012 (2016).
[Crossref] [PubMed]

K. Lee, H. D. Kim, K. Kim, Y. Kim, T. R. Hillman, B. Min, and Y. Park, “Synthetic Fourier transform light scattering,” Opt. Express 21(19), 22453–22463 (2013).
[Crossref] [PubMed]

Kim, T.

Kim, Y.

K. Lee, H. D. Kim, K. Kim, Y. Kim, T. R. Hillman, B. Min, and Y. Park, “Synthetic Fourier transform light scattering,” Opt. Express 21(19), 22453–22463 (2013).
[Crossref] [PubMed]

Kim, Y.-J.

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
[Crossref]

Kurowski, P.

Lee, K.

K. Lee, H. D. Kim, K. Kim, Y. Kim, T. R. Hillman, B. Min, and Y. Park, “Synthetic Fourier transform light scattering,” Opt. Express 21(19), 22453–22463 (2013).
[Crossref] [PubMed]

Lee, K. J.

Lee, S.

K. Kim, J. Yoon, S. Shin, S. Lee, S.-A. Yang, and Y. Park, “Optical diffraction tomography techniques for the study of cell pathophysiology,” J. Biomed. Photonics Eng. 2, 020201 (2016).

Mallahi, A. E.

Millet, L.

Min, B.

K. Lee, H. D. Kim, K. Kim, Y. Kim, T. R. Hillman, B. Min, and Y. Park, “Synthetic Fourier transform light scattering,” Opt. Express 21(19), 22453–22463 (2013).
[Crossref] [PubMed]

Minetti, C.

Mir, M.

B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012).
[Crossref] [PubMed]

Miwa, M.

Monnom, O.

Mudanyali, O.

Narayanan, S. K.

S. K. Narayanan and R. Wahidabanu, “A view on despeckling in ultrasound imaging,” IJSIP 2, 85–98 (2009).

Nguyen, T.

Novella Requena, M.-L.

Oh, C.

Ou, X.

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
[Crossref] [PubMed]

Ozcan, A.

Oztoprak, C.

Park, H.

Park, Y.

K. Kim, J. Yoon, S. Shin, S. Lee, S.-A. Yang, and Y. Park, “Optical diffraction tomography techniques for the study of cell pathophysiology,” J. Biomed. Photonics Eng. 2, 020201 (2016).

J. Jung, K. Kim, J. Yoon, and Y. Park, “Hyperspectral optical diffraction tomography,” Opt. Express 24(3), 2006–2012 (2016).
[Crossref] [PubMed]

J. Jung and Y. Park, “Experimental observations of spectral changes produced by individual microscopic spheres,” Opt. Lett. 40(6), 1093–1096 (2015).
[Crossref] [PubMed]

J. Jung and Y. Park, “Spectro-angular light scattering measurements of individual microscopic objects,” Opt. Express 22(4), 4108–4114 (2014).
[Crossref] [PubMed]

K. Lee, H. D. Kim, K. Kim, Y. Kim, T. R. Hillman, B. Min, and Y. Park, “Synthetic Fourier transform light scattering,” Opt. Express 21(19), 22453–22463 (2013).
[Crossref] [PubMed]

S. K. Debnath and Y. Park, “Real-time quantitative phase imaging with a spatial phase-shifting algorithm,” Opt. Lett. 36(23), 4677–4679 (2011).
[Crossref] [PubMed]

Y. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. S. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17(15), 12285–12292 (2009).
[Crossref] [PubMed]

Pham, H.

B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012).
[Crossref] [PubMed]

Pitter, M. C.

M. C. Pitter, C. W. See, and M. G. Somekh, “Full-field heterodyne interference microscope with spatially incoherent illumination,” Opt. Lett. 29(11), 1200–1202 (2004).
[Crossref] [PubMed]

Popescu, G.

B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012).
[Crossref] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[Crossref] [PubMed]

Redding, B.

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

Rodrigo, J. A.

