Abstract

Spiral phase contrast (SPC) imaging offers a vital, convenient tool for edge detection in image processing. Despite significant experimental and theoretical progress in this area, SPC imaging with invisible light is still lacking. In contrast to the general SPC scheme, here we construct a nonlinear spatial filter by equivalently imprinting the vortex phase plate onto the potassium titanyl phosphate crystal using second harmonic generation (SHG). The phase or intensity objects are displayed by a spatial light modulator (SLM) and illuminated with 1064 nm infrared light. Then the combination of our nonlinear filter with SHG in the Fourier domain enables concise, yet highly efficient SPC imaging, leading to a visible edge enhancement with invisible illumination. By programming a running dog cartoon with SLM, we also demonstrate the capacity of our scheme to detect edges and contours in real time. Our present scheme could find direct applications in infrared monitoring.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (3)

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

M. Ritsch-Marte, “Orbital angular momentum light in microscopy,” Philos. Trans. R. Soc. London A 375, 20150437 (2017).
[Crossref]

G. Gibson, B. Sun, M. Edgar, D. Phillips, N. Hempler, G. Maker, G. Malcolm, and M. Padgett, “Real-time imaging of methane gas leaks using a single-pixel camera,” Opt. Express 25, 2998–3005 (2017).
[Crossref]

2016 (2)

Y. Zhou, S. Feng, S. Nie, J. Ma, and C. Yuan, “Image edge enhancement using Airy spiral phase filter,” Opt. Express 24, 25258–25268 (2016).
[Crossref]

K. Haase, N. Kröger-Lui, A. Pucci, A. Schönhals, and W. Petrich, “Real-time mid-infrared imaging of living microorganisms,” J. Biophoton. 9, 61–66 (2016).
[Crossref]

2015 (1)

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref]

2014 (1)

D. E. Chang, V. Vuletic, and M. D. Lukin, “Quantum nonlinear optics—photon by photon,” Nat. Photonics 8, 685–694 (2014).
[Crossref]

2012 (1)

K. Huang, X. Gu, H. Pan, E. Wu, and H. Zeng, “Few-photon-level two-dimensional infrared imaging by coincidence frequency upconversion,” Appl. Phys. Lett. 100, 151102 (2012).
[Crossref]

2011 (1)

B. Klein, E. Plis, M. Kutty, N. Gautam, A. Albrecht, S. Myers, and S. Krishna, “Varshni parameters for InAs/GaSb strained layer superlattice infrared photodetectors,” J. Phys. D 44, 075102 (2011).
[Crossref]

2009 (2)

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[Crossref]

G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 26, 1788–1797 (2009).
[Crossref]

2007 (1)

2006 (2)

2005 (2)

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689–694 (2005).
[Crossref]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref]

2004 (1)

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency upconversion,” J. Mod. Opt. 51, 1433–1445 (2004).
[Crossref]

2000 (1)

1996 (1)

K. Kohlmann, “Corner detection in natural images based on the 2-D Hilbert transform,” Signal Process. 48, 225–234 (1996).
[Crossref]

1993 (1)

Z. Jaroszewicz and A. Koodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102, 391–396 (1993).
[Crossref]

1992 (1)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[Crossref]

1986 (1)

V. Torre and T. A. Poggio, “On edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 147–163 (1986).
[Crossref]

1980 (1)

D. Marr and E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London B 207, 187–217 (1980).
[Crossref]

1975 (1)

R. Hudson and J. Hudson, “The military applications of remote sensing by infrared,” Proc. IEEE 63, 104–128 (1975).
[Crossref]

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1970).

Albrecht, A.

B. Klein, E. Plis, M. Kutty, N. Gautam, A. Albrecht, S. Myers, and S. Krishna, “Varshni parameters for InAs/GaSb strained layer superlattice infrared photodetectors,” J. Phys. D 44, 075102 (2011).
[Crossref]

Bernet, S.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Campos, J.

Chang, D. E.

D. E. Chang, V. Vuletic, and M. D. Lukin, “Quantum nonlinear optics—photon by photon,” Nat. Photonics 8, 685–694 (2014).
[Crossref]

Chen, L.

