Abstract

A recently proposed concept suggests that a matched periodic modulation of both the refractive index and the gain/loss of the media breaks the coupling symmetry of the two co-propagating modes and allows only a unidirectional coupling from the i-th mode to j-the mode but not the opposite. This concept has been used to design a ring resonator coupled through a complex grating composed of both real (index) and imaginary (loss/gain) parts according to Euler relation: Δn=n0 exp(-jkx)=n0 (cos(kx)−j sin(kx)). Such asymmetrical coupling allows light to be coupled into the ring without letting it out. We present a detailed theoretical analysis of the ring resonator in the linear regime, and we investigate its linear temporal dynamics. Three possible states of the complex grating leads to the possibility of developing a dynamic optical memory cell where, for example, a data modulated train of optical pulses can be stored. This data can be accessed without destroying it, and can also be erased thus permitting the storage of a new bit. Finally, the ring can be used for pulse retiming.

© 2005 Optical Society of America

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References

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Appl. Opt. (1)

J. Opt. B (1)

V.S.C. M. Rao, S. D. Gupta, G.S. Agarwal, �??Study of asymmetric multilayered structures by means of nonreciprocity in phases,�?? J. Opt. B: Quantum Semiclass. Opt. 6, 555-562 (2004).
[CrossRef]

Letters to Nature (1)

A. Rebane, J. Feinberg, �??Time-resolved holography,�?? Letters to Nature 351, 378-380 (1991).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. B (1)

A. Armitage, S. Skolnik, A.V. Kavokin, D.M. Whittaker, V.N. Astratov, G.A. Gehring, J.S. Roberts, �??Polariton-induced optical asymmetry in semiconductor microcavities,�?? Phys. Rev. B 58, 15367-15370 (1998).
[CrossRef]

Phys. Rev. E (1)

L. Poladian, �??Resonance mode expansions and exact solutions for nonuniform gratings,�?? Phys. Rev. E 54, 2963-2975 (1996).
[CrossRef]

Rep. Prog. Phys. (1)

R.J. Potton, �??Reciprocity in optics,�?? Rep. Prog. Phys. 67, 717-754 (2004).
[CrossRef]

Other (1)

M. Greenberg, �??Unidirectional mode devices based on irreversible mode coupling,�?? MSc. Thesis, Israel Institute of Technology, Haifa, March, 2004.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Grating assisted co-directional coupler (GACC) and (b) a ring resonator coupled with the guide by the complex grating. The propagation constants of the guides are given by β1 and β2 , the guides are asynchronous (β1 >β2 ). The grating length is given by L.

Fig. 2.
Fig. 2.

Transmission spectra of an A-GACC for bar-state (solid, red) and cross-state (dash, blue) for κL=π/2, L=25 mm, Λ=80 µm. The signal is launched into Port A

Fig. 3.
Fig. 3.

Pulse train in the ring at Port C (red) and the train released from the ring after four full roundtrips inside the ring (blue) compared to the input signal (green).

Fig. 4.
Fig. 4.

Demonstration of pulse train retiming from a repetition rate of tR =280 ps (in Port A) to tR ≈20 ps inside the ring (red). The retimed train of six pulses duplicated into output Port D (blue) along with the input pulse (green).

Equations (34)

