Abstract

The effect of the coherence of a beam traveling through a photonic 1D structure coupled with a rough glass is studied. The analysis is made for the case of spatial coherence showing the possibility to determine the coherence characteristics of the beam by an examination of the output field. We have shown that starting from a simple plane wave with Gaussian partial spatial coherence, we still obtain a partial coherence process on the output of the system, which, however, is no more Gaussian. The output process is no more a stationary Gaussian process. The output field correlation now depends not only on the distance between two points, but also on their position.

© 2007 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  3. L. Mandel and E. Wolf, "Spectral coherence and the concept of cross-spectral purity," J. Opt. Soc. Am. 66, 529-535 (1976).
    [CrossRef]
  4. G. P. Agrawal and E. Wolf, "Propagation-induced polarization changes in partially coherent optical beams," J. Opt. Soc. Am. A 17, 2019-2023 (2000).
    [CrossRef]
  5. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  6. G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  7. H. Roychowdhury, A. S. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
    [CrossRef]
  8. B. J. Hoenders and M. Bertolotti, "Theory of partial coherence for weakly periodic media," J. Opt. Soc. Am. A 22, 2682-2690 (2005).
    [CrossRef]
  9. B. J. Hoenders and M. Bertolotti, "Coherence theory of electromagnetic wave propagation through stratified N-layer media," J. Opt. Soc. Am. A 22, 1143-1150 (2005).
    [CrossRef]
  10. A. Mandatori, M. Bertolotti, C. Sibilia, B. Hoenders, and M. Scalora, "Coherence effects in propagation through photonic crystals," J. Eur. Opt. Soc. 1, 06005 (2006).
    [CrossRef]

2006

A. Mandatori, M. Bertolotti, C. Sibilia, B. Hoenders, and M. Scalora, "Coherence effects in propagation through photonic crystals," J. Eur. Opt. Soc. 1, 06005 (2006).
[CrossRef]

2005

2003

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

2002

2000

1976

J. Eur. Opt. Soc.

A. Mandatori, M. Bertolotti, C. Sibilia, B. Hoenders, and M. Scalora, "Coherence effects in propagation through photonic crystals," J. Eur. Opt. Soc. 1, 06005 (2006).
[CrossRef]

J. Mod. Opt.

H. Roychowdhury, A. S. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Lett. A

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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Figures (11)

Fig. 1
Fig. 1

Generic system obtained adding a simple 1D PBG to a thin rough glass.

Fig. 2
Fig. 2

Simple PBG 1D with, on the right side, a semispace filled by a material with refractive index n g .

Fig. 3
Fig. 3

Simple rough glass with, on the left side, a semispace filled by a material with refractive index n g .

Fig. 4
Fig. 4

(a) Example of the phase function. (b) Zoom of a portion of the phase function.

Fig. 5
Fig. 5

Decomposition of the input field in plane wave transmitted and reflected by the surface between the PBG 1D and the rough glass.

Fig. 6
Fig. 6

Example of a single-layer 1D PBG attached to the rough glass.

Fig. 7
Fig. 7

(a) Simple system obtained by putting the vacuum on the left side and a semispace with refractive index n = 3 on the right side. (b) Transmission spectrum of the system on the left.

Fig. 8
Fig. 8

Comparison between the single rough glass output intensity [(a) z = 20 μ m , σ = 10 μ m , λ = 5 μ m ; (b) z = 50 μ m , σ = 10 μ m , λ = 5 μ m ; (c) z = 100 μ m , σ = 10 μ m , λ = 5 μ m ; (d) z = 200 μ m , σ = 10 μ m , λ = 5 μ m ] with the intensity of the rough glass + PBG 1D system [(e) z = 20 μ m , σ = 10 μ m , λ = 5 μ m ; (f) z = 50 μ m , σ = 10 μ m , λ = 5 μ m ; (g) z = 100 μ m , σ = 10 μ m , λ = 5 μ m ; (h) z = 200 μ m , σ = 10 μ m , λ = 5 μ m ] for different planes of observation. The upper curves refer to the complete coherent input plane wave; the lower ones (the more regular) refer to the partial coherent input plane wave.