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref]

Cottrell, D.

Davis, J.

Ding, J.

Edgar, M.

Fan, S.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Feng, S.

Fürhapter, S.

Gautam, N.

B. Klein, E. Plis, M. Kutty, N. Gautam, A. Albrecht, S. Myers, and S. Krishna, “Varshni parameters for InAs/GaSb strained layer superlattice infrared photodetectors,” J. Phys. D 44, 075102 (2011).
[Crossref]

Gibson, G.

Gu, X.

K. Huang, X. Gu, H. Pan, E. Wu, and H. Zeng, “Few-photon-level two-dimensional infrared imaging by coincidence frequency upconversion,” Appl. Phys. Lett. 100, 151102 (2012).
[Crossref]

Guo, C.

Haase, K.

K. Haase, N. Kröger-Lui, A. Pucci, A. Schönhals, and W. Petrich, “Real-time mid-infrared imaging of living microorganisms,” J. Biophoton. 9, 61–66 (2016).
[Crossref]

Hadfield, R. H.

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[Crossref]

Han, Y.

Hempler, N.

Hildreth, E.

D. Marr and E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London B 207, 187–217 (1980).
[Crossref]

Huang, K.

K. Huang, X. Gu, H. Pan, E. Wu, and H. Zeng, “Few-photon-level two-dimensional infrared imaging by coincidence frequency upconversion,” Appl. Phys. Lett. 100, 151102 (2012).
[Crossref]

Hudson, J.

R. Hudson and J. Hudson, “The military applications of remote sensing by infrared,” Proc. IEEE 63, 104–128 (1975).
[Crossref]

Hudson, R.

R. Hudson and J. Hudson, “The military applications of remote sensing by infrared,” Proc. IEEE 63, 104–128 (1975).
[Crossref]

Jaroszewicz, Z.

Z. Jaroszewicz and A. Koodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102, 391–396 (1993).
[Crossref]

Jesacher, A.

Joffre, M.

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[Crossref]

Klein, B.

B. Klein, E. Plis, M. Kutty, N. Gautam, A. Albrecht, S. Myers, and S. Krishna, “Varshni parameters for InAs/GaSb strained layer superlattice infrared photodetectors,” J. Phys. D 44, 075102 (2011).
[Crossref]

Kohlmann, K.

K. Kohlmann, “Corner detection in natural images based on the 2-D Hilbert transform,” Signal Process. 48, 225–234 (1996).
[Crossref]

Koodziejczyk, A.

Z. Jaroszewicz and A. Koodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102, 391–396 (1993).
[Crossref]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[Crossref]

Krishna, S.

B. Klein, E. Plis, M. Kutty, N. Gautam, A. Albrecht, S. Myers, and S. Krishna, “Varshni parameters for InAs/GaSb strained layer superlattice infrared photodetectors,” J. Phys. D 44, 075102 (2011).
[Crossref]

Kröger-Lui, N.

K. Haase, N. Kröger-Lui, A. Pucci, A. Schönhals, and W. Petrich, “Real-time mid-infrared imaging of living microorganisms,” J. Biophoton. 9, 61–66 (2016).
[Crossref]

Kubarych, K.

Kutty, M.

B. Klein, E. Plis, M. Kutty, N. Gautam, A. Albrecht, S. Myers, and S. Krishna, “Varshni parameters for InAs/GaSb strained layer superlattice infrared photodetectors,” J. Phys. D 44, 075102 (2011).
[Crossref]

Kwiat, P. G.

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency upconversion,” J. Mod. Opt. 51, 1433–1445 (2004).
[Crossref]

Lou, Y.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Lukin, M. D.

D. E. Chang, V. Vuletic, and M. D. Lukin, “Quantum nonlinear optics—photon by photon,” Nat. Photonics 8, 685–694 (2014).
[Crossref]

Ma, J.

Maker, G.

Malcolm, G.

Marr, D.

D. Marr and E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London B 207, 187–217 (1980).
[Crossref]

Maurer, C.