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Δ n = Δ n DC + j Δ α DC k 0 + Δ n AC cos ( 2 π Λ z ) j Δ α AC k 0 sin ( 2 π Λ z ) ,
[ E ¯ D E ¯ C ] = M [ E ¯ A E ¯ B ] = [ M 11 M 12 M 21 M 22 ] [ E ¯ A E ¯ B ] ,
M = T avg ψ avg [ a ˜ + ψ + g j ( κ n κ α ) L sinc ( γ ˜ L ) ψ + g j ( κ n + κ α ) L sinc ( γ ˜ L ) ψ g a ˜ ψ g ]
T i = exp [ α i L ] ,
ψ i = exp [ j β i L ] .
β i ( ω ) = n eff , i ω 0 c + n g , i ω ω 0 c .
ψ i = ψ i , 0 exp [ j ( ω ω 0 ) τ i ] ,
ψ i , 0 = exp [ j n eff , i ω 0 L c ] ,
τ i = n g , i L 2 c ,
Re { Δ β ˜ } L = θ + ( ω ω 0 ) τ ,
θ = [ ( n eff , 1 n eff , 2 ) ω 0 2 c π Λ ] L ,
τ = n g , 1 n g , 2 2 c L .
M S = [ T 1 ψ 1 0 0 T 2 ψ 2 ] + j κ L sinc ( Δ β ˜ L ) T avg ψ avg [ 0 ( 1 S ) ψ + g ( 1 + S ) ψ g 0 ]
T r = T 2 T 3 = exp [ ( α 2 L + α 3 L 3 ) ] ,
ψ r = ψ 2 ψ 3 = ψ r , 0 exp [ j ( ω ω 0 ) τ r ]
ψ r , 0 = exp [ j ( n eff , 2 L + n eff , 3 L 3 ) ω 0 c ] ,
τ r = n g , 2 L + n g , 3 L 3 c
E ¯ D = T 1 ψ 1 E ¯ A .
E ¯ C = 2 j κ L sinc ( Δ β ˜ L ) T avg ψ avg ψ g n = 0 N ( T r ψ r ) n E ¯ A ( n ) ,
E ¯ D = j κ L sinc ( γ ˜ L ) T avg ψ avg ψ + g E ¯ B ,
E ¯ C ( p ) = T 2 ψ 2 E ¯ B ( p ) ,
E ¯ D ( P ) = 2 j κ L sinc ( Δ β ˜ L ) T avg ψ avg ψ + g E ¯ B ( 0 ) p = 0 P ( T r ψ r ) p ,
E A ( t ) = m ϕ ( t m t R ) exp [ j ω 0 ( t m t R ) ] ,
E ¯ A ( ω ) = m ϕ ¯ ( ω ω 0 ) exp [ j ω m t R ] .
E D ( t ) = T 1 E A ( t τ 1 ) .
E ¯ C = 2 j κ L sinc ( Δ β ˜ L ) T avg ψ avg ψ g ϕ ¯ ( ω ω 0 ) m ( T r ψ r , 0 ) n m exp [ j ω m t R + j ( ω ω 0 ) n m τ r ] ,
E C ( t ) = m ( T r ψ r 0 ) n m ϕ inj ( t m t R n m τ r τ avg ) exp [ j m ω 0 t R ] , .
ϕ inj ( t ) = j 2 κ L 2 π T avg ψ avg , 0 ψ g sinc ( Δ β ˜ L ) ϕ ¯ ( ω ω 0 ) exp [ j ω t ] d ω .
ϕ inj ( t ) j ψ ± g ψ avg , 0 { 2 κ L Γ inj T avg exp [ θ inj 2 6 Γ inj 2 ] } G ( t ; Γ inj t 0 , δ ω inj ) ,
E D ( dup ) ( t ) = m p = 0 P ( T r ψ r 0 ) p + n m ϕ dup ( t m t R ( p + n m ) τ r τ 1 ) exp [ j m ω 0 t 0 ] ,
ϕ dup ( t ) = 4 κ 2 L 2 2 π T 1 ψ 1 , 0 sinc 2 ( Δ β ˜ L ) ϕ ¯ ( ω ω 0 ) exp [ j ω t ] d ω .
E D ( ext ) ( t ) = m ( T r ψ r 0 ) n m ϕ ext ( t m t R n m τ r τ 1 ) exp [ j m ω 0 t R ] .
ϕ ext ( t ) = 2 κ 2 L 2 2 π T 1 ψ 1 , 0 sinc ( Δ β ˜ L ) sinc ( γ ˜ L ) ϕ ¯ ( ω ω 0 ) exp [ j ω t ] d ω .
ϕ out ( t ; T out , f ) ψ 1 , 0 { 4 κ 2 L 2 Γ ext T 1 T out exp [ ( 1 + f ) θ 2 6 Γ out 2 ] } G ( t ; Γ out t o , δ ω out ) ,

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