Abstract

An alternative model to N-coupled wave theory of the spatially multiplexed finite thickness volume holographic reflection grating is developed from the parallel stacked mirrors (PSM) model in terms of N infinite arrays of parallel stacked mirrors each characterized by a different grating vector. A plane reference wave interacts with each of the N sets of stacked mirrors, producing N signal waves. First-order coupled partial differential equations describing the detailed process of Fresnel reflection within the grating are derived for the reference and N signal waves. These equations can be solved analytically at Bragg resonance where agreement with conventional N-coupled wave theory is exact. The new model is compared for the case of some simple multiplexed volume phase reflection gratings at and away from Bragg resonance with a rigorous coupled-wave solution of the Helmholtz equation. Good agreement is attained for even rather high values of index modulation. For lower modulations more characteristic of modern holographic materials, agreement appears extremely good at and around Bragg resonance, although differences inevitably appear in the higher-order diffractive sideband structure. The analytic model is extended to cover polychromatic spatially multiplexed volume phase gratings at Bragg resonance, where once again agreement with rigorous coupled-wave calculations is very good for index modulations typical for modern holographic gratings. Finally, the model is extended to cover the case of the lossless multicolor phase-reflection hologram, where analytic and graphical results are presented concerning diffractive efficiency.

© 2012 Optical Society of America

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References

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  1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  2. M. G. Moharam and T. K. Gaylord, “Rigorous coupled wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  3. E. N. Glytis and T. K. Gaylord, “Rigorous 3D coupled wave diffraction analysis of multiple superposed gratings in ansiotropic media,” Appl. Opt. 28, 2401–2421 (1989).
    [CrossRef]
  4. L. Solymar, “Two-dimensional N-coupled wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).
  5. D. Brotherton-Ratcliffe, “A treatment of the general volume holographic grating as an array of parallel stacked mirrors,” J. Mod. Opt. 59, 1113–1132 (2012).
    [CrossRef]
  6. F. Abeles, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).
  7. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1966), pp. 247–286.
  8. O. S. Heavens, “Optical properties of thin films,” Rep. Prog. Phys. 23, 1–65 (1960).
    [CrossRef]
  9. M. G. Moharam and T. K. Gaylord, “Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings,” J. Opt. Soc. Am. 72, 187–190 (1982).
    [CrossRef]
  10. M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. (Paris) Ser. II, 7, 291–384(1937).
  11. C. G. Darwin, “The theory of x-ray reflection,” Philos. Mag. 27, 315–333 (1914).
    [CrossRef]

2012 (1)

D. Brotherton-Ratcliffe, “A treatment of the general volume holographic grating as an array of parallel stacked mirrors,” J. Mod. Opt. 59, 1113–1132 (2012).
[CrossRef]

1989 (1)

1982 (1)

1981 (1)

1977 (1)

L. Solymar, “Two-dimensional N-coupled wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1960 (1)

O. S. Heavens, “Optical properties of thin films,” Rep. Prog. Phys. 23, 1–65 (1960).
[CrossRef]

1950 (1)

F. Abeles, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

1937 (1)

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. (Paris) Ser. II, 7, 291–384(1937).

1914 (1)

C. G. Darwin, “The theory of x-ray reflection,” Philos. Mag. 27, 315–333 (1914).
[CrossRef]

Abeles, F.

F. Abeles, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Brotherton-Ratcliffe, D.

D. Brotherton-Ratcliffe, “A treatment of the general volume holographic grating as an array of parallel stacked mirrors,” J. Mod. Opt. 59, 1113–1132 (2012).
[CrossRef]

Darwin, C. G.

C. G. Darwin, “The theory of x-ray reflection,” Philos. Mag. 27, 315–333 (1914).
[CrossRef]

Gaylord, T. K.

Glytis, E. N.

Heavens, O. S.

O. S. Heavens, “Optical properties of thin films,” Rep. Prog. Phys. 23, 1–65 (1960).
[CrossRef]

Jacobsson, R.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1966), pp. 247–286.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Moharam, M. G.

Rouard, M. P.

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. (Paris) Ser. II, 7, 291–384(1937).

Solymar, L.

