Abstract

Investigation of physics on two-dimensional curved surface has significant meaning in study of general relativity, inasmuch as its realizability in experimental analogy and verification of faint gravitational effects in laboratory. Several phenomena about dynamics of particles and electromagnetic waves have been explored on curved surfaces. Here we consider Wolf effect, a phenomenon of spectral shift due to the fluctuating nature of light fields, on an arbitrary surface of revolution (SOR). The general expression of the propagation of partially coherent beams propagating on arbitrary SOR is derived and the corresponding evolution of light spectrum is also obtained. We investigate the extra influence of surface topology on spectral shift by defining two quantities, effective propagation distance and effective transverse distance, and compare them with longitudinal and transverse proper lengths. Spectral shift is accelerated when the defined effective quantities are greater than real proper lengths, and vice versa. We also employ some typical SORs, cylindrical surfaces, conical surfaces, SORs generated by power function and periodic peanut-shell shapes, as examples to provide concrete analyses. This work generalizes the research of Wolf effect to arbitrary SORs, and provides a universal method for analyzing properties of propagation compared with that in flat space for any SOR whose topology is known.

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

Full Article  |  PDF Article
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References

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  5. U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A. 60, 4301–4312 (1999).
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    [Crossref]
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  33. A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev.X 8, 011001 (2018).
  34. E. Lustig, M. I. Cohen, R. Bekenstein, G. Harari, M. A. Bandres, and M. Segev, “Curved-space topological phases in photonic lattices,” Phys. Rev. A 96, 041804(R) (2017).
    [Crossref]
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    [Crossref]
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    [Crossref]
  39. V. H. Schultheiss, S. Batz, and U. Peschel, “Hanbury Brown and Twiss measurements in curved space,” Nat. Photon. 10, 106–110 (2016).
    [Crossref]
  40. C. Xu, A. Abbas, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, “Wolf effect of partially coherent light fields in two-dimensional curved space,” Phys. Rev. A 97, 063827 (2018).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  43. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Reports on Progress in Physics 59, 771–818 (1996).
    [Crossref]
  44. T. A. Leskova, A. A. Maradudin, A. V. Shchegrov, and E.R. Méndez, “Spectral changes of light scattered from a bounded medium with a random surface,” Phys. Rev. Lett. 79, 1010–1013 (1997).
    [Crossref]
  45. M. F. Bocko, D. H. Douglass, and R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
    [Crossref] [PubMed]
  46. R. W. Schoonover, R. Lavarello, M. L. Oelze, and P.S. Carney, “Observation of generalized Wolf shifts in short pulse spectroscopy,” Appl. Phys. Lett. 98, 251107 (2011).
    [Crossref]
  47. R. Zhu, S. Sridharan, K. Tangella, A. Balla, and G. Popescu, “Correlation-induced spectral changes in tissues,” Opt. Lett. 36, 4209–4211 (2011).
    [Crossref] [PubMed]
  48. D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483–3485 (2007).
    [Crossref] [PubMed]
  49. G. M. Morris and D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
    [Crossref]
  50. D. Faklis and G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
    [Crossref] [PubMed]
  51. J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
    [Crossref] [PubMed]
  52. H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A. 3, 296–299 (2001).
    [Crossref]
  53. S. Anand, B. K. Yadav, and H. C. Kandpal, “Experimental study of the phenomenon of 1 × n spectral switch due to diffraction of partially coherent light,” J. Opt. Soc. Am. A. 19, 2223–2228 (2002).
    [Crossref]

2018 (2)

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev.X 8, 011001 (2018).

