Abstract

In this paper, we model and experimentally observe the far-field radiation produced by interfering beams propagating in two-dimensional (2D) slab waveguides. Using a transmission-line analogy, we compare the 2D propagation with standard three-dimensional (3D) far-field representations and derive the 2D conditions for using standard far-field approximations. Then we test our theoretical results by experimentally observing the 2D far-field pattern produced by a 1×3 multimode interference (MMI) coupler on a silicon nanomembrane. The MMI outputs are connected to a slab silicon waveguide, and the far field is observed at the edge of the silicon slab. This represents the observation of 2D far-field pattern produced by an array of on-chip radiators.

© 2011 Optical Society of America

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  1. M. Jarrahi, R. F. W. Pease, D. A. B. Miller, and T. H. Lee, “Optical spatial quantization for higher performance analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 56, 2143–2150 (2008).
    [CrossRef]
  2. R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429. (2007).
    [CrossRef]
  3. A. W. Lohmann, A. Pe’er, D. Wang, and A. A. Friesem, “Flatland optics: fundamentals,” J. Opt. Soc. Am. A 17, 1755–1762 (2000).
    [CrossRef]
  4. A. W. Lohmann, D. Wang, A. Pe’er, and A. A. Friesem, “Flatland optics. II. basic experiments,” J. Opt. Soc. Am. A 18, 1056–1061 (2001).
    [CrossRef]
  5. A. W. Lohmann, A. Pe’er, D. Wang, and A. A. Friesem, “Flatland optics. III. Achromatic diffraction,” J. Opt. Soc. Am. A 18, 2095–2097.
    [CrossRef]
  6. A. Sommerfeld, Partial Differential Equations in Physics(Academic, 1949).
  7. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).
  8. C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).
  9. A. Alù and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. B 23, 571–583 (2006).
    [CrossRef]
  10. A. Hosseini, H. Subbaraman, D. Kwong, Y. Zhang, and R. T. Chen, “Optimum access waveguide width for 1×N multimode interference couplers on silicon nanomembrane,” Opt. Lett. 35, 2864–2866 (2010).
    [CrossRef] [PubMed]
  11. E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. 33, 1299–1307 (1997).
    [CrossRef]
  12. J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
    [CrossRef]
  13. A. Hosseini, D. N. Kwong, C.-Y. Lin, B. S. Lee, and R. T. Chen, “Output formulation for symmetrically excited one-to-Nmultimode interference coupler,” IEEE J. Sel. Top. Quant. Electron. 6, 53–60 (2010).
    [CrossRef]

2010

A. Hosseini, D. N. Kwong, C.-Y. Lin, B. S. Lee, and R. T. Chen, “Output formulation for symmetrically excited one-to-Nmultimode interference coupler,” IEEE J. Sel. Top. Quant. Electron. 6, 53–60 (2010).
[CrossRef]

A. Hosseini, H. Subbaraman, D. Kwong, Y. Zhang, and R. T. Chen, “Optimum access waveguide width for 1×N multimode interference couplers on silicon nanomembrane,” Opt. Lett. 35, 2864–2866 (2010).
[CrossRef] [PubMed]

2008

M. Jarrahi, R. F. W. Pease, D. A. B. Miller, and T. H. Lee, “Optical spatial quantization for higher performance analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 56, 2143–2150 (2008).
[CrossRef]

2007

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429. (2007).
[CrossRef]

2006

2001

2000

1999

J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
[CrossRef]

1997

E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. 33, 1299–1307 (1997).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Alù, A.

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).

Brongersma, M. L.

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429. (2007).
[CrossRef]

Chen, R. T.

A. Hosseini, H. Subbaraman, D. Kwong, Y. Zhang, and R. T. Chen, “Optimum access waveguide width for 1×N multimode interference couplers on silicon nanomembrane,” Opt. Lett. 35, 2864–2866 (2010).
[CrossRef] [PubMed]

A. Hosseini, D. N. Kwong, C.-Y. Lin, B. S. Lee, and R. T. Chen, “Output formulation for symmetrically excited one-to-Nmultimode interference coupler,” IEEE J. Sel. Top. Quant. Electron. 6, 53–60 (2010).
[CrossRef]

Donnelly, J. P.

E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. 33, 1299–1307 (1997).
[CrossRef]

Engheta, N.

Friesem, A. A.

Heaton, J. M.

J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
[CrossRef]

Hosseini, A.

