Abstract

The effect of damping on the Goos–Hänchen (GH) shift from weakly absorbing anisotropic metamaterials is investigated. Explicit formulas of the GH shifts are derived and analyzed at three particular angles of incidence: critical angle, pseudo-Brewster angle, and grazing incidence, near which the reflection phases exhibit strong variations and large GH shifts are likely to occur. The damping in the anisotropic metamaterials may result in GH shifts not available in ordinary isotropic media. In particular, a larger GH shift can be associated with a larger rather than a smaller damping, and a small change of damping may even reverse the direction of the GH shift near the pseudo-Brewster angle. This feature is characterized by a parabolic relation determined by the complex components of the permittivity tensor. The GH shifts are also illustrated with the incidence of Gaussian beams based on Fourier integral formulation.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013 (3)

2012 (1)

2009 (2)

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[CrossRef]

W. Dong, L. Gao, and C.-W. Qiu, “Goos-Hänchen shift at the surface of chiral negative refractive media,” Prog. Electromagn. Res. 90, 255–268 (2009).
[CrossRef]

2008 (2)

W. Ding, L. Chen, and C.-H. Liang, “Numerical study of Goos-Hänchen shift on the surface of anisotropic left-handed materials,” Prog. Electromagn. Res. 2, 151–164 (2008).
[CrossRef]

M. Miri, A. Naqavi, A. Khavasi, K. Mehrany, S. Khorasani, and B. Rashidian, “Geometrical approach in physical understanding of the Goos-Hänchen shift in one- and two-dimensional periodic structures,” Opt. Lett. 33, 2940–2942 (2008).
[CrossRef]

2007 (2)

2006 (6)

2005 (4)

L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann-Raether configuration with left-handed materials,” Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

L.-G. Wang and S.-Y. Zhu, “Large positive and negative Goos-Hänchen shifts from a weakly absorbing left-handed slab,” J. Appl. Phys. 98, 043522 (2005).
[CrossRef]

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B.-I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hänchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30, 2936–2938 (2005).
[CrossRef]

2003 (3)

C.-F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef]

D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28, 1633–1635 (2003).
[CrossRef]

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003).
[CrossRef]

2002 (2)

2001 (1)

2000 (1)

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

1991 (1)

I. J. Singh and V. P. Nayyar, “Lateral displacement of a light beam at a ferrite interface,” J. Appl. Phys. 69, 7820–7824 (1991).
[CrossRef]

1986 (1)

1982 (1)

W. J. Wild and C. L. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

1978 (2)

B. A. Anicin, R. Fazlic, and M. Kopric, “Theoretical evidence for negative Goos-Hänchen shifts,” J. Phys. A 11, 1657–1662 (1978).
[CrossRef]

D. J. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. 43, 377–407 (1978).
[CrossRef]

1977 (1)

1976 (1)

1972 (1)

K. W. Chiu and J. J. Quinn, “On the Goos-Hänchen effect: a simple example of a time delay scattering process,” Am. J. Phys. 40, 1847–1851 (1972).
[CrossRef]

1971 (2)

1964 (1)

1949 (1)

F. Goos and H. Hänchen, “New measurement of the beam displacement at total reflection effect,” Ann. Phys. 440, 251–252 (1949).
[CrossRef]

1948 (1)

K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 437, 87–102 (1948).
[CrossRef]

1947 (1)

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

’t Hooft, G. W.

Aiello, A.

Anicin, B. A.

B. A. Anicin, R. Fazlic, and M. Kopric, “Theoretical evidence for negative Goos-Hänchen shifts,” J. Phys. A 11, 1657–1662 (1978).
[CrossRef]

Artmann, K.

K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 437, 87–102 (1948).
[CrossRef]

Belov, P. A.

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003).
[CrossRef]

Bergman, D. J.

D. J. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. 43, 377–407 (1978).
[CrossRef]

Berman, P. R.

P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E 66, 067603 (2002).
[CrossRef]

Bertoni, H. L.

Birman, J. L.

Bonnet, C.

Bretenaker, F.

Carniglia, C. K.