C. Xu, A. Abbas, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, “Wolf effect of partially coherent light fields in two-dimensional curved space,” Phys. Rev. A 97, 063827 (2018).
[Crossref]

2017 (3)

E. Lustig, M. I. Cohen, R. Bekenstein, G. Harari, M. A. Bandres, and M. Segev, “Curved-space topological phases in photonic lattices,” Phys. Rev. A 96, 041804(R) (2017).
[Crossref]

K.-B. Hong, C.-Y. Lin, T.-C. Chang, W.-H. Liang, Y.-Y. Lai, C.-M. Wu, Y.-L. Chuang, T.-C. Lu, C. Conti, and R.-K. Lee, “Lasing on nonlinear localized waves in curved geometry,” Opt. Exp. 25, 29068–29077 (2017).
[Crossref]

L. C. B. da Silva, C. C. Bastos, and F. G. Ribeiro, “Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential,” Ann. Phys. 379, 13–33 (2017).
[Crossref]

2016 (2)

C. Conti, “Localization and shock waves in curved manifolds,” Sci. Bull. 61, 570–575 (2016).
[Crossref]

V. H. Schultheiss, S. Batz, and U. Peschel, “Hanbury Brown and Twiss measurements in curved space,” Nat. Photon. 10, 106–110 (2016).
[Crossref]

2015 (2)

R. Spittel, P. Uebel, H. Bartelt, and M. A. Schmidt, “Curvature-induced geometric momenta: the origin of waveguide dispersion of surface plasmons on metallic wires,” Opt. Exp. 23, 12174–12188 (2015).
[Crossref]

E. O. Silva, S. C. Ulhoa, F. M. Andrade, C. Filgueiras, and R. G. G. Amorim, “Quantum motion of a point particle in the presence of the Aharonov–Bohm potential in curved space,” Ann. Phys. 362, 739–751 (2015).
[Crossref]

2014 (1)

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wave packets in curved space,” Phys. Rev. X 4, 011038 (2014).

2013 (1)

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nature Photon. 7, 902–906 (2013).
[Crossref]

2012 (1)

J. Onoe, T. Ito, H. Shima, H. Yoshioka, and S. Kimura, “Observation of Riemannian geometric effects on electric states,” Europhys. Lett. 98, 27001 (2012).
[Crossref]

2011 (2)

R. W. Schoonover, R. Lavarello, M. L. Oelze, and P.S. Carney, “Observation of generalized Wolf shifts in short pulse spectroscopy,” Appl. Phys. Lett. 98, 251107 (2011).
[Crossref]

R. Zhu, S. Sridharan, K. Tangella, A. Balla, and G. Popescu, “Correlation-induced spectral changes in tissues,” Opt. Lett. 36, 4209–4211 (2011).
[Crossref] [PubMed]

2010 (3)

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

S. Batz and U. Peschel, “Solitons in curved space of constant curvature,” Phys. Rev. A 81, 053806 (2010).
[Crossref]

G. D. Valle and S. Longhi, “Geometric potential for plasmon polaritons on curved surfaces,” J. Phys. B: At. Mol. Opt. Phys. 43, 051002 (2010).
[Crossref]

2009 (4)

H. Shima, H. Yoshioka, and J. Onoe, “Geometry–driven shift in the Tomonaga–Luttinger exponent of deformed cylinders,” Phys. Rev. B 79, 201401(R) (2009).
[Crossref]

G. Cuoghi, G. Ferrari, and A. Bertoni, “Surface carrier transport in Y nanojunctions: Signatures of the geometric potential, ”Phys. Rev. B 79, 073410 (2009).
[Crossref]

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nature Phys. 5, 687–692 (2009).
[Crossref]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[Crossref]

2008 (4)

U. Leonhardt, T. G. Philbin, C. E. Kuklewicz, S. Robertson, S. Hill, and F. Konig, “Fibre-optical analogue of the event horizon,” Science 319, 1367–1370 (2008).
[Crossref]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[Crossref]

R. C. T. da Costa, “Quantum mechanics of a constrained particle,” Phys. Rev. A 78, 043821 (2008).

G. Ferrari and G. Cuoghi, “Schrödinger equation for a particle on a curved surface in an electric and magnetic field,” Phy. Rev. Lett. 100, 230403 (2008).
[Crossref]

2007 (2)

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett. 99, 183901 (2007).
[Crossref] [PubMed]