Fig. 9
Fig. 9

Example of a four-layer 1D PBG attached to the rough glass.

Fig. 10
Fig. 10

(a) Simple system obtained by putting the vacuum on the left side and a semispace with refractive index n = 2 on the left side. (b) Transmission spectrum of the system on the left.

Fig. 11
Fig. 11

Comparison between the intensity of the rough glass + PBG 1D for different planes of observation. (a) z = 20 μ m , σ = 20 μ m , λ = 5 μ m ; (b) z = 50 μ m , σ = 20 μ m , λ = 5 μ m ; (c) z = 100 μ m , σ = 20 μ m , λ = 5 μ m ; (d) z = 200 μ m , σ = 20 μ m , λ = 5 μ m . The upper curves refer to the complete coherent input plane wave; the lower ones (the more regular) refer to the partial coherent input plane wave.

Equations (53)

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E r ( k x 0 ) = R 1 D ( k x 0 ) E 1 ( k x 0 ) ,
E t ( k x 0 ) = T 1 D ( k x 0 ) E 1 ( k x 0 ) ,
E r ( k x ) = R V ( k x ) E 2 ( k x 0 ) ,
E t ( k x ) = T V ( k x ) E 2 ( k x 0 ) .
E r ( k x ) = R V ( k x ) E p ( k x 0 = 0 ) ,
E t ( k x ) = T V ( k x ) E p ( k x 0 = 0 ) ,
E r ( k x ) = R V ( k x k x 0 ) E p ( k x 0 ) ,
E t ( k x ) = T V ( k x k x 0 ) E p ( k x 0 ) ,
E r ( k x ) = R V ( k x k x 0 ) E p ( k x 0 ) d k x 0 ,
E t ( k x ) = T V ( k x k x 0 ) E p ( k x 0 ) d k x 0 .
E t 1 D 1 ( k x 0 ) = T 1 D ( k x 0 ) E i ( k x 0 ) ,
E r V 1 ( k x ) = R V ( k x k x 0 ) E t 1 D 1 ( k x 0 ) d k x 0 = R V ( k x k x 0 ) T 1 D ( k x 0 ) E i ( k x 0 ) d k x 0 = R V * ( T 1 D E i ) ,
E t V 1 ( k x ) = T V ( k x k x 0 ) E t 1 D 1 ( k x 0 ) d k x 0 = T V ( k x k x 0 ) T 1 D ( k x 0 ) E i ( k x 0 ) d k x 0 = T V * ( T 1 D E i ) .
E r 1 D 2 ( k x ) = R 1 D ( k x ) E r V 1 ( k x ) .
E r 1 D 2 ( k x 0 ) = R 1 D ( k x 0 ) E r V 1 ( k x 0 ) = R 1 D E r V 1 = R 1 D [ R V * ( T 1 D E i ) ] .
E r V 2 ( k x ) = R V ( k x k x 0 ) E r 1 D 2 ( k x 0 ) d k x 0 = R V * E r 1 D 2 = R V * { R 1 D [ R V * ( T 1 D E i ) ] } ,
E t V 2 ( k x ) = T V ( k x k x 0 ) E r 1 D 2 ( k x 0 ) d k x 0 = T V * E r 1 D 2 = T V * { R 1 D [ R V * ( T 1 D E i ) ] } .
E r 1 D 3 ( k x 2 ) = R 1 D ( k x 2 ) E r V 2 ( k x 2 ) = R 1 D E r V 2 ,
E r V 3 ( k x ) = R V * { R 1 D { R V * { R 1 D [ R V * ( T 1 D E i ) ] } } } ,
E t V 3 ( k x ) = T V * { R 1 D { R V * { R 1 D [ R V * ( T 1 D E i ) ] } } } .
E t V 1 ( k x ) = T V * ( T 1 D E i ) ,
E t V 2 ( k x ) = T V * { R 1 D [ R V * ( T 1 D E i ) ] } ,