Chan, S. W.

Chang, P.-H.

R.-L. Chern and P.-H. Chang, “Negative refraction and backward wave in pseudochiral mediums: illustrations of Gaussian beams,” Opt. Express 21, 2657–2666 (2013).
[CrossRef]

R.-L. Chern and P.-H. Chang, “Negative refraction and backward wave in chiral mediums: illustrations of Gaussian beams,” J. Appl. Phys. 113, 153504 (2013).
[CrossRef]

Chauvat, D.

Chen, C. W.

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[CrossRef]

Chen, H.

Chen, J.

N.-H. Shen, J. Chen, Q.-Y. Wu, T. Lan, Y.-X. Fan, and H.-T. Wang, “Large lateral shift near pseudo-Brewster angle on reflection from a weakly absorbing double negative medium,” Opt. Express 14, 10574–10579 (2006).
[CrossRef]

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B.-I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hänchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Chen, L.

W. Ding, L. Chen, and C.-H. Liang, “Numerical study of Goos-Hänchen shift on the surface of anisotropic left-handed materials,” Prog. Electromagn. Res. 2, 151–164 (2008).
[CrossRef]

Chen, X.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B.-I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hänchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Cheng, Q.

Q. Cheng and T. J. Cui, “Lateral shifts of optical beams on the interface of anisotropic metamaterial,” J. Appl. Phys. 99, 066114 (2006).
[CrossRef]

Chern, R.-L.

R.-L. Chern and P.-H. Chang, “Negative refraction and backward wave in chiral mediums: illustrations of Gaussian beams,” J. Appl. Phys. 113, 153504 (2013).
[CrossRef]

R.-L. Chern and P.-H. Chang, “Negative refraction and backward wave in pseudochiral mediums: illustrations of Gaussian beams,” Opt. Express 21, 2657–2666 (2013).
[CrossRef]

Chiang, H. P.

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[CrossRef]

Chiu, K. W.

K. W. Chiu and J. J. Quinn, “On the Goos-Hänchen effect: a simple example of a time delay scattering process,” Am. J. Phys. 40, 1847–1851 (1972).
[CrossRef]

Cui, T. J.

Q. Cheng and T. J. Cui, “Lateral shifts of optical beams on the interface of anisotropic metamaterial,” J. Appl. Phys. 99, 066114 (2006).
[CrossRef]

Ding, W.

W. Ding, L. Chen, and C.-H. Liang, “Numerical study of Goos-Hänchen shift on the surface of anisotropic left-handed materials,” Prog. Electromagn. Res. 2, 151–164 (2008).
[CrossRef]

Dong, W.

W. Dong, L. Gao, and C.-W. Qiu, “Goos-Hänchen shift at the surface of chiral negative refractive media,” Prog. Electromagn. Res. 90, 255–268 (2009).
[CrossRef]

Dutriaux, L.

Eliel, E. R.

Emile, O.

Fan, Y.-X.

Fang, A.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[CrossRef]

Fazlic, R.

B. A. Anicin, R. Fazlic, and M. Kopric, “Theoretical evidence for negative Goos-Hänchen shifts,” J. Phys. A 11, 1657–1662 (1978).
[CrossRef]

Felbacq, D.

Gao, L.

Y. Huang, B. Zhao, and L. Gao, “Goos-Hänchen shift of the reflected wave through an anisotropic metamaterial containing metal/dielectric nanocomposites,” J. Opt. Soc. Am. A 29, 1436–1444 (2012).
[CrossRef]

W. Dong, L. Gao, and C.-W. Qiu, “Goos-Hänchen shift at the surface of chiral negative refractive media,” Prog. Electromagn. Res. 90, 255–268 (2009).
[CrossRef]

Giles, C. L.

W. J. Wild and C. L. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

Goos, F.

F. Goos and H. Hänchen, “New measurement of the beam displacement at total reflection effect,” Ann. Phys. 440, 251–252 (1949).
[CrossRef]

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Grzegorczyk, T. M.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B.-I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hänchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, “New measurement of the beam displacement at total reflection effect,” Ann. Phys. 440, 251–252 (1949).
[CrossRef]

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

He, J.