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483–3485 (2007).
[Crossref] [PubMed]

2005 (5)

R. Schützhold and W. G. Unruh, “Hawking radiation in an electromagnetic waveguide?” Phys. Rev. Lett. 95, 031301 (2005).
[Crossref] [PubMed]

N. Fujita and O. Terasaki, “Band structure of the P, D, and G surfaces,” Phys. Rev. B 72, 085459 (2005).
[Crossref]

M. Koshino and H. Aoki, “Electronic structure of an electron on the gyroid surface: A helical labyrinth,” Phys. Rev. B 71, 073405 (2005).
[Crossref]

J. Gravesen and M. Willatzen, “Eigenstates of Möbius nanostructures including curvature effects,” Phys. Rev. A 72, 032108 (2005).
[Crossref]

A. Marchi, S. Reggiani, and M. Rudan, “Coherent electron transport in bent cylindrical surfaces,” Phys. Rev. B 72, 035403 (2005).
[Crossref]

2003 (4)

J. Onoe, T. Nakayama, M. Aono, and T. Hara, “Structural and electrical properties of an electron-beam-irradiated c60 film,” Appl. Phys. Lett. 82, 595–597 (2003).
[Crossref]

M. Encinosa and L. Mott, “Curvature-induced toroidal bound states,” Phys. Rev. A 68, 014102 (2003).
[Crossref]

I. I. Smolyaninov, “Surface plasmon toy model of a rotating black hole,” New J. Phys. 5, 147 (2003).
[Crossref]

P. O. Fedichev and U. R. Fischer, “Gibbons–Hawking effect in the sonic de sitter space-time of an expanding Bose–Einstein–condensed gas,” Phys. Rev. Lett. 91240407 (2003).
[Crossref]

2002 (2)

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

S. Anand, B. K. Yadav, and H. C. Kandpal, “Experimental study of the phenomenon of 1 × n spectral switch due to diffraction of partially coherent light,” J. Opt. Soc. Am. A. 19, 2223–2228 (2002).
[Crossref]

2001 (2)

H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A. 3, 296–299 (2001).
[Crossref]

H. Aoki, M. Koshino, D. Takeda, H. Morise, and K. Kuroki, “Electronic structure of periodic curved surfaces: Topological band structure,” Phys. Rev. B 65, 035102 (2001).
[Crossref]

2000 (3)

U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys.Rev. Lett. 84, 822–825 (2000).
[Crossref]

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[Crossref] [PubMed]

G. Cantele, D. Ninno, and G. Iadonisi, “Topological surface states in deformed quantum wires,” Phys. Rev. B 61, 13730–13736 (2000).
[Crossref]

1999 (1)

U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A. 60, 4301–4312 (1999).
[Crossref]

1998 (1)

M. Encinosa and B. Etemadi, “Energy shifts resulting from surface curvature of quantum nanostructures,” Phys. Rev. A 58, 77–81 (1998).
[Crossref]

1997 (1)

T. A. Leskova, A. A. Maradudin, A. V. Shchegrov, and E.R. Méndez, “Spectral changes of light scattered from a bounded medium with a random surface,” Phys. Rev. Lett. 79, 1010–1013 (1997).
[Crossref]

1996 (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Reports on Progress in Physics 59, 771–818 (1996).
[Crossref]

1988 (1)

1987 (3)

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

M. F. Bocko, D. H. Douglass, and R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[Crossref] [PubMed]

G. M. Morris and D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[Crossref]

1986 (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref] [PubMed]

1957 (1)

B. S. DeWitt, “Dynamical theory in curved spaces. I. A review of the classical and quantum action principles,” Rev. Mod. Phys. 29, 377–397 (1957).
[Crossref]

Abbas, A.

C. Xu, A. Abbas, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, “Wolf effect of partially coherent light fields in two-dimensional curved space,” Phys. Rev. A 97, 063827 (2018).
[Crossref]

Amorim, R. G. G.

