Abstract

The unidirectional grating-assisted codirectional coupler (U-GACC) has recently been proposed. This unique structure permits irreversible coupling between orthogonal waveguide eigenmodes by means of simultaneous modulation of both the real and imaginary parts of the refractive index in the coupling region. Analysis of the U-GACC has until now relied on coupled mode theory, which can be restrictive in its application as a design tool. We analyze the U-GACC by the transfer matrix method, which demonstrates in a simple fashion why the device operates in a unidirectional manner. In addition, we show that for all practical designs, there is a limit to the minimum cross talk between outputs, a phenomenon that has not been previously identified.

© 2007 Optical Society of America

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References

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  1. D. Marcuse, "Directional couplers made of nonidentical asymmetrical slabs. Part II: grating-assisted couplers," J. Lightwave Technol. LT-5, 268-273 (1987).
    [CrossRef]
  2. A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. QE-13, 233-253 (1977).
    [CrossRef]
  3. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).
  4. M. Greenberg and M. Orenstein, "Irreversible coupling by use of dissipative optics," Opt. Lett. 29, 451-453 (2004).
    [CrossRef] [PubMed]
  5. T. Galstian, "Non-reciprocal optical element for photonic devices," U.S. patent 6,611,644 B2 (23 August 2003).
  6. M. Greenberg and M. Orenstein, "Filterless 'add' multiplexer based on novel complex gratings assisted coupler," IEEE Photon. Technol. Lett. 17, 1450-1452 (2005).
    [CrossRef]
  7. M. Greenberg, "Unidirectional mode devices based on irreversible mode coupling," M.Sc. thesis (Israel Institute of Technology, 2004).
  8. M. Kulishov, J. M. Laniel, N. Belanger, and D. V. Plant, "Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission," Opt. Express 13, 3567-3578 (2005).
    [CrossRef] [PubMed]
  9. L. Poladian, "Resonance mode expansions and exact solutions for nonuniform gratings," Phys. Rev. E 54, 2963-2975 (1996).
    [CrossRef]
  10. M. Kulishov, J. M. Laniel, N. Belanger, J. Azana, and D. V. Plant, "Nonreciprocal waveguide Bragg gratings," Opt. Express 13, 3068-3078 (2005).
    [CrossRef] [PubMed]
  11. M. Greenberg and M. Orenstein, "Optical unidirectional devices by complex spatial single sideband perturbation," IEEE J. Quantum Electron. 41, 1013-1023 (2005).
    [CrossRef]
  12. W. Huang and J. Hong, "A transfer matrix approach based on local normal modes for coupled waveguides with periodic perturbations," J. Lightwave Technol. 10, 1367-1375 (1992).
    [CrossRef]
  13. W. K. Burns and A. F. Milton, "Mode conversion in planar-dielectric separating waveguides," IEEE J. Quantum Electron. QE-11, 32-39 (1975).
    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, 1980).
    [PubMed]
  15. M. S. Stern, "Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles," IEE Proc.-J.: Optoelectron. 135, 56-63 (1988).
    [CrossRef]
  16. C. M. Kim and R. V. Ramaswamy, "Modeling of graded-index channel waveguides using nonuniform finite difference method," J. Lightwave Technol. 7, 1581-1589 (1989).
    [CrossRef]
  17. Optiwave Systems, Inc., OptiBPM v. 8.1, http://www.optiwave.com.

2005 (4)

M. Greenberg and M. Orenstein, "Filterless 'add' multiplexer based on novel complex gratings assisted coupler," IEEE Photon. Technol. Lett. 17, 1450-1452 (2005).
[CrossRef]

M. Kulishov, J. M. Laniel, N. Belanger, and D. V. Plant, "Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission," Opt. Express 13, 3567-3578 (2005).
[CrossRef] [PubMed]

M. Kulishov, J. M. Laniel, N. Belanger, J. Azana, and D. V. Plant, "Nonreciprocal waveguide Bragg gratings," Opt. Express 13, 3068-3078 (2005).
[CrossRef] [PubMed]

M. Greenberg and M. Orenstein, "Optical unidirectional devices by complex spatial single sideband perturbation," IEEE J. Quantum Electron. 41, 1013-1023 (2005).
[CrossRef]

2004 (1)

1996 (1)

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

1992 (1)

