Abstract

We study the diffraction produced by a PT -symmetric volume Bragg grating that combines modulation of refractive index and gain/loss of the same periodicity with a quarter-period shift between them. Such a complex grating has a directional coupling between the different diffraction orders, which allows us to find an analytic solution for the first three orders of the full Maxwell equations without resorting to the paraxial approximation. This is important, because only with the full equations can the boundary conditions, allowing for reflections, be properly implemented. Using our solution we analyze the properties of such a grating in a wide variety of configurations.

© 2015 Optical Society of America

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References

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  1. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
    [Crossref]
  2. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13, 3068–3078 (2005).
    [Crossref] [PubMed]
  3. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29, 451–453 (2004).
    [Crossref] [PubMed]
  4. S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
    [Crossref]
  5. H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A 45, 135306 (2012).
    [Crossref]
  6. M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31, 3493–3502 (1998).
    [Crossref]
  7. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  8. M. Kulishov and B. Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20, 29319–29328 (2012).
    [Crossref]
  9. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014).
    [Crossref] [PubMed]
  10. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1976).
    [Crossref]
  11. T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
    [Crossref]

2014 (1)

2012 (2)

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

2005 (1)

2004 (1)

1998 (1)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31, 3493–3502 (1998).
[Crossref]

1996 (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
[Crossref]

1982 (1)

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[Crossref]

1976 (1)

Azaña, J.

Bélanger, N.

Berry, M. V.

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31, 3493–3502 (1998).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Christodoulides, D.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Feng, L.

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[Crossref]

Greenberg, M.

Jones, H. F.

H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A 45, 135306 (2012).
[Crossref]

Kong, J. A.

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Kress, B.

Kulishov, M.

Laniel, J. M.

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Longhi, S.

S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[Crossref]

Orenstein, M.

Plant, D. V.

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
[Crossref]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Wang, Y.

Yang, S.

Yin, X.

Zhang, P.

Zhang, X.

Zhu, H.

Zhu, X.

Appl. Phys. B (1)

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. A (2)

S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A 45, 135306 (2012).
[Crossref]

J. Phys. Math. Gen. (1)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31, 3493–3502 (1998).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. E (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
[Crossref]

Phys. Rev. Lett. (1)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 (a) Planar slanted grating of the index (black color fringes) and gain/loss (red color fringes) modulation and (b) non-slanted grating
Fig. 2
Fig. 2 Two-mode solution for incidence at the first Bragg angle θ′B: transmission and reflection coefficients as functions of the grating strength for different values of ξ = ξ 1 / ξ 2, where ξ = 1 (magenta, dot-dashed) corresponds to a traditional index grating and ξ = 0 (red, solid) describes an ideal balanced PT -symmetric grating. The other values shown are ξ = 0.5 (green, dashed) and ξ = 0.25 (blue, dotted).The remaining parameters are ε1 = 1, ε2 = 2.4, ε3 = 1, d = 8 μm, Λ= 0.5 μm, λ0=0.6328 μm.
Fig. 3
Fig. 3 Two-mode solution for incidence at the first Bragg angle θ′B: transmission and reflection coefficients as functions of the grating strength for the filled-space configuration ε1 = ε2 = ε3 = 2.4. The remaining quantities are the same as in Fig. 2.
Fig. 4
Fig. 4 Filled-space configuration (ε1 = ε2 = ε3 = 2.4) : diffraction efficiency in (a) first and (b) second orders in transmission as a function of the internal angle of incidence for Λ= 0.5 μm (red, solid), Λ= 0.75 μm (blue, dashed), Λ= 1.0 μm (magenta, dot-dashed). The other parameters are d = 8 μm, λ0=0.633 μm.
Fig. 5
Fig. 5 Symmetric vs. filled-space configuration: diffraction efficiency in transmission ((a), (c), (e)) and reflection ((b), (d), (f)) as a function of the internal angle of incidence θ for ε1 = ε3 = 1, ε2 = 2.4 (blue, dashed), ε1 = ε2 = ε3 = 2.4 (red, solid). The other parameters are d = 8μm, Λ= 0.75 μm and λ0=0.633 μm.
Fig. 6
Fig. 6 Asymmetric configurations when the input light comes from (a) the substrate side and (b) the air side.
Fig. 7
Fig. 7 Light incident from substrate: transmission (red, solid) and reflection (blue, dashed) angular spectra for zeroth order (a) and (b), first-order (c) and (d) and second-order light (e) and (f) as functions of the internal angle of incidence. In the left-hand panels ε1 = ε2 = 2.4, while in the right-hand panels ε1 = 2.0, ε2 = 2.4. The remaining parameters are ε3 = 1, d = 8 μm; Λ = 0.75 μm; λ =0.633 μm and ξ = 0.04.
Fig. 8
Fig. 8 Light incident from the air: transmission (red, solid) and reflection (blue, dashed) angular spectra for first-order (a) and (b) and second-order diffracted light (c) and (d) as functions of the internal angle of incidence. In the left-hand panels ε2 = ε3 = 2.4, while in the right-hand panels ε2 = 2.4, ε3 = 2.0. The remaining parameters are ε1 = 1, d = 8 μm; Λ = 0.75 μm; λ =0.633 μm and ξ = 0.04.
Fig. 9
Fig. 9 Reflective set-up (ε1 = 1.0, ε2 = 2.4, ε3 = −54.7+21.83 j): transmission (red, solid) and reflection (blue, dashed) angular spectra for zeroth-order (a) and first-order diffracted light (c) as functions of the internal angle of incidence for d = 8 μm, Λ = 0.75 μm, λ0=0.633 μm, ξ = 0.004.
Fig. 10
Fig. 10 Prominent modes of the PT -symmetric grating for incidence at different angles and from different sides: (a) from the left near the first Bragg angle θB; (b) from the left near −θB; (c) from the right near θB; (d) from the right near −θB.