E. O. Silva, S. C. Ulhoa, F. M. Andrade, C. Filgueiras, and R. G. G. Amorim, “Quantum motion of a point particle in the presence of the Aharonov–Bohm potential in curved space,” Ann. Phys. 362, 739–751 (2015).
[Crossref]

Anand, S.

S. Anand, B. K. Yadav, and H. C. Kandpal, “Experimental study of the phenomenon of 1 × n spectral switch due to diffraction of partially coherent light,” J. Opt. Soc. Am. A. 19, 2223–2228 (2002).
[Crossref]

Andrade, F. M.

E. O. Silva, S. C. Ulhoa, F. M. Andrade, C. Filgueiras, and R. G. G. Amorim, “Quantum motion of a point particle in the presence of the Aharonov–Bohm potential in curved space,” Ann. Phys. 362, 739–751 (2015).
[Crossref]

Anglin, J. R.

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[Crossref] [PubMed]

Aoki, H.

M. Koshino and H. Aoki, “Electronic structure of an electron on the gyroid surface: A helical labyrinth,” Phys. Rev. B 71, 073405 (2005).
[Crossref]

H. Aoki, M. Koshino, D. Takeda, H. Morise, and K. Kuroki, “Electronic structure of periodic curved surfaces: Topological band structure,” Phys. Rev. B 65, 035102 (2001).
[Crossref]

Aono, M.

J. Onoe, T. Nakayama, M. Aono, and T. Hara, “Structural and electrical properties of an electron-beam-irradiated c60 film,” Appl. Phys. Lett. 82, 595–597 (2003).
[Crossref]

Balla, A.

Bandres, M. A.

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev.X 8, 011001 (2018).

E. Lustig, M. I. Cohen, R. Bekenstein, G. Harari, M. A. Bandres, and M. Segev, “Curved-space topological phases in photonic lattices,” Phys. Rev. A 96, 041804(R) (2017).
[Crossref]

Bartelt, H.

R. Spittel, P. Uebel, H. Bartelt, and M. A. Schmidt, “Curvature-induced geometric momenta: the origin of waveguide dispersion of surface plasmons on metallic wires,” Opt. Exp. 23, 12174–12188 (2015).
[Crossref]

Bastos, C. C.

L. C. B. da Silva, C. C. Bastos, and F. G. Ribeiro, “Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential,” Ann. Phys. 379, 13–33 (2017).
[Crossref]

Batz, S.

V. H. Schultheiss, S. Batz, and U. Peschel, “Hanbury Brown and Twiss measurements in curved space,” Nat. Photon. 10, 106–110 (2016).
[Crossref]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

S. Batz and U. Peschel, “Solitons in curved space of constant curvature,” Phys. Rev. A 81, 053806 (2010).
[Crossref]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[Crossref]

Bekenstein, R.

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev.X 8, 011001 (2018).

E. Lustig, M. I. Cohen, R. Bekenstein, G. Harari, M. A. Bandres, and M. Segev, “Curved-space topological phases in photonic lattices,” Phys. Rev. A 96, 041804(R) (2017).
[Crossref]

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wave packets in curved space,” Phys. Rev. X 4, 011038 (2014).

Bertoni, A.

G. Cuoghi, G. Ferrari, and A. Bertoni, “Surface carrier transport in Y nanojunctions: Signatures of the geometric potential, ”Phys. Rev. B 79, 073410 (2009).
[Crossref]

Bocko, M. F.

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A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev.X 8, 011001 (2018).