W. Huang and J. Hong, "A transfer matrix approach based on local normal modes for coupled waveguides with periodic perturbations," J. Lightwave Technol. 10, 1367-1375 (1992).
[CrossRef]

1989 (1)

C. M. Kim and R. V. Ramaswamy, "Modeling of graded-index channel waveguides using nonuniform finite difference method," J. Lightwave Technol. 7, 1581-1589 (1989).
[CrossRef]

1988 (1)

M. S. Stern, "Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles," IEE Proc.-J.: Optoelectron. 135, 56-63 (1988).
[CrossRef]

1987 (1)

D. Marcuse, "Directional couplers made of nonidentical asymmetrical slabs. Part II: grating-assisted couplers," J. Lightwave Technol. LT-5, 268-273 (1987).
[CrossRef]

1977 (1)

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. QE-13, 233-253 (1977).
[CrossRef]

1975 (1)

W. K. Burns and A. F. Milton, "Mode conversion in planar-dielectric separating waveguides," IEEE J. Quantum Electron. QE-11, 32-39 (1975).
[CrossRef]

IEE Proc.-J.: Optoelectron. (1)

M. S. Stern, "Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles," IEE Proc.-J.: Optoelectron. 135, 56-63 (1988).
[CrossRef]

IEEE J. Quantum Electron. (3)

W. K. Burns and A. F. Milton, "Mode conversion in planar-dielectric separating waveguides," IEEE J. Quantum Electron. QE-11, 32-39 (1975).
[CrossRef]

M. Greenberg and M. Orenstein, "Optical unidirectional devices by complex spatial single sideband perturbation," IEEE J. Quantum Electron. 41, 1013-1023 (2005).
[CrossRef]

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. QE-13, 233-253 (1977).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. Greenberg and M. Orenstein, "Filterless 'add' multiplexer based on novel complex gratings assisted coupler," IEEE Photon. Technol. Lett. 17, 1450-1452 (2005).
[CrossRef]

J. Lightwave Technol. (3)

W. Huang and J. Hong, "A transfer matrix approach based on local normal modes for coupled waveguides with periodic perturbations," J. Lightwave Technol. 10, 1367-1375 (1992).
[CrossRef]

D. Marcuse, "Directional couplers made of nonidentical asymmetrical slabs. Part II: grating-assisted couplers," J. Lightwave Technol. LT-5, 268-273 (1987).
[CrossRef]

C. M. Kim and R. V. Ramaswamy, "Modeling of graded-index channel waveguides using nonuniform finite difference method," J. Lightwave Technol. 7, 1581-1589 (1989).
[CrossRef]

Opt. Express (2)

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]

Other (5)

M. Greenberg, "Unidirectional mode devices based on irreversible mode coupling," M.Sc. thesis (Israel Institute of Technology, 2004).

T. Galstian, "Non-reciprocal optical element for photonic devices," U.S. patent 6,611,644 B2 (23 August 2003).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).

Optiwave Systems, Inc., OptiBPM v. 8.1, http://www.optiwave.com.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, 1980).
[PubMed]

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Figures (7)

Fig. 1
Fig. 1

Asynchronous coupler embodiment of the U-GACC (three periods shown). The real index grating is in waveguide 1 (WG1) and the gain–loss grating is in waveguide 2 (WG2). White lines are drawn to show each grating period, indicating the ± π / 2 phase offset between the gratings.

Fig. 2
Fig. 2

Dual-mode waveguide embodiment of the U-GACC (three periods shown). Both gratings coexist within one half of the waveguide cross section. Inset, one period of the grating, with the labeling used in this paper.

Fig. 3
Fig. 3

Modal power evolution in an ideal U-GACC ( 2 1 coupling permitted, ε X = 0.01 ). •, mode 1; ∘, mode 2. (a) Power input to mode 1. (b) Power input to mode 2.

Fig. 4
Fig. 4

Cross section of example U-GACC analyzed in this work. Δ n R = Δ n I = 5 × 10 4 .

Fig. 5
Fig. 5

Geometry of the adiabatic asymmetric y-branch multiplexer.

Fig. 6
Fig. 6

(Color online) Optical power in the U-GACC example from Section 4. Top row, order of grating segments: HH, HL, LL, LH. Bottom row, order of grating segments: HH, LH, LL, HL. Left column, power input to mode 2. Right column, power input to mode 1. The curves to the right of each plot show the power at the output plane of the demultiplexer (arbitrary units).