He, S.

Horowitz, B. R.

Huang, Y.

Khavasi, A.

Khorasani, S.

Kong, J. A.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B.-I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hänchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Kopric, M.

B. A. Anicin, R. Fazlic, and M. Kopric, “Theoretical evidence for negative Goos-Hänchen shifts,” J. Phys. A 11, 1657–1662 (1978).
[CrossRef]

Koschny, T.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[CrossRef]

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Lai, H. M.

H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27, 680–682 (2002).
[CrossRef]

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Lan, T.

Le Floch, A.

Leung, P. T.

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[CrossRef]

Li, C.-F.

C.-F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef]

Liang, C.-H.

W. Ding, L. Chen, and C.-H. Liang, “Numerical study of Goos-Hänchen shift on the surface of anisotropic left-handed materials,” Prog. Electromagn. Res. 2, 151–164 (2008).
[CrossRef]

Liu, F.

Liu, N.-H.

Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

McGuirk, M.

Mehrany, K.

Merano, M.

Miri, M.

Moreau, A.

Naqavi, A.

Nayyar, V. P.

I. J. Singh and V. P. Nayyar, “Lateral displacement of a light beam at a ferrite interface,” J. Appl. Phys. 69, 7820–7824 (1991).
[CrossRef]

Pacheco, J.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B.-I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hänchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

Puri, A.

Qiu, C.-W.

W. Dong, L. Gao, and C.-W. Qiu, “Goos-Hänchen shift at the surface of chiral negative refractive media,” Prog. Electromagn. Res. 90, 255–268 (2009).
[CrossRef]

Quinn, J. J.

K. W. Chiu and J. J. Quinn, “On the Goos-Hänchen effect: a simple example of a time delay scattering process,” Am. J. Phys. 40, 1847–1851 (1972).
[CrossRef]

Rashidian, B.

Renard, R. H.

Shen, N.-H.

Singh, I. J.

I. J. Singh and V. P. Nayyar, “Lateral displacement of a light beam at a ferrite interface,” J. Appl. Phys. 69, 7820–7824 (1991).
[CrossRef]

Smaâli, R.

Song, G.

Soukoulis, C. M.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[CrossRef]

Tamir, T.

Tsai, D. P.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

van Exter, M. P.

Wang, H.-T.

Wang, L.-G.

Wild, W. J.

W. J. Wild and C. L. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

Woerdman, J. P.

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

Wu, B.-I.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B.-I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hänchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Wu, Q.-Y.

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Xu, J.

Yang, Y.

Yi, J.

Zhao, B.

Zhu, S.-Y.

Am. J. Phys. (1)

K. W. Chiu and J. J. Quinn, “On the Goos-Hänchen effect: a simple example of a time delay scattering process,” Am. J. Phys. 40, 1847–1851 (1972).
[CrossRef]

Ann. Phys. (3)

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

F. Goos and H. Hänchen, “New measurement of the beam displacement at total reflection effect,” Ann. Phys. 440, 251–252 (1949).
[CrossRef]

K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 437, 87–102 (1948).
[CrossRef]

Appl. Phys. Lett. (1)

L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann-Raether configuration with left-handed materials,” Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

J. Appl. Phys. (4)

L.-G. Wang and S.-Y. Zhu, “Large positive and negative Goos-Hänchen shifts from a weakly absorbing left-handed slab,” J. Appl. Phys. 98, 043522 (2005).
[CrossRef]

Q. Cheng and T. J. Cui, “Lateral shifts of optical beams on the interface of anisotropic metamaterial,” J. Appl. Phys. 99, 066114 (2006).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

(a) Parallel and (b) perpendicular components of the effective permittivity for the anisotropic metamaterial composed of dielectric metal layers with f=0.75, ε1=3+0.01i, and Γ=0.05.

Fig. 2.
Fig. 2.

(a) Reflection amplitude and phase and (b) estimated GH shift as a function of the angle of incidence for a weakly absorbing anisotropic metamaterial with εx=0.5+0.005i and εz=0.5+0.01i.

