Abstract

This paper examines dispersion caused by diffraction through uniform volume holographic gratings. Of interest is the impact of this dispersion on the spatial and temporal fidelity of an optical communications signal. To this end, a holographic grating is illuminated by a Gaussian beam with 1/e 2 diameter large compared to the optical wavelength. Coupled-wave analysis is used to calculate the temporal response of the grating to transmitted symbols encoded in time as a train of Gaussian-shaped pulses. It is shown that temporal dispersion due to diffraction impacts bit-error performance, yielding increased power penalty for larger diffraction angles and beam diameters.

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References

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  1. G. B. Venus, A. Sevian, V. I. Smirnov, and L. B. Glebov, "Stable coherent coupling of laser diodes by a volume Bragg grating in photothermorefractive glass," Opt. Lett. 31, 1453-1455 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. J.-P. Herriau, A. Delboulb’e, J.-P. Huignard, G. Roosen, and G. Pauliat, "Optical-Beam Steering for Fiber Array Using Dynamic Holography," J. Lightwave Technol. LT-4, 905-907 (1986).
    [CrossRef]
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    [CrossRef]
  5. J. M. Tsui, C. Thompson, V. Mehta, J. M. Roth, V. I. Smirnov, and L. B. Glebov, "Coupled-wave analysis of apodized volume gratings," Opt. Express 12, 6642-6653 (2004).
    [CrossRef] [PubMed]
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  9. P. M. Morse and H. F. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953).
  10. L. G. Kazovsky and O. K. Tonguz, "Sensitivity of Direct-Detection Lightwave Receivers Using Optical Preamplifiers," IEEE Photon. Technol. Lett. 3, 53-55 (1991).
    [CrossRef]
  11. P. A. Humblet and M. Azizo˜glu, "On the Bit Error Rate of Lightwave Systems with Optical Amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
    [CrossRef]

2006 (2)

2005 (1)

2004 (1)

1991 (2)

L. G. Kazovsky and O. K. Tonguz, "Sensitivity of Direct-Detection Lightwave Receivers Using Optical Preamplifiers," IEEE Photon. Technol. Lett. 3, 53-55 (1991).
[CrossRef]

P. A. Humblet and M. Azizo˜glu, "On the Bit Error Rate of Lightwave Systems with Optical Amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

1990 (1)

1986 (1)

J.-P. Herriau, A. Delboulb’e, J.-P. Huignard, G. Roosen, and G. Pauliat, "Optical-Beam Steering for Fiber Array Using Dynamic Holography," J. Lightwave Technol. LT-4, 905-907 (1986).
[CrossRef]

1981 (1)

1973 (1)

Gaylord, T. K.

Glebov, L. B.

Glytsis, E. N.

Herriau, J.-P.

J.-P. Herriau, A. Delboulb’e, J.-P. Huignard, G. Roosen, and G. Pauliat, "Optical-Beam Steering for Fiber Array Using Dynamic Holography," J. Lightwave Technol. LT-4, 905-907 (1986).
[CrossRef]

Humblet, P. A.

P. A. Humblet and M. Azizo˜glu, "On the Bit Error Rate of Lightwave Systems with Optical Amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

Kazovsky, L. G.

L. G. Kazovsky and O. K. Tonguz, "Sensitivity of Direct-Detection Lightwave Receivers Using Optical Preamplifiers," IEEE Photon. Technol. Lett. 3, 53-55 (1991).
[CrossRef]

Kraus, H. G.

Lumeau, J.

Mehta, V.

Moharam, M. G.

Roth, J. M.

Sevian, A.

Smirnov, V.

Smirnov, V. I.

Thompson, C.

Tonguz, O. K.

L. G. Kazovsky and O. K. Tonguz, "Sensitivity of Direct-Detection Lightwave Receivers Using Optical Preamplifiers," IEEE Photon. Technol. Lett. 3, 53-55 (1991).
[CrossRef]

Tsui, J. M.

Venus, G. B.

Williams, C. S.

Wu, S.-D.

Wu, Y.-M.

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (1)

L. G. Kazovsky and O. K. Tonguz, "Sensitivity of Direct-Detection Lightwave Receivers Using Optical Preamplifiers," IEEE Photon. Technol. Lett. 3, 53-55 (1991).
[CrossRef]

J. Lightwave Technol. (2)

P. A. Humblet and M. Azizo˜glu, "On the Bit Error Rate of Lightwave Systems with Optical Amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

J.-P. Herriau, A. Delboulb’e, J.-P. Huignard, G. Roosen, and G. Pauliat, "Optical-Beam Steering for Fiber Array Using Dynamic Holography," J. Lightwave Technol. LT-4, 905-907 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (2)

Other (1)

P. M. Morse and H. F. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic of the HOE. The diffraction orders are denoted by subscripts. and are the incident and output spatial beam dimensions, respectively. The refractive indexes are n 1 = n 3 = 1.