Fig. 7
Fig. 7

Modal power evolution in the U-GACC example of Section 4. Order of grating segments: HH, HL, LL, LH. •, mode 1; ∘, mode 2. (a) Power input to mode 1: see Fig. 6, upper right figure. (b) Power input to mode 2: see Fig. 6, upper left figure.

Tables (1)

Tables Icon

Table 1 Calculated Effective Mode Indices, Segment Lengths, and Segment Gain for the Example in Section 4

Equations (60)

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β p β q = K g = 2 π / Λ ,
Δ n ( z ) = n 0   exp ( ± i K g z ) = n 0 [ cos ( K g z ) ± i   sin ( K g z ) ] .
[ E 1 out E 2 out ] = T N [ E 1 in E 2 in ] ,
n ( x , y ) = n ˜ ( x , y ) ± Δ n R ( x , y ) ± Δ n I ( x , y ) ,
1 4 ( E p × H q + E q × H p ) d a = δ p q
δ p q = { 1 , p = q 0 , p q
E p , q = E ˜ p , q + Δ E p , q R + i Δ E p , q I ,
H p , q = H ˜ p , q + Δ H p , q R + i Δ H p , q I .
1 4 ( E ˜ p × H ˜ q + E ˜ q × H ˜ p ) d a = δ p q ,
1 4 [ ( E ˜ p ± Δ E p R ) × ( H ˜ q ± Δ H q R ) ] + [ ( E ˜ q ± Δ E q R ) × ( H ˜ p ± Δ H p R ) ] d a = δ p q ,
1 4 [ ( E ˜ p ± i Δ E p I ) × ( H ˜ q ± i Δ H q I ) ] + [ ( E ˜ q ± i Δ E q I ) × ( H ˜ p ± i Δ H p I ) ] d a = δ p q .
E p , q H H = E ˜ p , q + Δ E p , q R + i Δ E p , q I ,
H p , q H H = H ˜ p , q + Δ H p , q R + i Δ H p , q I ,
E p , q H L = E ˜ p , q + Δ E p , q R i Δ E p , q I ,
H p , q H L = H ˜ p , q + Δ H p , q R i Δ H p , q I ,
E p , q L L = E ˜ p , q Δ E p , q R i Δ E p , q I ,
H p , q L L = H ˜ p , q Δ H p , q R i Δ H p , q I ,
E p , q L H = E ˜ p , q Δ E p , q R + i Δ E p , q I ,
H p , q L H = H ˜ p , q Δ H p , q R + i Δ H p , q I .
T p q ( R I ) R I = 1 4 [ ( E p R I × H q R I ) + ( E q R I × H p R I ) ] d a .
T p q ( H L ) H H = 1 4 [ ( E p H L × H q H H ) + ( E q H H × H p H L ) ] d a .
T 11 ( H L ) H H = 1 + 1 2 ( Δ E 1 I × Δ H 1 I ) d a ,
T 22 ( H L ) H H = 1 + 1 2 ( Δ E 2 I × Δ H 2 I ) d a ,
T 21 ( H L ) H H = i 2 [ ( Δ E 2 I × H ˜ 1 ) + ( E ˜ 1 × Δ H 2 I ) ] d a ,
T 12 ( H L ) H H = i 2 [ ( Δ E 2 I × H ˜ 1 ) + ( E ˜ 1 × Δ H 2 I ) ] d a .
T ( H L ) H H = [ 1 + ( ε 1 I ) 2 2 i ε X I i ε X I 1 + ( ε 2 I ) 2 2 ] ,
( ε 1 I ) 2 = ( Δ E 1 I × Δ H 1 I ) d a ,
( ε 2 I ) 2 = ( Δ E 2 I × Δ H 2 I ) d a ,
ε X I = 1 2 [ ( Δ E 2 I × H ˜ 1 ) + ( E ˜ 1 × Δ H 2 I ) ] d a .