Equations (63)

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ε ( x , z ) = ε 2 + Δ ε cos ( K ( x sin φ + z cos φ ) )
σ ( x , z ) = Δ σ sin ( K ( x sin φ + z cos φ ) )
k 2 ( x , z ) = k 0 2 ε ( x , z ) j ω μ σ ( x , z ) ,
k 2 ( x , z ) = k 2 2 + 2 k 2 ( κ exp ( j K . r ) + κ + exp ( j K . r ) ) ,
κ ± = 1 4 ( ε 2 ) 1 2 ( k 0 Δ ε ± c μ Δ σ )
E 1 ( x , z ) = exp [ j k 1 ( x sin θ + z cos θ ) ] + + m = R m exp [ j { ( k 2 sin θ m K sin φ ) x ( k 1 2 ( k 2 sin θ m K sin φ ) 2 ) 1 2 z } ]
E 3 ( x , z ) = m = T m exp [ j { ( k 2 sin θ m K sin φ ) x + ( k 3 2 ( k 2 sin θ m K sin φ ) 2 ) 1 2 ( z d ) } ]
E 2 ( x , z ) = m = S m ( z ) exp [ j ( k 2 sin θ m K sin φ ) x ] ,
2 E 2 ( x , z ) + k 0 2 ε ( x , z ) E 2 ( x , z ) = 0
d 2 S m ( z ) d z 2 + [ k 2 2 ( k 2 sin θ m K sin φ ) 2 ] S m ( z ) + + 2 k 2 [ κ e j K z cos φ S m + 1 ( z ) + κ + e j K z cos φ S m 1 ( z ) ] = 0
d 2 S 0 ( z ) d z 2 + k 2 2 cos 2 θ S 0 ( z ) + 2 k 2 κ S 1 ( z ) = 0 d 2 S 1 ( z ) d z 2 + [ k 2 2 ( k 2 sin θ K ) 2 ] S 1 ( z ) + 2 k 2 κ + S 0 ( z ) = 0
d 2 S 0 ( u ) d u 2 + cos 2 θ B S 0 ( u ) + ξ 1 S 1 ( u ) = 0 d 2 S 1 ( u ) d u 2 + cos 2 θ B S 1 ( u ) + ξ 2 S 0 ( u ) = 0
ξ 1 = 2 κ / k 2 and ξ 2 = 2 κ + / k 2
V 0 = S 0 + ξ 1 / ξ 2 S 1 and V 1 = S 0 ξ 1 / ξ 2 S 1 ,
d 2 V 0 ( u ) d u 2 + ρ 1 2 V 0 ( u ) = 0 d 2 V 1 ( u ) d u 2 + ρ 2 2 V 1 ( u ) = 0
ρ 1 = ( cos 2 θ B + ξ 1 ξ 1 ) 1 2 and ρ 2 = ( cos 2 θ B ξ 1 ξ 1 ) 1 2 .
V 0 ( u ) = A e j ρ 1 u + B e j ρ 1 u V 1 ( u ) = C e j ρ 2 u + D e j ρ 2 u
1 + R 0 = S 0 ( 0 ) , R 1 = S 1 ( 0 )
k 2 S 0 ( 0 ) = j ( k 1 2 k 2 2 sin 2 θ ) 1 2 ( R 0 1 ) , k 2 S 1 ( 0 ) = j [ k 1 2 ( k 2 sin θ K ) 2 ] 1 2 R 1
T 0 = S 0 ( d ) , T 1 = S 1 ( d )
k 2 S 0 ( d ) = j ( k 3 2 k 2 2 sin 2 θ ) 1 2 T 0 , k 2 S 1 ( d ) = j [ k 1 2 ( k 2 sin θ K ) 2 ] 1 2 T 1
R 0 + ξ R 1 + 1 = A + B