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T. A. Leskova, A. A. Maradudin, A. V. Shchegrov, and E.R. Méndez, “Spectral changes of light scattered from a bounded medium with a random surface,” Phys. Rev. Lett. 79, 1010–1013 (1997).
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C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nature Photon. 7, 902–906 (2013).
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J. Onoe, T. Ito, H. Shima, H. Yoshioka, and S. Kimura, “Observation of Riemannian geometric effects on electric states,” Europhys. Lett. 98, 27001 (2012).
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V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
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Figures (7)

Fig. 1
Fig. 1 Sketch of surface of revolution. SOR is generated by rotating generatrix (denoted by red solid line) in YZ plane with respect to Z axis, and r(t) is radius of rotation and θ is rotation angle. Different expression of r(t) will lead to different topology of SOR.
Fig. 2
Fig. 2 Behaviors of longitudinal (a) and transverse (b) spectral shifts on conical surfaces with different slopes at (a) x=0 and (b) t=1.3 m. On each surface, propagation starts from positions where ROR r0 = 1 m. Other parameters are ω0 = 2π × 500 THz, δ = 0.1ω0, σs = 1 mm, σg = 0.5 mm, and n0 = 1.51. Note that black dot lines denote the case in flat space.
Fig. 3
Fig. 3 (a) Effective propagation distance and (b) longitudinal spectral shift versus propagation distance t for five typical SORs generated by power function (p = −3, −1, 2, 4 and 8). For better comparison, situation of flat space is also plotted by black dot line. Other parameters are r0 = 1 m, ω0 = 2π × 500 THz, δ = 0.1ω0, σs = 1 mm, σg = 0.5 mm, n0 = 1.51.
Fig. 4
Fig. 4 Comparison about distributions of spectral shifts between three typical SORs generated by generalized power function. (a1)–(d1) Shape of such SORs with p = −1 (a1), 1 (b1), 2 (c1) together with corresponding generatrices. For better comparison, situation on cylindrical surface, which is proved to be equivalent to flat space, is also plotted (d1). (a2–d2) The corresponding longitudinal spectral shifts with respect to propagation distance t at different transverse coordinates x = 0, 1 mm, 1.5 mm, 2 mm, 2.5 mm on the corresponding surfaces. Both the on-axis longitudinal spectral shift and situation of transverse spectral shift can be analyzed from these figures. In (a2) – (d2), other parameters are the same as Fig. 3.
Fig. 5
Fig. 5 Schematic illustration of periodic peanut-shell shape. Curvilinear coordinates t and x are marked on surface by blue lines. Parameters a and b denote average radius of surface and roughness on surface, respectively.
Fig. 6
Fig. 6 (a) Effective propagation distance and (b) on-axis longitudinal spectral shift of PPSSs with three different initial phases φ0 = 0, 0.39π and π. For better comparison, situation in flat space is plotted in black dashed line. Other parameters are a = 2 m, b = 1 m, R = 1 m, ω0 = 2π × 500 THz, δ = 0.1ω0, σs = 1 mm, σg = 0.5 mm, n0 = 1.51.
Fig. 7
Fig. 7 Distribution of spectral shift Δω (THz) with different undulation on PPSSs with three different initial phases (a1–a3) φ0 = 0, (b1–b3) φ0 = 0.39π, (c1–c3) φ0 = π, Here a = 2 m, R = 1.9 m, and undulation parameter b are (a1–c1) b = 0.3 m, (a2–c2) b = 1 m, (a3–c3) b = 1.7 m. Other parameters are same as those in Fig. 6.

Equations (30)