J. A. Rodrigo and T. Alieva, “Illumination coherence engineering and quantitative phase imaging,” Opt. Lett. 39(19), 5634–5637 (2014).
[Crossref] [PubMed]

Rogers, J.

Schmitt, J. M.

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Fig. 1 Preceding studies classified by degree of spatiotemporal coherence of the illumination. Reports in which the degree of coherence was not specified are classified as “partially coherent”.

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Fig. 2 Schematics of interferometric imaging of a sample with a dust particle which is set apart from the sample along the optical axis. The right-side images are example phase images of a 3 μm−diameter polymethyl methacrylate bead with respect to the DSTCI. The degrees of coherence are (a) spatiotemporally coherent, (b) spatially coherent and temporally low coherent, (c) temporally coherent and spatially low coherent, and (d) spatiotemporally low coherent.

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Fig. 3 (a) Experimental setup. Common-path interferometry with varying spatiotemporal frequency of the illumination. The temporal frequency of the illumination is controlled by using the prism, the galvanometric mirror (GM1), and the pinhole (P1); the prism disperses an incident super-continuum laser, and the temporal frequency of light passing through P1 is controlled by the GM1. The dual-axis galvanometric mirror (GM2), which is located at the conjugate plane of a sample, controls the spatial frequency of the illumination. The other galvanometric mirror (GM3) compensates the tilted angle by the GM2 for common-path interferometric geometry. P1: pinhole (diameter = 100 μm); TL1: tube lens (f = 200 mm); CL: condenser lens, 0.9 NA; OL: objective lens, 1.42 NA; TL2: tube lens (f = 180 mm) L1: lens (f = 100 mm); L2: lens (f = 200 mm); L3,7: lenses (f = 75 mm); L4,8: lenses (f = 150 mm); L5,6: lenses (f = 100 mm); G: grating; P2: pinhole (diameter = 25 μm). (b) Temporal power spectra: center wavelength is changed from 525 nm to 667 nm with 61 steps, the mean bandwidth is 4.6 nm. (c) Spatial frequencies controlled by the GM2 in (a) are uniformly scanned within the numerical aperture (NA) 0.75 with 441 steps.

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Fig. 4 Representative optical field images of a PMMA microsphere illuminated with various DSTCI. The inset shows magnified images. The degree of spatial and temporal coherence of the illumination decreases along the row and column, respectively.

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Fig. 5 Spatial and temporal power spectra, autocorrelation functions (ACF), and coherence lengths with respect to bandwidths. (a) Representative azimuthally averaged spatial power spectra of the illuminations. Solid black, blue, and red lines correspond to the spatial frequency bandwidth (Δk_s) = 1.24, 1.72, and 2.37 μm⁻¹, respectively. (b) The absolute of the spatial ACF corresponding to (a). (c) Spatial coherence lengths (l_s) as a function of spatial frequency bandwidths. The fitted curve is l_s = 1.22/Δk_s. The spatial power spectra in (a) are marked as black, blue, and red arrows, respectively. (d) Representative temporal power spectra with the center wavelength of 579 nm. Solid black, blue, and red lines correspond to the wavelength bandwidth (Δλ) of 13.7, 21.9, and 44.7 nm, respectively. (e) The absolute of the temporal ACF corresponding to (d). (f) Temporal coherence lengths (l_t) with respect to wavelength bandwidths. The fitted curve is l_t = λ²/(n_mediumΔλ), where n_medium is the refractive index of the medium. The temporal power spectra in (d) are marked as black, blue, and red arrows, respectively.

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Fig. 6 Speckle noise quantified as the spatial phase standard deviation (σ_φ) with respect to the spatial and temporal coherence lengths (l_s, l_t). (a) Measured σ_φ are presented by black dots corresponding to the spatial and temporal coherence lengths, and the surface plot is obtained by the linear interpolation from the measured results for visualization purposes. (b-c) A cross-sectional view of (a) at the l_t = 63.3 μm and l_s = 11.8 μm, respectively. Error bars represent standard errors of the mean at each point. In (b), error bars are too small to be shown. The red circle marker in (c) corresponds to a coherent illumination.

