Abstract

The numerical modeling of a directional coupler composed of four identical slab waveguides reveals that the ratio of the field extrema of the four propagating modes is close to the golden ratio ϕ. This is confirmed by an analytical derivation leading to the exact dispersion equation of the eigenmodes of the four-waveguide coupler and to the expression of the ratio between field extrema. A coupled wave analysis gives the analytical expression of the propagation constants of the coupler eigenmodes and of the ratio between extrema in terms of the coupling coefficients between waveguides; it shows that this ratio is strictly equal to ϕ in weakly coupled structures where the coupling coefficient between next-nearest neighboring waveguides is negligible.

© 2017 Optical Society of America

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References

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  1. R. Herz-Fischler, A Mathematical History of Division in Extreme and Mean Ratio (Wilfrid Laurier University, 1987).
  2. M. Livio, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number (Broadway, 2002).
  3. M. Neveux and H. E. Huntley, Le nombre d’or: radiographie d’un mythe (Editions du Seuil, Points Sciences, 1995).
  4. P. Hemenway, Divine Proportion: ϕ Phi in Art, Nature, and Science (Sterling Publishing, 2005).
  5. P. Roth and G. Voirin, “Integrated optic coupler for interferometric mixer,” Proc. SPIE 1014, 35–41 (1989).
    [Crossref]
  6. D. Michel, O. Parriaux, and G. Voirin, “Defraction photoelectric position measuring system,” U.S. patent4,923,300 (May 8, 1990).
  7. O. Parriaux, “Guided wave electromagnetism and opto-chemical sensors,” in Fiber Optic Chemical Sensors and Biosensors, O. S. Wolfbeis, ed. (CRC Press, 1991).
  8. N. Lyndin, https://mcgrating.com .
  9. G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
    [Crossref]
  10. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125–127 (1984).
    [Crossref]
  11. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
    [Crossref]
  12. C. M. Moorman and J. E. Goff, “Golden ratio in a coupled-oscillator problem,” Eur. J. Phys. 28, 897–902 (2007).
    [Crossref]
  13. W. Givens, “Computation of plane unitary rotations transforming a general matrix to triangular form,” J. Soc. Ind. Appl. Math. 6, 26–50 (1958).
    [Crossref]

2007 (1)

C. M. Moorman and J. E. Goff, “Golden ratio in a coupled-oscillator problem,” Eur. J. Phys. 28, 897–902 (2007).
[Crossref]

1989 (1)

P. Roth and G. Voirin, “Integrated optic coupler for interferometric mixer,” Proc. SPIE 1014, 35–41 (1989).
[Crossref]

1985 (1)

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[Crossref]

1984 (1)

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

1958 (1)

W. Givens, “Computation of plane unitary rotations transforming a general matrix to triangular form,” J. Soc. Ind. Appl. Math. 6, 26–50 (1958).
[Crossref]

Givens, W.

W. Givens, “Computation of plane unitary rotations transforming a general matrix to triangular form,” J. Soc. Ind. Appl. Math. 6, 26–50 (1958).
[Crossref]

Goff, J. E.

C. M. Moorman and J. E. Goff, “Golden ratio in a coupled-oscillator problem,” Eur. J. Phys. 28, 897–902 (2007).
[Crossref]

Golubenko, G. A.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[Crossref]

Hemenway, P.

P. Hemenway, Divine Proportion: ϕ Phi in Art, Nature, and Science (Sterling Publishing, 2005).

Herz-Fischler, R.

R. Herz-Fischler, A Mathematical History of Division in Extreme and Mean Ratio (Wilfrid Laurier University, 1987).

Huntley, H. E.

M. Neveux and H. E. Huntley, Le nombre d’or: radiographie d’un mythe (Editions du Seuil, Points Sciences, 1995).

Kapon, E.

Katz, J.

Livio, M.

M. Livio, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number (Broadway, 2002).

Michel, D.

D. Michel, O. Parriaux, and G. Voirin, “Defraction photoelectric position measuring system,” U.S. patent4,923,300 (May 8, 1990).

Moorman, C. M.

C. M. Moorman and J. E. Goff, “Golden ratio in a coupled-oscillator problem,” Eur. J. Phys. 28, 897–902 (2007).
[Crossref]

Neveux, M.

M. Neveux and H. E. Huntley, Le nombre d’or: radiographie d’un mythe (Editions du Seuil, Points Sciences, 1995).

Parriaux, O.

D. Michel, O. Parriaux, and G. Voirin, “Defraction photoelectric position measuring system,” U.S. patent4,923,300 (May 8, 1990).

O. Parriaux, “Guided wave electromagnetism and opto-chemical sensors,” in Fiber Optic Chemical Sensors and Biosensors, O. S. Wolfbeis, ed. (CRC Press, 1991).

Roth, P.

P. Roth and G. Voirin, “Integrated optic coupler for interferometric mixer,” Proc. SPIE 1014, 35–41 (1989).
[Crossref]

Svakhin, A. S.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[Crossref]

Sychugov, V. A.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[Crossref]

Tishchenko, A. V.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[Crossref]

Voirin, G.

P. Roth and G. Voirin, “Integrated optic coupler for interferometric mixer,” Proc. SPIE 1014, 35–41 (1989).
[Crossref]

D. Michel, O. Parriaux, and G. Voirin, “Defraction photoelectric position measuring system,” U.S. patent4,923,300 (May 8, 1990).

Yariv, A.