Fig. 2.
Fig. 2.

Diagram of axes rotations for input (a) and output (b) beams.

Fig. 3.
Fig. 3.

Block diagram for numerical simulation of Gaussian beam propagating through a holographic optical element.

Fig. 4.
Fig. 4.

The spatial spectrum of the first-order, forward-propagating diffracted wave for the Gaussian incident beam at z = d. The different solid curves are for different 1/e 2 beam diameters (2wx ). T 1 is the amplitude of the first-order, forward-diffracted beam, plotted along different components of the spatial wavenumber. For high diffraction efficiency, the spatial frequency content of the beam should be well-contained within the HOE spectrum, T 1. Grating period: Λ = 1420 nm; grating thickness: d = 1.125 mm; optical wavelength: λo = 1550 nm.

Fig. 5.
Fig. 5.

The 3-dB spatial and spectral (time-frequency) bandwidth of the volume grating designed for optical wavelengths in the 1550-nm band. Horizontal axis given in both grating wavelength and diffraction angle.

Fig. 6.
Fig. 6.

Comparison of approximate model (Eqs. (32 – 34)) and exact model (Eqs. (7, 12 – 13)): (a) ε 3 y (x,0,d,ωo ) as a function of x at y = 0, (b) ε 3 y (0,y,d,ωo ) as a function of y at x = 0, wx = wy = 10mm. Both models match very closely.

Fig. 7.
Fig. 7.

Incident and first-order, forward-propagating diffracted waves as a function of or at z = -z 1 and z = d + z 2. The incident Gaussian beam has 1 /e 2 diameter 2wx and temporal full-width, half-maximum (FWHM) 33-ps (σo = 27 ps). The plots show different values of 2wx . Refractive indices for the three regions are n = n 3 = 1 and n 2 = 1.5.

Fig. 8.
Fig. 8.

Normalized output waveforms, po , for first-order, forward-propagating diffracted waves as a function of time at z = z 1 + d + z 2. The incident wave is a Gaussian beam with 1/e 2 diameter 2wx and a Gaussian temporal pulse with 33-ps FWHM (σo = 27 ps). z 1 = 10wx and z 2 = 10wx

Fig. 9.
Fig. 9.

Simulation approach for HOE bit-error-ratio analysis.

Fig. 10.
Fig. 10.

Bit-error ratio (BER) of HOE for 10-Gb/s simulation (RZ-OOK). θ 1 = 33° and λo = 1550 nm. Horizontal line shows a 10-5 BER.

Fig. 11.
Fig. 11.

Power penalty of the HOE as a function of effective beam diameter 2 w x ˜ , generalized for varying diffraction angle. λo = 1550 nm, BER = 10-5, 2 w x ˜ = 2 w x cos θ .

Equations (46)