T ( L H ) L L = [ 1 + ( ε 1 I ) 2 2 i ε X I i ε X I 1 + ( ε 2 I ) 2 2 ] = ( T ( H L ) H H ) T ,
T ( L L ) H L = [ 1 ( ε 1 R ) 2 2 ε X R ε X R 1 ( ε 2 R ) 2 2 ] ,
T ( H H ) L H = ( T ( L L ) H L ) T ,
( ε 1 R ) 2 = ( Δ E 1 R × Δ H 1 R ) d a ,
( ε 2 R ) 2 = ( Δ E 2 R × Δ H 2 R ) d a ,
ε X R = 1 2 [ ( Δ E 2 R × H ˜ 1 ) + ( E ˜ 1 × Δ H 2 R ) ] d a .
| ε X R | = | ε X I | | ε X | .
ϕ p , q R I ( z ) = β p , q R I ( z z 0 ) ,
Λ H = π 2   Re ( β 1 H β 2 H ) ,   ( H H , H L   segments ) ,
Λ L = π 2   Re ( β 1 L β 2 L ) ,   ( L H , L L   segments ) ,
Λ = 2 ( Λ H + Λ L ) .
g p , q H = exp [ Im ( β p , q H ) Λ H ] ,
g p , q L = exp [ Im ( β p , q L ) Λ L ] .
P R I = [ exp ( i ϕ 1 R I ) 0 0 exp ( i ϕ 2 R I ) ] .
P H H = [ g 1 H 0 0 i g 2 H ] ,     P H L = [ ( g 1 H ) 1 0 0 i ( g 2 H ) 1 ] ,
P L H = [ g 1 L 0 0 i g 2 L ] ,     P L L = [ ( g 1 L ) 1 0 0 i ( g 2 L ) 1 ] .
T = T ( H H ) L H P L H T ( L H ) L L P L L T ( L L ) H L P H L T ( H L ) H H P H H = [ T 11 T 12 T 21 T 22 ] ,
T 11 = 1 ε X 2 ( g 1 L g 2 L 2 g 1 H g 2 H g 2 L g 1 L g 1 L g 1 H g 2 L g 2 H + 1 ) ,
T 12 = ε X ( g 2 H g 1 H + g 1 L g 2 L + 2 ) ,
T 21 = ε X ( g 1 H g 2 H + g 2 L g 1 L 2 ) ,
T 22 = 1 ε X 2 ( g 1 L g 2 L + 2 g 2 H g 1 H g 2 L g 1 L g 2 L g 2 H g 1 L g 1 H + 1 ) .
T [ 1 + 2 ε X 2 4 ε X 0 1 2 ε X 2 ] .
T rev = P H H T ( H H ) H L P H L T ( H L ) L L P L L T ( L L ) L H P L H T ( L H ) H H .
T ( R I ) R I = ( T ( R I ) R I ) T .
T rev = ( P H H ) T ( T ( H L ) H H ) T ( P H L ) T ( T ( L L ) H L ) T ( P L L ) T × ( T ( L H ) L L ) T ( P L H ) T ( T ( H H ) L H ) T = ( T ( H H ) L H P L H T ( L H ) L L P L L T ( L L ) H L P H L T ( H L ) H H P H H ) T = T T .
T N = [ 1 N ε X 2 ( g 1 L g 2 L 2 g 1 H g 2 H g 2 L g 1 L g 1 L g 1 H g 2 L g 2 H + 1 ) N ε X ( g 2 H g 1 H + g 1 L g 2 L + 2 ) N ε X ( g 1 H g 2 H + g 2 L g 1 L 2 ) 1 N ε X 2 ( g 1 L g 2 L + 2 g 2 H g 1 H g 2 L g 1 L g 2 L g 2 H g 1 L g 1 H + 1 ) ] .
T N = [ 1 + 2 N ε X 2 4 N ε X 0 1 2 N ε X 2 ] [ 1 4 N ε X 0 1 ] ,
[ E 1 out E 2 out ] = T N [ E 1 in E 2 in ] = [ E 1 in 4 N ε X E 2 in E 2 in ] .
N eq = 1 4 ε X ,
cross   talk = | T 21 / T 12 | 2 = | ( g 1 H / g 2 H ) + ( g 2 L / g 1 L ) 2 ( g 2 H / g 1 H ) + ( g 1 L / g 2 L ) + 2 | 2 ,
T = [ 1.000106 2.91 × 10 2 1.13 × 10 6 0 . 9 9 9 8 9 4 ] .

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