McCanne, R.

McNamara, D.

Myers, S.

B. Klein, E. Plis, M. Kutty, N. Gautam, A. Albrecht, S. Myers, and S. Krishna, “Varshni parameters for InAs/GaSb strained layer superlattice infrared photodetectors,” J. Phys. D 44, 075102 (2011).
[Crossref]

Nee, J.

Nie, S.

Osten, W.

Padgett, M.

Pan, H.

K. Huang, X. Gu, H. Pan, E. Wu, and H. Zeng, “Few-photon-level two-dimensional infrared imaging by coincidence frequency upconversion,” Appl. Phys. Lett. 100, 151102 (2012).
[Crossref]

Pedrini, G.

Petrich, W.

K. Haase, N. Kröger-Lui, A. Pucci, A. Schönhals, and W. Petrich, “Real-time mid-infrared imaging of living microorganisms,” J. Biophoton. 9, 61–66 (2016).
[Crossref]

Phillips, D.

Plis, E.

B. Klein, E. Plis, M. Kutty, N. Gautam, A. Albrecht, S. Myers, and S. Krishna, “Varshni parameters for InAs/GaSb strained layer superlattice infrared photodetectors,” J. Phys. D 44, 075102 (2011).
[Crossref]

Poggio, T. A.

V. Torre and T. A. Poggio, “On edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 147–163 (1986).
[Crossref]

Pucci, A.

K. Haase, N. Kröger-Lui, A. Pucci, A. Schönhals, and W. Petrich, “Real-time mid-infrared imaging of living microorganisms,” J. Biophoton. 9, 61–66 (2016).
[Crossref]

Qi, Q.

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref]

Qiu, M.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Ritsch-Marte, M.

M. Ritsch-Marte, “Orbital angular momentum light in microscopy,” Philos. Trans. R. Soc. London A 375, 20150437 (2017).
[Crossref]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[Crossref]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref]

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689–694 (2005).
[Crossref]

Ruan, Z.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Schönhals, A.

K. Haase, N. Kröger-Lui, A. Pucci, A. Schönhals, and W. Petrich, “Real-time mid-infrared imaging of living microorganisms,” J. Biophoton. 9, 61–66 (2016).
[Crossref]

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[Crossref]

Situ, G.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1970).

Sun, B.

Torre, V.

V. Torre and T. A. Poggio, “On edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 147–163 (1986).
[Crossref]

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[Crossref]

Vandevender, A. P.

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency upconversion,” J. Mod. Opt. 51, 1433–1445 (2004).
[Crossref]

Vuletic, V.

D. E. Chang, V. Vuletic, and M. D. Lukin, “Quantum nonlinear optics—photon by photon,” Nat. Photonics 8, 685–694 (2014).
[Crossref]

Wang, J.

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref]

Wu, E.

K. Huang, X. Gu, H. Pan, E. Wu, and H. Zeng, “Few-photon-level two-dimensional infrared imaging by coincidence frequency upconversion,” Appl. Phys. Lett. 100, 151102 (2012).
[Crossref]

Xu, J.

Ye, H.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Yuan, C.

Zeng, H.

K. Huang, X. Gu, H. Pan, E. Wu, and H. Zeng, “Few-photon-level two-dimensional infrared imaging by coincidence frequency upconversion,” Appl. Phys. Lett. 100, 151102 (2012).
[Crossref]

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]

Zhang, W.

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref]

Zheng, S.