α B ( R 0 + ξ R 1 1 ) = ρ 1 ( A B )
R 0 ξ R 1 + 1 = C + D
α B ( R 0 ξ R 1 1 ) = ρ 2 ( C + D )
T 0 + ξ T 1 = A e j ρ 1 u d + B e j ρ 1 u d
T 0 ξ T 1 = C e j ρ 2 u d + D e j ρ 2 u d
β B ( T 0 + ξ T 1 ) = ρ 1 ( A e j ρ 1 u d B e j ρ 1 u d )
β B ( T 0 ξ T 1 ) = ρ 2 ( C e j ρ 2 u d D e j ρ 2 u d )
R 0 = 1 2 ( F ( ρ 1 ) + F ( ρ 2 ) ) , R 1 = 1 2 ξ ( F ( ρ 1 ) F ( ρ 2 ) )
F ( ρ m ) = ( ρ m β B ) ( α B + ρ m ) e j ρ m u d + ( ρ m + β B ) ( α B ρ m ) e j ρ m u d ( ρ m β B ) ( α B ρ m ) e j ρ m u d + ( ρ m + β B ) ( α B + ρ m ) e j ρ m u d
T 0 = 1 2 ( G ( ρ 1 ) + G ( ρ 2 ) ) , T 1 = 1 2 ξ ( G ( ρ 1 ) G ( ρ 2 ) ) ,
G ( ρ m ) = 4 ρ m α B ( ρ m β B ) ( α B ρ m ) e j ρ m u d + ( ρ m + β B ) ( α B + ρ m ) e j ρ m u d
d 2 S 0 ( u ) d u 2 + cos 2 θ S 0 ( u ) = 0
d 2 S 1 ( u ) d u 2 + [ 1 ( 2 sin θ B sin θ ) 2 ] S 1 ( u ) + ξ 2 S 0 ( u ) = 0
S 0 ( u ) = A 0 e j u cos θ + B 0 e j u cos θ
T 0 = 4 α 0 cos θ ( α 0 + cos θ ) ( β 0 + cos θ ) e j u d cos θ ( α 0 cos θ ) ( β 0 cos θ ) e j u d cos θ
R 0 = ( α 0 cos θ ) ( β 0 + cos θ ) e j u d cos θ + ( α 0 + cos θ ) ( β 0 cos θ ) e j u d cos θ ( α 0 + cos θ ) ( β 0 + cos θ ) e j u d cos θ ( α 0 cos θ ) ( β 0 cos θ ) e j u d cos θ
A 0 = T 0 2 ( cos θ β 0 cos θ ) e j u d cos θ B 0 = T 0 2 ( cos θ + β 0 cos θ ) e j u d cos θ
( S 1 ( u ) ) H = C 1 e j η 1 u + D 1 e j η 1 u
( S 1 ( u ) ) I = A 1 e j u cos θ + B 1 e j u cos θ
x 1 = ξ 2 4 sin θ B ( sin θ B sin θ )
f ( a , b , c ) : = ( a + b ) ( a c ) ( 1 e j ( a b ) u d ) ( a b ) ( a + c ) ( 1 e j ( a + b ) u d )
g ( a , b , c ) : = ( a + b ) ( a c ) ( e j b u d e j a u d ) ( a b ) ( a + c ) ( e j b u d e j a u d )
h ( a , b , c ) : = ( a + b ) ( a + c ) e j a u d ( a b ) ( a c ) e j a u d
T 1 = 1 h ( η 1 , α 1 , β 1 ) [ f ( η 1 , η 0 , α 1 ) A 1 + f ( η 1 , η 0 , α 1 ) B 1 ]