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Δ g Φ + ( k 2 + H 2 K ) Φ = 0 ,
2 Φ ( ε , t ) t 2 + r ( t ) r ( t ) Φ ( ε , t ) t + r 0 2 r 2 ( t ) 2 Φ ( ε , t ) ε 2 + k 2 Φ ( ε , t ) = 0 ,
2 i k u ( t , ε ) t + V eff ( t ) u ( t , ε ) + r 0 2 r 2 ( t ) 2 u ( t , ε ) ε 2 = 0 ,
V eff ( t ) = 1 4 r 2 ( t ) [ d r ( t ) d t ] 2 1 2 r ( t ) d 2 r ( t ) d t 2
Ξ ( t ) = 0 t r 0 2 r 2 ( t ) d t .
2 i k v ( t , ε ) Ξ + 2 v ( t , ε ) ε 2 = 0 ,
h ( ε , ε , t ) = k r 0 i 2 π Ξ r ( t ) exp [ i k ( ε ε ) 2 2 Ξ ] exp [ i k t + i 2 k 0 t V eff ( t ) d t ] .
h f ( ε f ε f z f ) = k i 2 π z f exp ( i k z f ) exp [ i k 2 z f ( ε f 2 2 ε f ε f + ε f 2 ) ] ,
W out ( ε 1 , ε 2 , t , ω ) = Φ out ( ε 1 , t , ω ) Φ out * ( ε 2 , t , ω ) = W in ( ε 1 , ε 2 , 0 , ω ) h ( ε 1 , ε 1 , t ) h * ( ε 2 , ε 2 , t ) d ε 1 d ε 2 .
W in ( ε 1 , ε 2 , 0 , ω ) = S i ( ω ) exp ( ε 1 2 + ε 2 2 4 σ s 2 ) exp [ ( ε 1 ε 2 ) 2 2 σ g 2 ] ,
S out ( x , t , ω ) W out ( ε , ε , t , ω ) = S i ( ω ) Ω ( t , ω ) exp [ x 2 2 σ s 2 Ω 2 ( t , ω ) ] ,
Ω ( t , ω ) = α [ 1 + Ξ 2 / Z R 2 ( ω ) ] 1 / 2
S i ( ω 0 ) S i ( ω 0 ) + Ω ( t , ω 0 ) Ω ( t , ω 0 ) [ x 2 σ s 2 1 Ω 2 ( t , ω 0 ) 1 ] = 0 ,
x 2 α 2 σ s 2 Λ 2 ( Ξ , ω 0 ) = 1 S i ( ω 0 ) S i ( ω 0 ) Λ ( Ξ , ω 0 ) Λ ( Ξ , ω 0 ) ,
x f 2 σ s 2 Λ f 2 ( z f , ω 0 ) = 1 S i ( ω 0 ) S i ( ω 0 ) Λ f ( z f , ω 0 ) Λ f ( z f , ω 0 ) ,
ε = x / α .
Ξ 2 / Z R 2 ( ω 0 ) 1 .
1 ω 0 + S i ( ω 0 ) S i ( ω 0 ) = 0 .
Ξ = 0 t r 0 2 r 2 ( t ) d t = t ,
Case 1 : r ( t ) = r 0 + 1 1 + m 2 t for t [ 0 , )
Case 2 : r ( t ) = r 0 1 1 + m 2 t for t [ 0 , r 0 1 + m 2 )
Ξ = 0 t r 0 2 r 2 ( t ) d t = r 0 t r 0 ± 1 1 + m 2 t .
t = r 0 r 1 + p 2 Y 2 p 2 d Y G ( r ) ,
r ( t ) = a b cos ( t / R + φ 0 )
Ξ ( t ) = γ 1 [ m π + tan 1 β t tan 1 β 0 ] Q sin ( t / 2 R ) r ( t ) ,
Φ ± g ( t , ε ) = A ± g r 1 / 2 ( t ) u ± g ( t , ε ) exp [ ± i k ( t t 0 ) ] ,
h ± g ( ε , ε , t ) = k r 0 i 2 π Ξ ± r ( t ) exp [ i k ( ε ε ) 2 2 Ξ ± ] exp [ ± i k ( t t 0 ) ± i 2 k t 0 t V eff ( t ) d t ] ,
W in ( ρ 1 , ρ 2 , t 0 , ω ) = S i ( ω ) exp ( ρ 1 2 + ρ 2 2 4 σ s 2 ) exp [ ( ρ 1 ρ 2 ) 2 2 σ g 2 ] ,
S out , ± g ( x , t , ω ) = S i ( ω ) Ω ± g ( t , ω ) exp [ x 2 2 σ s 2 Ω ± g ( t , ω ) 2 ] ,
Ω ± g ( t , ω ) = α [ 1 + Ξ ± 2 r 0 4 Z R 2 ( ω ) r 0 4 ] 1 / 2 ,

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