L. Solymar, “Two-dimensional N-coupled wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).

Ann. Phys. (1)

F. Abeles, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Ann. Phys. (Paris) Ser. II (1)

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. (Paris) Ser. II, 7, 291–384(1937).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Mod. Opt. (1)

D. Brotherton-Ratcliffe, “A treatment of the general volume holographic grating as an array of parallel stacked mirrors,” J. Mod. Opt. 59, 1113–1132 (2012).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

L. Solymar, “Two-dimensional N-coupled wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).

Philos. Mag. (1)

C. G. Darwin, “The theory of x-ray reflection,” Philos. Mag. 27, 315–333 (1914).
[CrossRef]

Rep. Prog. Phys. (1)

O. S. Heavens, “Optical properties of thin films,” Rep. Prog. Phys. 23, 1–65 (1960).
[CrossRef]

Other (1)

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1966), pp. 247–286.

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Figures (8)

Fig. 1.
Fig. 1.

PSM model of the unslanted reflection grating. The grating is treated as an infinity of infinitesimal layers of infinitesimally differing refractive index. A plane reference wave is incident to the bottom boundary at angle θ c . Fresnel transmission and reflection at the infinitesimal layer boundaries synthesize the signal or image wave.

Fig. 2.
Fig. 2.

Geometry for rotating the unslanted reflection grating in the ( x , y ) system to a slanted grating in the primed system.

Fig. 3.
Fig. 3.

Example of a spatially multiplexed phase reflection grating. The grating, whose ( x , y ) index distribution is shown in (c), is formed by the sequential recording of the two simple gratings shown in (a) and (b). Each diagram shows a section of size 0.5 μm by 2 μm. Each simple grating has been recorded with a reference beam angle of Φ c = 30 ° and with a wavelength of 532 nm. One grating has a slope of ψ 1 = 20 ° and the other has a slope of ψ 2 = 20 ° . Note that the form of the multiplexed grating in (c) is fundamentally different from the characteristic linear shape of its component simple gratings of (a) and (b). Note also that identical index modulations for each of the two component gratings have been assumed in (c).

Fig. 4.
Fig. 4.

Diffractive efficiency, η σ versus normalized grating thickness, d / Λ as predicted by the PSM model and by an RCW calculation for the case of the twin multiplexed reflection grating of Fig. 3(c) at Bragg resonance. The grating is replayed using light of 532 nm at an incidence angle of Φ c = 30 ° . The grating index modulation of each of the component twin gratings has been taken to be n 1 = 0.3 in (a), n 1 = 0.2 in (b), n 1 = 0.1 in (c), and n 1 = 0.05 in (d). In each case the average index inside and outside the grating has been set to n 0 = 1.5 . The dotted lines indicate the S 1 and S 2 modes of the PSM model and the full lines indicate the modes of the rigorous coupled wave calculation. The most prominent RCW modes are the 01 and 10 modes, which correspond to the S 1 and S 2 modes in PSM. Note that Λ refers to the larger of the two grating periods, i.e., to that of the component grating shown in Fig. 3(b).

Fig. 5.
Fig. 5.

Diffractive efficiency, η σ , versus replay wavelength, λ c , as predicted by the PSM model and by a RCW calculation for the case of the twin multiplexed reflection grating of Fig. 3(c) at and away from Bragg resonance. The grating is illuminated at its recording angle of Φ c = 30 ° . The average index inside and outside the grating has been set to n 0 = 1.5 . In (a) a grating index modulation of n 1 = 0.03 for each component grating is assumed and a grating thickness of d = 7 μm is used. The dotted lines indicate the S 1 and S 2 modes of the PSM model and the full lines indicate the corresponding 10 and 01 modes of the rigorous coupled wave calculation. (b) and (c) show the PSM diffractive response expected when one or other of the component gratings is deleted from the diffractive element. (d) shows a similar case to (a) but with higher index modulations ( n 1 = 0.15 for each grating) and with smaller thickness: d = 2 μm .

Fig. 6.
Fig. 6.

Example of a panchromatic spatially multiplexed phase reflection grating. The grating, whose ( x , y ) index distribution is shown in (d), is formed by the sequential recording of the three simple gratings shown in (a) to (b). Each diagram shows a section of size 0.5 μm by 2 μm. Each simple grating has been recorded with a reference beam angle of Φ c = 30 ° . Gratings 1 and 2 have been recorded at 532 nm, whereas grating 3 has been recorded at 660 nm. The gratings have slopes of, respectively, ψ 1 = 10 ° , ψ 1 = 0 ° and ψ 3 = 15 ° and index modulations of 0.03, 0.02, and 0.035.

Fig. 7.
Fig. 7.