R 1 = 1 h ( η 1 , α 1 , β 1 ) [ g ( η 1 , η 0 , β 1 ) A 1 + g ( η 1 , η 0 , β 1 ) B 1 ]
C 1 = 1 h ( η 1 , α 1 , β 1 ) { A 1 [ ( α 1 η 0 ) ( β 1 η 1 ) e j η 1 u d ( α 1 + η 1 ) ( β 1 + η 0 ) e j η 0 u d ] + B 1 [ ( α 1 + η 0 ) ( β 1 η 1 ) e j η 1 u d ( α 1 + η 1 ) ( β 1 η 0 ) e j η 0 u d ] }
D 1 = 1 h ( η 1 , α 1 , β 1 ) { A 1 [ ( α 1 η 0 ) ( β 1 + η 1 ) e j η 1 u d ( α 1 η 1 ) ( β 1 + η 0 ) e j η 0 u d ] + B 1 [ ( α 1 + η 0 ) ( β 1 + η 1 ) e j η 1 u d ( α 1 η 1 ) ( β 1 η 0 ) e j η 0 u d ] }
d 2 S 2 ( u ) d u 2 + [ 1 ( 4 sin θ B sin θ ) 2 ] S 2 ( u ) + ξ 2 S 1 ( u ) = 0
( S 2 ( u ) ) H = E 2 e j u η 2 + F 2 e j u η 2
( S 2 ( u ) ) I = C 2 e j u η 1 + D 2 e j u η 1 + A 2 e j u cos θ + B 2 e j u cos θ ,
x 2 = ξ 2 8 sin θ B ( 2 sin θ B sin θ ) , x 3 = ξ 2 4 sin θ B ( 3 sin θ B sin θ )
T 2 = 1 h ( η 2 , α 2 , β 2 ) [ f ( η 2 , η 0 , α 2 ) A 2 + f ( η 2 , η 0 , α 2 ) B 2 + f ( η 2 , η 1 , α 2 ) C 2 + f ( η 2 , η 1 , α 2 ) D 2 ]
R 2 = 1 h ( η 2 , α 2 , β 2 ) [ g ( η 2 , η 0 , α 2 ) A 2 + g ( η 2 , η 0 , α 2 ) B 2 + g ( η 2 , η 1 , α 2 ) C 2 + g ( η 2 , η 1 , α 2 ) D 2 ]
DER m = Re ( α m α 0 ) | R m | 2 DET m = Re ( β m α 0 ) | T m | 2
T 1 = x 1 ( η 1 + η 0 ) 2 η 1 ( e j η 1 u d e j η 0 u d ) = ξ 2 ( η 1 + cos θ ) ( e j η 1 u d e j η 0 u d ) 8 η 1 sin θ B ( sin θ B sin θ )
R 1 = x 1 η 1 η 0 2 η 1 ( 1 e j ( η 0 + η 1 ) u d ) = ξ 2 ( η 1 cos θ ) ( 1 e j ( η 0 + η 1 ) u d ) 8 η 1 sin θ B ( sin θ B sin θ )
T 1 = j ξ 2 u d 2 cos θ B e j u d cos θ B R 1 = j ξ 2 sin ( u d cos θ B ) 2 cos 2 θ B e j u d cos θ B
T 0 = ( 2 cos θ cos θ + β 0 ) e j u d cos θ
R 0 = ( cos θ β 0 cos θ + β 0 ) e 2 j u d cos θ ,
d 2 S 0 ( u ) d u 2 + cos 2 θ S 0 ( u ) = 0
d 2 S 1 ( u ) d u 2 + [ 1 ( 2 sin θ B + sin θ ) 2 ] S 1 ( u ) + ξ 1 S 0 ( u ) = 0 ,

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