E t V 3 ( k x ) = T V * { R 1 D { R V * { R 1 D [ R V * ( T 1 D E i ) ] } } } ,
.
E t V 1 ( x ) = t V I 1 { T 1 D E i } ,
E t V 2 ( x ) = t V r 1 D r V I 1 { T 1 D E i } ,
E t V 3 ( x ) = t V ( r 1 D r V ) 2 I 1 { T 1 D E i } ,
.
E T ( x ) = t V ( x ) 1 r 1 D r V ( x ) I 1 { T 1 D E i } .
E T ( k x ) = I { t V ( x ) 1 r 1 D r V ( x ) } k x * [ T 1 D ( k x ) E i ( k x ) ] = T 0 ( k x ) * [ T 1 D ( k x ) E i ( k x ) ] = T 0 ( k x τ ) [ T 1 D ( τ ) E i ( τ ) ] d τ ,
T 0 ( k x ) = I { t V ( x ) 1 r 1 D r V ( x ) } k x .
E t V 1 ( k x ) = T V * ( T 1 D E i ) ,
E t V 2 ( k x ) = T V * { R 1 D R ¯ V ( T 1 D E i ) } ,
E t V 3 ( k x ) = T V * { ( R 1 D R ¯ V ) 2 ( T 1 D E i ) } ,
.
E T ( k x ) = T V ( k x ) * T 1 D ( k x ) E i ( k x ) 1 R 1 D ( k x ) R ¯ V .
E T ( k x ) = T V ( k x τ ) [ T 1 D ( τ ) 1 R 1 D ( τ ) R ¯ V E i ( τ ) ] d τ .
E T ( k x ) = F V ( k x τ ) [ F 1 D ( τ ) E i ( τ ) ] d τ ,
F V ( k x ) = { I { t V ( x ) 1 r 1 D r V ( x ) } k x for narrow PBG T V ( k x ) for light rough glass ,
F 1 D ( k x ) = { T 1 D ( k x ) for narrow PBG T 1 D ( k x ) 1 R 1 D ( k x ) R ¯ V for light rough glass .
E out ( x , z ) = E T ( k x ) e i k x x e i n 0 2 k 0 2 k x 2 z d k x ,
E out ( x , z ) = [ F 1 D ( τ ) E i ( τ ) ] F V ( k x τ ) e i k x x e i n 0 2 k 0 2 k x 2 z d k x d τ = F 1 D ( τ ) E i ( τ ) F ̃ V ( τ , x , z ) d τ ,
F ̃ V ( τ , x , z ) = F V ( k x τ ) e i k x x e i n 0 2 k 0 2 k x 2 z d k x .
Γ out ( ν ) ( x 1 , x 2 ) = E out ( x 1 ) E out ( x 2 ) * = F 1 D ( τ ) E i ( τ ) F ̃ V ( τ , x 1 , z ) d τ F 1 D ( τ ) * E i ( τ ) * F ̃ V ( τ , x 2 , z ) * d τ
Γ out ( ν ) ( x 1 , x 2 ) = F 1 D ( τ ) F 1 D ( τ ) * F ̃ V ( τ , x 1 , z ) F ̃ V ( τ , x 2 , z ) * E i ( τ ) E i ( τ ) * d τ d τ .
Γ out ( ν ) ( x 1 , x 2 ) = F 1 D ( τ ) 2 F ̃ V ( τ , x 1 , z ) F ̃ V ( τ , x 2 , z ) * Γ i ( τ ) d τ .
I out ( x ) = Γ out ( ν ) ( x , x ) = F 1 D ( τ ) 2 F ̃ V ( τ , x , z ) 2 Γ i ( τ ) d τ .
I out ( x ) = F 1 D ( k x 0 ) 2 F ̃ V ( k x 0 , x , z ) 2 .
F ̃ ( τ , x , z ) rough glass = T V ( k x τ ) e i k x x e i n 0 2 k 0 2 k x 2 z d k x ,
F ̃ V ( τ , x , z ) rough glass + PBG 1 D = F V ( k x τ ) e i k x x e i n 0 2 k 0 2 k x 2 z d k x ,
t V ( x ) = rect T ( x ) e i k 0 Φ ( x ) ,
r V ( x ) = rect T ( x ) e i k 0 α Φ ( x ) ,
t V + PBG 1 D ( x ) = t V ( x ) 1 r 1 D r V ( x ) .

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