Diffractive efficiency, η σ , at 532 nm and 660 nm versus normalized grating thickness, d / Λ , as predicted by the N-PSM model and by a RCW calculation for the case of the triple multiplexed bicolor reflection grating of Fig. 6(d) at Bragg resonance. The grating is replayed using laser light at 532 nm and 660 nm at an incidence angle of Φ c = 30 ° . In each case the average index inside and outside the grating has been set to n 0 = 1.5 . The full lines indicate the modes of the N-PSM model, and the markers indicate the modes of the rigorous coupled-wave calculation. By far the most prominent RCW modes are the 100 and 010 modes at 532 nm and the 001 mode at 660 nm. Note that Λ refers to the larger of the three grating periods, i.e., to that of the component grating shown in Fig. 6(c).

Fig. 8.
Fig. 8.

Diffractive efficiency versus grating thickness for typical lossless three-color reflection volume phase holograms according to the N-PSM theory ( σ -polarization). All holograms are recorded at 660 nm, 532 nm, and 440 nm. Dashed lines indicate a 45° reference illumination beam. Solid lines indicate the case of normal incidence illumination. (a)  n 1 ( 660 ) = n 1 ( 532 ) = n 1 ( 440 ) = 0.01 and (b)  n 1 ( 660 ) = n 1 ( 532 ) = n 1 ( 440 ) = 0.02 .

Equations (63)