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ϵ ( x ¯ ) = ϵ 20 + ϵ 21 cos [ K ¯ · x ¯ ] ,
[ x ̂ z ̂ ] = [ C 1 S 1 S 1 C 1 ] [ x z ]
[ x ̃ z ̃ ] = [ C 3 1 S 3 1 S 3 1 C 3 1 ] [ x z ]
2 ε n y ( x ¯ , ω ) + k o 2 μ ϵ ( x ¯ ) ε n y ( x ¯ , ω ) = 0 ,
e 1 y i ( x ̂ ¯ , t ) = ( 1 2 π ) 3 E 1 y i ( x ̂ ¯ , k ¯ , ω ) e jωt d k x d k y
E 1 y i ( x ̂ ¯ , k ¯ , ω ) = E ( k x , k y , ω ) e j k x x ̂ e j k y y e j k 1 2 k x 2 k y 2 z ̂ .
E 2 y = m = + E ( k x , k y , ω ) A m ( k x , k y , z , ω ) e j Ψ m ( x , z ) e j k y y .
2 A m z 2 2 j [ m k x C ̂ k x S 2 + k 2 2 k x 2 k y 2 C 2 ] A m z
2 mK [ mK 2 + k x sin ( ϕ θ 2 ) + k 2 2 k x 2 k y 2 cos ( ϕ θ 2 ) ] A m
+ k 2 2 ϵ 2 A m + 1 + k 2 2 ϵ 2 A m 1 = 0 ,
Ψ m = Ψ 0 + mK ( S ̂ x + C ̂ z )
Ψ 0 = k x [ C 2 x S 2 z ] + k 2 2 k x 2 k y 2 [ S 2 x + C 2 z ]
E 1 y i = E ( k x , k y , ω ) e j k x [ C 1 x S 1 z ] e j k 1 2 k x 2 k y 2 [ S 1 x + C 1 z ] e j k y y
E 1 y r = E ( k x , k y , ω ) R m ( k x , k y , ω ) e j [ k x C r + k 1 2 k x 2 k y 2 S r mK S ̂ ] x e j k y y e j γ 1 m z
E 3 y = E ( k x , k y , ω ) T m ( k x , k y , ω ) e j [ k x C 3 + k 1 2 k x 2 k y 2 S 3 mK S ̂ ] x e j k y y ×
e j γ 3 m ( z d ) e j ( Ψ 0 + mK C ̂ ) d
k x C 1 + S 1 k 1 2 k x 2 k y 2 = k x C r + S r k 1 2 k x 2 k y 2 ,
k x C 1 + S 1 k 1 2 k x 2 k y 2 = k x C 2 + S 2 k 2 2 k x 2 k y 2 = k x C 3 + S 3 k 3 2 k x 2 k y 2 .
γ 1 m 2 = k 1 2 { k x C r + S r k 1 2 k x 2 k y 2 mK S ̂ } 2
γ 3 m 2 = k 3 2 { k x C 3 + S 3 k 1 2 k x 2 k y 2 mK S ̂ } 2
( E ¯ ) t ( z = 0 ) : R m A m = δ m 0
( H ¯ ) t ( z = 0 ) : 1 m R m A m z + j Ψ m z A m
= j δ m 0 ( k S 1 + k 1 2 k x 2 k y 2 C 1 )
( E ¯ ) t ( z = d ) : A m T m = 0
( H ¯ ) t ( z = d ) : A m z Ψ m z A m j γ 3 m T m = 0
f ( x ¯ , t ) = d t o d S o [ f g z o ]
g z o = 1 2 π z R [ δ ( R / c t + t o ) R 2 + t o δ ( R / c t + t o ) Rc ]
f ( x ¯ , t ) = 1 2 π d S o cos ϑ [ f x ¯ o , t R c R 2 + 1 Rc f ( x ¯ o , t o ) t o t o = t R c ] ,
E ( k x , k y , ω ) = g ix ( x ̂ ) g iy ( y ) p ̂ i ( t ) e j k y y e j k x x ̂ e jωt d x ̂ dy dt
= G ix ( k x ) G iy ( k y ) p ̂ ( ω )
g ix = e ( x ̂ w x ) 2
g iy = e ( y w y ) 2
p i ( t ) = = e ( τ μ t ) 2 σ o 2
k n 2 k x 2 k y 2 k n 2 k x 2
sin ( θ 2 ) sin ( θ o ) + β cos ( θ o ) ,
e 1 y i ( x ¯ , t ) = ( 1 2 π ) 3 + G iy ( k y ) ) e j k y y d k y + G ix ( k x ) P ̂ ( ω ) e j k x [ C 1 x S 1 z ] e j k 1 2 k x 2 [ S 1 x + C 1 z ] d k x e jωt
= g iy ( 1 2 π ) 2 + G ix ( k x ) P ̂ ( ω ) e j k x [ C 1 x S 1 z ] e j k 1 2 k x 2 [ S 1 x + C 1 z ] d k x e jωt
e 1 y r ( x ¯ , t ) = g iy ( 1 2 π ) 2 + m = + G ix ( k x ) P ̂ ( ω ) R m ( k x , ω ) e j [ k x C r + k 1 2 k x 2 S r mK S ̂ ] x e j γ 1 m z e jωt d k x
e 2 y ( x ¯ , t ) = g iy ( 1 2 π ) 2 + m = + G ix ( k x ) P ̂ ( ω ) A m ( k x , z , ω ) e j Ψ m ( x , z ) e jωt d k x
e 3 y ( x ¯ , t ) = g iy ( 1 2 π ) 2 + m G ix ( k x ) P ̂ ( ω ) T m ( k x , ω ) e j [ k x C 3 + k 3 2 k x 2 S 3 mK S ̂ ] x ×
e j γ 3 m ( z d ) e j ( Ψ 0 + mK C ̂ ) d e jωt d k x
e 1 y i ( x ¯ , t ) t = g ix g iy ( d p i ( t ) dt cos ( ω o t ) + ω o p i ( t ) cos ( π 2 + ω o t ) )
[ e 1 y i ( x ¯ , t ) e 3 y ( x ¯ , t ) ] 1 2 π d S o cos ϑ ( 1 R 2 + ω o Rc ) [ e 1 y i ( x ¯ o , t R c ) e 3 y ( x ¯ o , t R c ) ] .
P ( t ) = n b = 1 N b Bit ( n b ) × p o ( t n b τ b ) , where τ b = 100 ps
σ N 2 = 2 × n sp × h × f × B o × ( G 1 ) ,
r ( t ) = n b = 1 N b ( Bit ( n b ) × p o ( t n b τ b ) + n ASE ( t ) ) 2

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