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Fig. 7 (a) Phasor sum representation for the sum of complex amplitudes of various illumination temporal frequencies. (b) Scheme of a resultant interference signal from two plane wave illuminations with randomly varying relative phase.

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Equations (8)

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I = {〈 {| S + R |}^{2} 〉}_{T} = \sum_{n = 1}^{2} Re [A_{n}^{2} + 1 + A_{n} e^{j φ_{n}} \times e^{- j θ} + A_{n} e^{- j φ_{n}} \times e^{j θ}],

S = Re [A_{1} e^{j φ_{1}} \times e^{j (k_{1} z - ω_{1} t + ϕ_{1} (\vec{k_{1}}, ω_{1}))} + A_{2} e^{j φ_{2}} \times e^{j (k_{2} z - ω_{2} t + ϕ_{2} (\vec{k_{2}}, ω_{2}))}],

R = Re [{e^{j (k_{1} z - ω_{1} t + ϕ_{1} (\vec{k_{1}}, ω_{1}))} + e^{j (k_{2} z - ω_{2} t + ϕ_{2} (\vec{k_{2}}, ω_{2}))}} \times e^{j θ}] .

\begin{array}{l} {〈 S R^{*} 〉}_{T} = Re [〈 (A_{1} e^{j φ_{1}} + A_{2} e^{j φ_{2}} + A_{1} e^{j φ_{1}} \times e^{j ((k_{1} - k_{2}) z - (ω_{1} - ω_{2}) t + (ϕ_{1} (\vec{k_{1}}, ω_{1}) - ϕ_{2} (\vec{k_{2}}, ω_{2})))} \\ {+ A_{2} e^{j φ_{2}} \times e^{- j ((k_{1} - k_{2}) z - (ω_{1} - ω_{2}) t + (ϕ_{1} (\vec{k_{1}}, ω_{1}) - ϕ_{2} (\vec{k_{2}}, ω_{2})))}) \times e^{- j θ} 〉}_{T}] = Re [(A_{1} e^{j φ_{1}} + A_{2} e^{j φ_{2}}) \times e^{- j θ}] . \end{array}

Δ ϕ = Δ (k_{n} z + φ_{n}) = Δ (\frac{2 π (z + n d)}{λ_{n}}) = 2 π (z + n d) \frac{Δ λ}{λ_{0}^{2}} \equiv 2 π \frac{z + n d}{l_{t}},

S = Re [A_{1} e^{j φ_{1}} \times e^{j (\vec{k_{1}} \cdot \vec{r} - ω t + ϕ_{1} (\vec{k_{1}}))} + A_{2} e^{j φ_{2}} \times e^{j (\vec{k_{2}} \cdot \vec{r} - ω t + ϕ_{2} (\vec{k_{2}}))}],

R = Re [{e^{j (\vec{k_{1}} \cdot \vec{r} - ω t + ϕ_{1} (\vec{k_{1}}))} + e^{j (\vec{k_{2}} \cdot \vec{r} - ω t + ϕ_{2} (\vec{k_{2}}))}} \times e^{j θ}] .

\begin{array}{l} {〈 S R^{*} 〉}_{T} = Re [〈 (A_{1} e^{j φ_{1}} + A_{2} e^{j φ_{2}} + A_{1} e^{j φ_{1}} \times e^{j ((\vec{k_{1}} - \vec{k_{2}}) \cdot \vec{r} + (ϕ_{1} (\vec{k_{1}}) - ϕ_{2} (\vec{k_{2}})))} \\ {+ A_{2} e^{j φ_{2}} \times e^{- j ((\vec{k_{1}} - \vec{k_{2}}) \cdot \vec{r} + (ϕ_{1} (\vec{k_{1}}) - ϕ_{2} (\vec{k_{2}})))}) \times e^{- j θ} 〉}_{T}] = Re [(A_{1} e^{j φ_{1}} + A_{2} e^{j φ_{2}}) \times e^{- j θ}] \end{array}

Abstract

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