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R = e i ( k c x x + k c y y ) ,
k c = 2 π n 0 λ c ( sin θ c cos θ c ) = β ( sin θ c cos θ c ) ,
n ( 0 ) = n 0 .
S = S 0 e i ( k i x x + k i y y ) ,
k i = β ( sin θ c cos θ c ) .
k c β · R = sin θ c R x + cos θ c R y = R 2 { 2 i β 1 n cos θ c n y } S 2 n cos θ c n y , k i β · S = sin θ c S x cos θ c S y = S 2 { 2 i β + 1 n cos θ c n y } + R 2 n cos θ c n y ,
k c β · R = sin θ c R x + cos θ c R y = R 2 { 2 i β 1 n cos 2 θ c cos θ c n y } S 2 n cos 2 θ c cos θ c n y , k i β · S = sin θ c S x cos θ c S y = S 2 { 2 i β + 1 n cos 2 θ c cos θ c n y } + R 2 n cos 2 θ c cos θ c n y .
( x y ) = ( cos ψ sin ψ sin ψ cos ψ ) ( x y ) .
k c = β ( sin θ c cos θ c ) ; k i = β ( sin θ c cos θ c ) ,
k c = β ( sin ( θ c ψ ) cos ( θ c ψ ) ) ; k i = β ( sin ( θ c + ψ ) cos ( θ c + ψ ) ) .
n = ( n 0 n 1 ) + n 1 cos ( 2 α β cos θ r y ) = ( n 0 n 1 ) + n 1 cos ( 2 α β cos θ r { sin ψ x cos ψ y } ) = ( n 0 n 1 ) + n 1 2 { e 2 i β α cos θ r K ^ . r + e 2 i β α cos θ r K ^ . r } ,
α = λ c λ r
k c β · R k c β · R = R x sin ( θ c ψ ) + R y cos ( θ c ψ ) , k i β · S k i β · S = S x sin ( θ c + ψ ) S y cos ( θ c + ψ ) ,
sin ( θ c ψ ) R x + cos ( θ c ψ ) R y i β R = i κ α ( 1 + 2 κ β ) cos θ r cos θ c [ S e 2 i β α ζ ( θ r , ψ ) + { R e 2 i β α ζ ( θ r , ψ ) S e 2 i β α ζ ( θ r , ψ ) R e 2 i β α ζ ( θ r , ψ ) } ] i κ α cos θ r cos θ c [ S e 2 i β α ζ ( θ r , ψ ) ] + Q R ,
sin ( θ c + ψ ) S x cos ( θ c + ψ ) S y i β S = i α κ ( 1 + 2 κ β ) cos θ r cos θ c [ R e 2 i β α ζ ( θ r , ψ ) + { S e 2 i β α ζ ( θ r , ψ ) R e 2 i β α ζ ( θ r , ψ ) S e 2 i β α ζ ( θ r , ψ ) } ] i κ α cos θ r cos θ c [ R e 2 i β α ζ ( θ r , ψ ) ] + Q S ,
κ = π n 1 λ c ,
ζ ( θ r , ψ ) = cos θ r ( y cos ψ x sin ψ ) .
Φ c = θ c ψ , Φ r = θ r ψ
Φ i = θ i ψ , Φ o = θ o ψ
sin Φ c R x + cos Φ c R y i β R = i κ α cos ( Φ r + ψ ) cos ( Φ c + ψ ) S e 2 i β α ζ ( Φ r , ψ ) + Q R , sin Φ i S x cos Φ i S y i β S = i α κ cos ( Φ r + ψ ) cos ( Φ c + ψ ) R e 2 i β α ζ ( Φ r , ψ ) + Q S .
n = ( n 0 μ = 1 N n μ ) + μ = 1 N n μ cos ( 2 α β cos θ r μ { sin ψ μ x cos ψ μ y } ) = ( n 0 μ = 1 N n μ ) + μ = 1 N n μ 2 { e 2 i β α cos θ r μ K ^ μ . r + e 2 i β α cos θ r μ K ^ μ . r } = ( n 0 μ = 1 N n μ ) + μ = 1 N n μ 2 { e i K μ . r + e i K μ . r } .
sin Φ c R x + cos Φ c R y i β R = i α μ = 1 N κ μ cos ( Φ r + ψ μ ) cos ( Φ c + ψ μ ) S μ e 2 i β α ζ ( Φ r , ψ μ ) + μ = 1 N Q R μ .
sin Φ i S μ x cos Φ i S μ y i β S μ = i α κ μ cos ( Φ r + ψ μ ) cos ( Φ c + ψ μ ) R e 2 i β α ζ ( Φ r , ψ μ ) + Q S μ .
R = R ( y ) e i k c · r , S μ = S μ ( x , y ) e i k i · r ,
R R , S S ,
cos Φ c R y = i α μ = 1 N κ μ cos ( Φ r + ψ μ ) cos ( Φ c + ψ μ ) S μ e 2 i β α ζ ( Φ r , ψ μ ) e i ( k i μ k c ) · r + = i α μ = 1 N κ μ cos ( Φ r + ψ μ ) cos ( Φ c + ψ μ ) S μ e 2 i β α ζ ( Φ r , ψ μ ) e i ( k i μ k c ) · r , sin Φ i S μ x cos Φ i S μ y = i α κ μ cos ( Φ r + ψ μ ) cos ( Φ c + ψ μ ) R e 2 i β α ζ ( Φ r , ψ μ ) e i ( k i μ k c ) · r + = i α κ μ cos ( Φ r + ψ μ ) cos ( Φ c + ψ μ ) R e 2 i β α ζ ( Φ r , ψ μ ) e i ( k i μ k c ) · r .
S μ = S ^ μ ( y ) e 2 i β α ζ ( Φ r , ψ μ ) i ( k i μ k c ) · r .
R y = i μ = 1 N κ μ c R μ S ^ μ , c S μ S ^ μ y = i ϑ μ S ^ μ i κ μ R ,