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref]

Zhou, Y.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Y. Zhou, S. Feng, S. Nie, J. Ma, and C. Yuan, “Image edge enhancement using Airy spiral phase filter,” Opt. Express 24, 25258–25268 (2016).
[Crossref]

Zhu, T.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Appl. Phys. Lett. (1)

K. Huang, X. Gu, H. Pan, E. Wu, and H. Zeng, “Few-photon-level two-dimensional infrared imaging by coincidence frequency upconversion,” Appl. Phys. Lett. 100, 151102 (2012).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

V. Torre and T. A. Poggio, “On edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 147–163 (1986).
[Crossref]

J. Biophoton. (1)

K. Haase, N. Kröger-Lui, A. Pucci, A. Schönhals, and W. Petrich, “Real-time mid-infrared imaging of living microorganisms,” J. Biophoton. 9, 61–66 (2016).
[Crossref]

J. Mod. Opt. (2)

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency upconversion,” J. Mod. Opt. 51, 1433–1445 (2004).
[Crossref]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

B. Klein, E. Plis, M. Kutty, N. Gautam, A. Albrecht, S. Myers, and S. Krishna, “Varshni parameters for InAs/GaSb strained layer superlattice infrared photodetectors,” J. Phys. D 44, 075102 (2011).
[Crossref]

Nat. Commun. (1)

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Nat. Photonics (2)

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[Crossref]

D. E. Chang, V. Vuletic, and M. D. Lukin, “Quantum nonlinear optics—photon by photon,” Nat. Photonics 8, 685–694 (2014).
[Crossref]

Opt. Commun. (1)

Z. Jaroszewicz and A. Koodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102, 391–396 (1993).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Philos. Trans. R. Soc. London A (1)

M. Ritsch-Marte, “Orbital angular momentum light in microscopy,” Philos. Trans. R. Soc. London A 375, 20150437 (2017).
[Crossref]

Phys. Rev. Lett. (1)

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref]

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]

Proc. IEEE (1)

R. Hudson and J. Hudson, “The military applications of remote sensing by infrared,” Proc. IEEE 63, 104–128 (1975).
[Crossref]

Proc. R. Soc. London B (1)

D. Marr and E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London B 207, 187–217 (1980).
[Crossref]

Sci. Rep. (1)

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref]

Signal Process. (1)

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[Crossref]

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Supplementary Material (2)

NameDescription
» Visualization 1       up-conversion image of a running dog
» Visualization 2       spiral phase contrast imaging of a running dog in the nonlinear optics

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Figures (5)

Fig. 1.
Fig. 1. Schematic illustration of spiral phase contrast imaging. (a) The generic case in linear optics. (b) Our case in nonlinear optics.
Fig. 2.
Fig. 2. Schematic overview of the experimental setup to implement nonlinear SPC imaging in type-II SHG (see text for details). Inset shows a typical holographic grating to generate a phase flower object.
Fig. 3.
Fig. 3. Experimental results of nonlinear spiral phase contrast imaging. (a) Input binary intensity object of a flower illuminated by 1064 nm infrared light, (b) bright field image, and (c) spiral phase contrast image borne by 532 nm green light. (d) Input phase object of the letters “optica,” where the phase jump on the edge of each letter is π. (e) Bright field image. (f) Spiral phase contrast image.
Fig. 4.
Fig. 4. Experimental observations for a running dog cartoon. Left panel: Selected typical frames of purely phased running dog illuminated by invisible 1064 nm fundamental wave. Middle panel: Bright field images recorded with the SHG visible signals. Right panel: Spiral phase contrast images (see Visualization 1 and Visualization 2).
Fig. 5.
Fig. 5. Experimental observation of contrast enhancement for a pure phase clover of different phase steps: (a1) π/3, (a2) 2π/3, and (a3) π. (b1)–(b3) bright field images. (c1)–(c3) phase contrast images. (d1)–(d3) Intensity curves corresponding to the red dashed lines across the upper leaf without and with a spatial filter.

Equations (5)

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E˜f(ρ,φ)=E˜in(ρ,φ)F(ρ,φ).
Eout(r,ϕ)=Ein(r,ϕ)*F[F(ρ,φ)],
F[circ(ρR)exp(iφ)]=πR2r[J0(kRrf)H1(kRrf)J1(kRrf)H0(kRrf)]exp(iϕ),
dE˜out(ρ,φ,λvis)dz=i2πdeffλvisnvisc×E˜in(ρ,φ,λinvis)F(ρ,φ,λinvis),
Eout(r,ϕ,λvis)Ein(r,ϕ,λinvis)*F[F(ρ,φ,λinvis)],

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