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125–127 (1984).
[Crossref]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

Eur. J. Phys. (1)

C. M. Moorman and J. E. Goff, “Golden ratio in a coupled-oscillator problem,” Eur. J. Phys. 28, 897–902 (2007).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

J. Soc. Ind. Appl. Math. (1)

W. Givens, “Computation of plane unitary rotations transforming a general matrix to triangular form,” J. Soc. Ind. Appl. Math. 6, 26–50 (1958).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (1)

P. Roth and G. Voirin, “Integrated optic coupler for interferometric mixer,” Proc. SPIE 1014, 35–41 (1989).
[Crossref]

Sov. J. Quantum Electron. (1)

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[Crossref]

Other (7)

D. Michel, O. Parriaux, and G. Voirin, “Defraction photoelectric position measuring system,” U.S. patent4,923,300 (May 8, 1990).

O. Parriaux, “Guided wave electromagnetism and opto-chemical sensors,” in Fiber Optic Chemical Sensors and Biosensors, O. S. Wolfbeis, ed. (CRC Press, 1991).

N. Lyndin, https://mcgrating.com .

R. Herz-Fischler, A Mathematical History of Division in Extreme and Mean Ratio (Wilfrid Laurier University, 1987).

M. Livio, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number (Broadway, 2002).

M. Neveux and H. E. Huntley, Le nombre d’or: radiographie d’un mythe (Editions du Seuil, Points Sciences, 1995).

P. Hemenway, Divine Proportion: ϕ Phi in Art, Nature, and Science (Sterling Publishing, 2005).

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Figures (3)

Fig. 1.
Fig. 1.

Upper half of a symmetrical four-waveguide directional coupler. The optogeometrical parameters, as well as the field components tangent to the interfaces, are explicitly identified and expressed.

Fig. 2.
Fig. 2.

Transverse modal field profile of the four TE modes in arbitrary units. The mode excitation plane wave impinges on the coupler from the left.

Fig. 3.
Fig. 3.

TM field profile of the four TM modes with waveguide layers of 200 nm thickness.

Tables (1)

Tables Icon

Table 1. Ratio R12 between Field Extrema and Propagation Constants βm of the Coupler’s Eigenmodes of Order m under the Hypothesis of Negligible Coupling Coefficient c between Outer Waveguides and Coupling Coefficient a between Nearest Neighbors Much Larger than b between Next-Nearest Neighbors

Equations (31)

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ξg=ξs=1forTEmodes,andξg=1/ng2andξs=1/ns2forTMmodes.
ϕx(y)=Aiejkgy+Biejkgy,
ϕz(y)=(ξgkg/(ωμ0))(Aiejkgy+Biejkgy),
ϕx(y)=Aieksy+Bieksy,
ϕz(y)=(ξsks/(jωμ0))(Aieksy+Bieksy).
2rsin(kgw)cos(kgw)[(1+r2)(1+e2kss)2(1r2)e3kss]+sin2(kgw)[(1r4)(e2kss1)±(1+r2)2ekss(1r2)2e3kss]4r2cos2(kgw)e3kss=0,
r=ξgkg/(ξsks).
tan(2kgw)=1/r
cot(2kgw)=1/r
|ϕx(y)|2=1+|A3|2+2Re[A3e2jkgy]
A3=1r2+2jr1+r2ejkg(3s+4w);thus  |A3|=1.
2ηe=3s+4w+[tan1(2r1r2)+mπ]/kg.
|ϕx(ηe)|2=4.
|ϕx(y)|2=|B1|2+|A1|2+2Re[A1B1*e2jkgy],
A1=[(2jr+r21)ekss{(1r2)sin(kgw)+2rcos(kgw)}+(1+r2)2eksssin(kgw)]ejkg(3w+2s)2r(1+r2)
B1=[{(1r2)sin(kgw)+2rcos(kgw)}ekss+(r212jr)eksssin(kgw)]ejkg(w+s)2r.
2ηi=2w+s[tan1(2rr21)+mπ]/kg.
|ϕx(ηi)|2={(1r2)sin(kgw)[4rcos(kgw)(e2ks1)+(1r2)sin(kgw)(e2kss2)]+4r2e2ksscos2(kgw)+(1+r2)2e2ksssin2(kgw)}/r2.
dV(z)/dz=MV(z),
M=(jβabcajβabbajβacbajβ),
V(z)=exp(M)V(0).
R12=D++(ac)D+(ac)
R122=5a21+8ab5a2+a5a21+8ab5a2a5+151[12b5a].
R125+151[1b5a]=ϕ[1b5a].
βm=β±a+c±D±2.
β0=β+a+c+D+2.
β0β+1+52a+2b5=β+ϕa+2b5.
M=(jβabcajβabbajβacbajβ)=P1DP,
P1=12(D+(ac)D+D++(ac)D+D+(ac)DD(ac)DD++(ac)D+D+(ac)D+D(ac)DD+(ac)DD++(ac)D+D+(ac)D+D(ac)DD+(ac)DD+(ac)D+D++(ac)D+D+(ac)DD(ac)D),D=(jβ+(a+c)+D+20000jβ+(a+c)D+20000jβ(a+c)D20000jβ(a+c)+D2),P=12(D+(ac)D+D++(ac)D+D++(ac)D+D+(ac)D+D++(ac)D+D+(ac)D+D+(ac)D+D++(ac)D+D+(ac)DD(ac)DD(ac)DD+(ac)DD(ac)DD+(ac)DD+(ac)DD(ac)D),
exp(jβabcajβabbajβacbajβ)=P1DeP,
De=(exp[jβ+(a+c)+D+2]0000exp[jβ+(a+c)D+2]0000exp[jβ(a+c)D2]0000exp[jβ(a+c)+D2]).

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