c R μ = cos θ c μ cos ( θ c μ ψ μ ) α cos θ r μ , c S μ = cos θ c μ cos ( θ c μ + ψ μ ) α cos θ r μ , ϑ μ = 2 β ( 1 cos θ c μ α cos θ r μ ) cos 2 θ c μ ,
θ c μ ψ μ = Φ c θ r μ ψ μ = Φ r θ c μ + ψ μ = Φ i μ θ r μ + ψ μ = Φ o μ } μ N .
R ( 0 ) = 1 S ^ μ ( d ) = 0 μ N .
c R = c R μ = cos θ c μ cos ( θ c μ ψ μ ) α μ cos θ r μ = cos Φ c ,
η μ 1 c R | c s μ | S ^ μ ( 0 ) S ^ μ * ( 0 ) = 1 c s μ κ μ 2 k = 1 N κ k 2 c s k tanh 2 { d 1 c R k = 1 N κ k 2 c s k } .
η μ = 1 N η μ = tanh 2 { d 1 cos Φ c k = 1 N κ k 2 cos Φ i k } .
2 u x 2 + 2 u y 2 γ 2 u = 0 ,
γ 2 = β 2 2 β μ = 1 N κ μ { e i K μ · r + e i K μ · r }
u ( y < 0 ) = e i ( k x x + k y y ) ,
k x = β sin ( θ c μ ψ μ ) k y = β cos ( θ c μ ψ μ ) μ .
u ( x , y ) = l 1 = l 2 = l 3 = u l 1 l 2 l 3 ( y ) e i ( k x + l 1 K 1 x + l 2 K 2 x + ) x = l 1 = l 2 = l 3 = u l 1 l 2 l 3 ( y ) e i k x x σ = 1 N e i l σ K σ x x .
{ ( k x + σ = 1 N l σ K σ x ) 2 β 2 } u l 1 l 2 l 3 l N ( y ) 2 u l 1 l 2 l 3 l N y 2 ( y ) = 2 β σ = 1 N κ σ { u l 1 l 2 l 3 ( l σ 1 ) l N ( y ) e i K σ y y + u l 1 l 2 l 3 ( l σ + 1 ) l N ( y ) e i K σ y y } .
{ ( k x + l 1 K 1 x + l 2 K 2 x + ) 2 β 2 } u l 1 l 2 l 3 ( y ) 2 u l 1 l 2 l 3 y 2 ( y ) = 0 .
u l 1 l 2 = A e i { β 2 ( k x + l 1 K 1 x + l 2 K 2 x + ) 2 } y + B e i { β 2 ( k x + l 1 K 1 x + l 2 K 2 x + ) 2 } y ,
u ( x , y ) = e i k x x e i β 2 k x 2 y + l 1 = l 2 = l 3 = u l 1 l 2 l 3 e i { β 2 ( k x + l 1 K 1 x + l 2 K 2 x + ) 2 } y e i ( k x + l 1 K 1 x + l 2 K 2 x + ) x .
u ( x , y ) = l 1 = l 2 = l 3 = u l 1 l 2 l 3 e i { β 2 ( k x + l 1 K 1 x + l 2 K 2 x + ) 2 } y e i ( k x + l 1 K 1 x + l 2 K 2 x + ) x .
i β 2 k x 2 ( 2 u 000 ( 0 ) ) = d u 000 d y | y = 0 i β 2 ( k x + l 1 K 1 x + l 2 K 2 x + ) 2 u l 1 l 2 l 3 ( 0 ) = d u l 1 l 2 l 3 d y | y = 0 .
i β 2 ( k x + l 1 K 1 x + l 2 K 2 x + ) 2 u l 1 l 2 l 3 ( d ) = d u l 1 l 2 l 3 d y | y = d .
β 2 > ( k x + l 1 K 1 x + l 2 K 2 x + ) 2 .
η l 1 l 2 l 3 = β 2 ( k x + l 1 K 1 x + l 2 K 2 x + l 3 K 3 x + ) 2 k y u l 1 l 2 l 3 u l 1 l 2 l 3 * ,
n = ( n 0 m = 1 M μ = 1 N n μ ) + m = 1 M μ = 1 N n m μ cos ( 2 α m β cos θ r μ { sin ψ μ x cos ψ μ y } ) = ( n 0 m = 1 M μ = 1 N n μ ) + m = 1 M μ = 1 N n m μ 2 { e 2 i β α m cos θ r μ K ^ μ . r + e 2 i β α m cos θ r μ K ^ μ . r } .
η m j 1 c s j κ m j 2 k = 1 N κ m k 2 c s k tanh 2 { d 1 c R k = 1 N κ m k 2 c s k } .
η m j = 1 N η m j = tanh 2 { d 1 cos Φ c k = 1 N κ m k 2 cos Φ i k } .
cos Φ c R y = i m = 1 M α m μ = 1 N κ m μ cos ( Φ r + ψ μ ) cos ( Φ c + ψ μ ) S μ e 2 i β α m ζ ( Φ r , ψ μ ) e i ( k i μ k c ) · r , sin Φ i S μ x cos Φ i S μ y = i m = 1 M α m κ m μ cos ( Φ r + ψ μ ) cos ( Φ c + ψ μ ) R e 2 i β α m ζ ( Φ r , ψ μ ) e i ( k i μ k c ) · r .
S μ = S ^ μ ( y ) e i ( k i μ k c ) · r
η m ( Φ c , Φ i ) = κ m 2 ( Φ i ) L m cos Φ i tanh 2 { d L m cos Φ c } , η m = 1 Δ Φ κ m 2 ( Φ ) L m cos Φ tanh 2 { d L m cos Φ c } d Φ = tanh 2 { d L m cos Φ c } ,
L m = 1 Δ Φ κ m 2 ( Φ ) cos Φ d Φ ,
κ m 2 ( Φ ) κ m 2 cos Φ ,
η m = tanh 2 { d κ m cos Φ c } .
η m = tanh 2 { 1.06 d κ m cos Φ c } .
L m = 1 Δ Φ κ m 2 ( Φ i ) cos Φ i d Φ i 1 cos Φ i Δ Φ κ m 2 ( Φ ) d Φ 1 cos Φ i κ m 2 ( Φ ) ,
η m tanh 2 { d κ m 2 ( Φ ) 1 / 2 sec Φ c sec Φ i } .
k i = k c + K ,
k i = k c 2 K · k c | K | 2 K .
k i = ( k c x + K x ) x ^ β 2 ( k c x + K x ) 2 y ^ .

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