A. Hosseini, H. Subbaraman, D. Kwong, Y. Zhang, and R. T. Chen, “Optimum access waveguide width for 1×N multimode interference couplers on silicon nanomembrane,” Opt. Lett. 35, 2864–2866 (2010).
[CrossRef] [PubMed]

A. Hosseini, D. N. Kwong, C.-Y. Lin, B. S. Lee, and R. T. Chen, “Output formulation for symmetrically excited one-to-Nmultimode interference coupler,” IEEE J. Sel. Top. Quant. Electron. 6, 53–60 (2010).
[CrossRef]

Jarrahi, M.

M. Jarrahi, R. F. W. Pease, D. A. B. Miller, and T. H. Lee, “Optical spatial quantization for higher performance analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 56, 2143–2150 (2008).
[CrossRef]

Jenkins, R. M.

J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
[CrossRef]

Kwong, D.

Kwong, D. N.

A. Hosseini, D. N. Kwong, C.-Y. Lin, B. S. Lee, and R. T. Chen, “Output formulation for symmetrically excited one-to-Nmultimode interference coupler,” IEEE J. Sel. Top. Quant. Electron. 6, 53–60 (2010).
[CrossRef]

Lee, B. S.

A. Hosseini, D. N. Kwong, C.-Y. Lin, B. S. Lee, and R. T. Chen, “Output formulation for symmetrically excited one-to-Nmultimode interference coupler,” IEEE J. Sel. Top. Quant. Electron. 6, 53–60 (2010).
[CrossRef]

Lee, T. H.

M. Jarrahi, R. F. W. Pease, D. A. B. Miller, and T. H. Lee, “Optical spatial quantization for higher performance analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 56, 2143–2150 (2008).
[CrossRef]

Lin, C.-Y.

A. Hosseini, D. N. Kwong, C.-Y. Lin, B. S. Lee, and R. T. Chen, “Output formulation for symmetrically excited one-to-Nmultimode interference coupler,” IEEE J. Sel. Top. Quant. Electron. 6, 53–60 (2010).
[CrossRef]

Lohmann, A. W.

Miller, D. A. B.

M. Jarrahi, R. F. W. Pease, D. A. B. Miller, and T. H. Lee, “Optical spatial quantization for higher performance analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 56, 2143–2150 (2008).
[CrossRef]

Molter, L. A.

E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. 33, 1299–1307 (1997).
[CrossRef]

Pe’er, A.

Pease, R. F. W.

M. Jarrahi, R. F. W. Pease, D. A. B. Miller, and T. H. Lee, “Optical spatial quantization for higher performance analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 56, 2143–2150 (2008).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics(Academic, 1949).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Subbaraman, H.

Thoen, E. R.

E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. 33, 1299–1307 (1997).
[CrossRef]

Wang, D.

Zhang, Y.

Zia, R.

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429. (2007).
[CrossRef]

IEEE J. Quantum Electron.

E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. 33, 1299–1307 (1997).
[CrossRef]

IEEE J. Sel. Top. Quant. Electron.

A. Hosseini, D. N. Kwong, C.-Y. Lin, B. S. Lee, and R. T. Chen, “Output formulation for symmetrically excited one-to-Nmultimode interference coupler,” IEEE J. Sel. Top. Quant. Electron. 6, 53–60 (2010).
[CrossRef]

IEEE Photon. Technol. Lett.

J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

M. Jarrahi, R. F. W. Pease, D. A. B. Miller, and T. H. Lee, “Optical spatial quantization for higher performance analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 56, 2143–2150 (2008).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nat. Nanotechnol.

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429. (2007).
[CrossRef]

Opt. Lett.

Other

A. Sommerfeld, Partial Differential Equations in Physics(Academic, 1949).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).

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Figures (5)

Fig. 1
Fig. 1

(a) Geometry of the silicon slab waveguide. (b) Schematic of the waveguide array connected to a large slab waveguide.

Fig. 2
Fig. 2

(a) Schematic of the 1 × 3 MMI used for the far-field test. The inset shows the cross section of the waveguiding structure. Access waveguide width, W w = 2 μm ; MMI width, W MMI = 9.3 μm ; MMI length, L MMI = 52.5 μm ; MMI thickness, 2 h = 0.23 μm . (b) Eigenmode-decomposition-based simulation of the designed 1 × 3 MMI. (c) Calculated output uniformity and insertion loss of the designed MMI.