Fig. 3.
Fig. 3.

(a) Reflection amplitude and phase and (b) estimated GH shift as a function of the angle of incidence for a weakly absorbing anisotropic metamaterial with εx=0.5+0.03i and εz=0.5+0.01i.

Fig. 4.
Fig. 4.

(a) Allowed region of GH shift associated with the pseudo-Brewster angle as a function of εx and εz for εx=εz. (b) Parabolic relations in the εzεx plane [Eq. (9)] that separate the positive and negative GH shifts for different ratios of εx/εz.

Fig. 5.
Fig. 5.

Magnetic field (Hy) of TM-polarized Gaussian beams incident from vacuum near the pseudo-Brewster angle onto the weakly absorbing anisotropic metamaterial with (a) εx=0.5+0.005i and εz=0.5+0.01i and (b) εx=0.5+0.03i and εz=0.5+0.01i. White dashed lines are light rays predicted by geometric optics. White solid lines are the reflected beam centers.

Fig. 6.
Fig. 6.

Estimated GH shift as a function of the angle of incidence for moderately absorbing anisotropic metamaterials with (a) εx=0.5+0.1i, εz=0.5+0.1i and (b) εx=0.5+0.2i, εz=0.5+0.1i.

Fig. 7.
Fig. 7.

(a) Reflection amplitude and phase and (b) estimated GH shift as a function of the angle of incidence for a weakly absorbing anisotropic metamaterial with εx=10+0.01i and εz=1.1+0.01i.

Fig. 8.
Fig. 8.

(a) Reflection amplitude and phase and (b) estimated GH shift as a function of the angle of incidence for a weakly absorbing anisotropic metamaterial with εx=10+0.01i and εz=0.9+0.001i.

Equations (32)

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r=εxcosθεxεxεzsin2θεxcosθ+εxεxεzsin2θ.
S=λ2πdϕdθ,
dϕdθ=2εx(εz1)sinθεxεxεzsin2θ[εz(εxcos2θ1)+sin2θ],
θc=arcsin(εz),
Scλπ2εxεz.
Scλπ2εε,
θb=arctan(εz(εx1)εz1),
Sb=λεz(εz1)(εx1)|εxεz1|π[εxεz(εz1)εz(εx1)].
εxεx1=εzεz(εz1),
|r|2[εxεz(εz1)εz(εx1)]216(εx)2(εz)2(εz1)2.
Sbλε(ε+1)πε,
|r|2(ε)2(ε1)216(ε)4.
Sgλεxπεxεzεx.
Sgλεπ1ε,
f(x,z)=ψ(kx)eikxx+ikzzdkx,
ψ(kx)=w02cosθπexp[w024cos2θ(kxk0sinθ)2ikxx0+ikzh],
Hi=hiψ(kx)eikxx+ik0zzdkx,
Ei=η0eiψ(kx)eikxx+ik0zzdkx,
Hr=hrr(kx)ψ(kx)eikxxik0zzdkx,
Er=η0err(kx)ψ(kx)eikxxik0zzdkx,
Ht=htt(kx)ψ(kx)eikxx+ikzzdkx,
Et=η0ett(kx)ψ(kx)eikxx+ikzzdkx,
r=k0zε0kzεxk0zε0+kzεx,t=2k0zε0k0zε0+kzεx.
f(x,z)=IFT[FT[f0(x)]ϕ(kx)eiqzz],
f0(x)=exp[(xx0)2cos2θw02+ik0sinθ(xx0)+ikzh]
FT[f(x)]=F(kx)12πf(x)eikxxdx,
IFT[F(kx)]=f(x)12πF(kx)eikxxdkx.
ε=(1f)ε1+fε2,
ε=[(1f)ε11+fε21]1,
ε(1f)ε1+f(11Ω2)+i[(1f)ε1+fΓΩ3],
ε(1fε1fΩ21Ω2)1+i[(1f)ε1(ε1)2+fΓΩ(1Ω2)2](1fε1fΩ21Ω2)2,
εx=εxεz[(εz)2εz]+1.

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