Fig. 3
Fig. 3

SEM images of (a)  1 × 3 MMI coupler, (b)  1 × 3 MMI coupler output, (c) output waveguides’ width tapers, and (d) MMI output connections to the slab waveguide.

Fig. 4
Fig. 4

Optical test setup. The slab waveguide region on the chip is 8.0 mm long and is indicated by dashed lines.

Fig. 5
Fig. 5

Two-dimensional far-field pattern of a 1 × 3 MMI with 3.1 μm separation between the output waveguides.

Equations (19)

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f ( z ) = { A cos ( β z h ) exp [ α ( z h ) ] z > h A cos ( β z z ) | z | h A cos ( β z h ) exp [ α ( z + h ) ] z < h ,
( h ε μ ω 2 β ρ 2 ) tan ( h ε μ ω 2 β ρ 2 ) = i h ε c μ ω 2 β ρ 2 , where β ρ 2 = β x 2 + β y 2 ,
E y = j 2 β x f ( z ) exp ( j β y y ) exp ( j β x x ) .
E y = j 4 π β x f ( z ) exp ( j β x x ) exp ( j β y y ) d β y .
H 0 ( 2 ) ( β ρ ρ , φ ) = 1 π π / 2 j 3 π / 2 + j exp [ j β ρ ρ cos ( ζ φ ) ] d ζ ,
1 β x exp ( j β x x ) exp ( j β y y ) d β y = π H 0 ( 2 ) ( β ρ ρ ) .
E y = j 4 π β x f ( z ) exp ( j β x x ) exp ( j β y y ) d β y = j 4 f ( z ) H 0 ( 2 ) ( β ρ ρ ) .
H 0 ( 2 ) ( β ρ ρ ) 2 j π β ρ ρ exp ( j β ρ ρ ) for     β ρ ρ 0 ,
ρ = | ρ ¯ ρ ¯ | | ρ ¯ | | ρ ¯ | cos ( φ φ ) + | ρ ¯ | 2 2 | ρ ¯ | | ρ ¯ | | ρ ¯ | cos ( φ φ ) for     β ρ | ρ ¯ | 2 2 | ρ ¯ | 1 .
d V d x = E y x | z = 0 = j ω μ H z | z = 0 = j ω μ I d I d x = H z x | z = 0 = j ω ε E y | z = 0 H x z | z = 0 = j ω ε eff V ,
ε eff = ε H x z | z = 0 j ω ε E y | z = 0 .
β ρ 2 = ω 2 μ ε eff .
n = 1 N W e / 2 W e / 2 j A n 4 π exp ( j ϕ n ) H 0 ( 2 ) ( ω μ ε eff | ρ ¯ d ¯ n y | ) d y ,
H 0 ( 2 ) ( β ρ ρ , ϕ ) = 1 π π / 2 j 3 π / 2 + j exp [ j β ρ ρ cos ( ζ φ ) ] d ζ .
H 0 ( 2 ) ( β ρ ρ , φ ) = 1 π 0 j π + j exp [ j β ρ ρ sin ( ζ φ ) ] d ζ = 1 π 0 j π + j exp [ j β ρ ρ sin ( ζ ) cos ( φ ) ] exp [ j β ρ ρ cos ( ζ ) sin ( φ ) ] d ζ = 1 π 0 j π + j exp [ j β ρ x sin ( ζ ) ] exp [ j β ρ y cos ( ζ ) ] d ζ .
H 0 ( 2 ) ( β ρ ρ , φ ) = 1 π ς = 0 j ζ = π + j exp [ j β ρ x sin ( ζ ) ] exp [ j β ρ y cos ( ζ ) ] 1 β ρ 2 β x 2 d β x ,
π < ξ < π + j j < β x < 0 j and β ρ < β y < , π / 2 < ξ < π 0 < β x < β ρ and 0 < β y < β ρ , 0 < ξ < π / 2 0 < β x < β ρ and β ρ < β y < 0 , j < ξ < 0 j < β x < 0 j and < β y < β ρ .
H 0 ( 2 ) ( β ρ ρ , φ ) = 1 π β y = β y = exp [ j β x x ] exp [ j β y y ] 1 β ρ 2 β x 2 d β x .
1 β y exp ( j β x x ) exp ( j β y y ) d β x = π H 0 ( 2 ) ( β ρ ρ ) .

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