Abstract

We present a novel matrix approach to proving that the phase shift at a turning point in a planar optical waveguide is exactly equal to π rather than to π/2 or to some other value. We also show the existence of phase contributions from reflected subwaves, which to our knowledge have never been taken into account previously.

© 2001 Optical Society of America

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References

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  1. V. S. Popov, B. M. Karnakov, V. D. Mur, “On matching conditions in the WKB method,” Phys. Lett. A 210, 402–408 (1996).
    [CrossRef]
  2. S. Zivanovic, V. Milanovic, Z. Ikonic, “A modified WKB method for graded semiconductor quantum wells,” Phys. Status Solidi B 204, 713–719 (1997).
    [CrossRef]
  3. H. Friedrich, J. Trost, “Nonintegral Maslov indices,” Phys. Rev. 54, 1136–1145 (1996).
    [CrossRef]
  4. D. Marcuse, “Elementary derivation of the phase shift at a caustic,” Appl. Opt. 15, 2949–2950 (1976).
    [CrossRef] [PubMed]
  5. G. B. Hocker, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
    [CrossRef]
  6. Z. Cao, Y. Jiang, Q. Shen, X. Dou, Y. Chen, “Exact analytical method for planar optical waveguides with arbitrary index profile,” J. Opt. Soc. Am. A 16, 2209–2212 (1999).
    [CrossRef]
  7. M. J. Adams, An Introduction to Optical Waveguides (Vail-Ballou, Binghamton, New York, 1981), pp. 75–77.

1999

1997

S. Zivanovic, V. Milanovic, Z. Ikonic, “A modified WKB method for graded semiconductor quantum wells,” Phys. Status Solidi B 204, 713–719 (1997).
[CrossRef]

1996

H. Friedrich, J. Trost, “Nonintegral Maslov indices,” Phys. Rev. 54, 1136–1145 (1996).
[CrossRef]

V. S. Popov, B. M. Karnakov, V. D. Mur, “On matching conditions in the WKB method,” Phys. Lett. A 210, 402–408 (1996).
[CrossRef]

1976

1975

G. B. Hocker, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Vail-Ballou, Binghamton, New York, 1981), pp. 75–77.

Cao, Z.

Chen, Y.

Dou, X.

Friedrich, H.

H. Friedrich, J. Trost, “Nonintegral Maslov indices,” Phys. Rev. 54, 1136–1145 (1996).
[CrossRef]

Hocker, G. B.

G. B. Hocker, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[CrossRef]

Ikonic, Z.

S. Zivanovic, V. Milanovic, Z. Ikonic, “A modified WKB method for graded semiconductor quantum wells,” Phys. Status Solidi B 204, 713–719 (1997).
[CrossRef]

Jiang, Y.

Karnakov, B. M.

V. S. Popov, B. M. Karnakov, V. D. Mur, “On matching conditions in the WKB method,” Phys. Lett. A 210, 402–408 (1996).
[CrossRef]

Marcuse, D.

Milanovic, V.

S. Zivanovic, V. Milanovic, Z. Ikonic, “A modified WKB method for graded semiconductor quantum wells,” Phys. Status Solidi B 204, 713–719 (1997).
[CrossRef]

Mur, V. D.

V. S. Popov, B. M. Karnakov, V. D. Mur, “On matching conditions in the WKB method,” Phys. Lett. A 210, 402–408 (1996).
[CrossRef]

Popov, V. S.

V. S. Popov, B. M. Karnakov, V. D. Mur, “On matching conditions in the WKB method,” Phys. Lett. A 210, 402–408 (1996).
[CrossRef]

Shen, Q.

Trost, J.

H. Friedrich, J. Trost, “Nonintegral Maslov indices,” Phys. Rev. 54, 1136–1145 (1996).
[CrossRef]

Zivanovic, S.

S. Zivanovic, V. Milanovic, Z. Ikonic, “A modified WKB method for graded semiconductor quantum wells,” Phys. Status Solidi B 204, 713–719 (1997).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

G. B. Hocker, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Lett. A

V. S. Popov, B. M. Karnakov, V. D. Mur, “On matching conditions in the WKB method,” Phys. Lett. A 210, 402–408 (1996).
[CrossRef]

Phys. Rev.

H. Friedrich, J. Trost, “Nonintegral Maslov indices,” Phys. Rev. 54, 1136–1145 (1996).
[CrossRef]

Phys. Status Solidi B

S. Zivanovic, V. Milanovic, Z. Ikonic, “A modified WKB method for graded semiconductor quantum wells,” Phys. Status Solidi B 204, 713–719 (1997).
[CrossRef]

Other

M. J. Adams, An Introduction to Optical Waveguides (Vail-Ballou, Binghamton, New York, 1981), pp. 75–77.

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Figures (2)

Fig. 1
Fig. 1

Four-layer index waveguide.

Fig. 2
Fig. 2

Planar waveguide with arbitrary index profile.

Tables (1)

Tables Icon

Table 1 Results of Numerical Calculations of Φ(r) for V=2.0

Equations (42)

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κ1d1=mπ+tan-1(P0/κ1)+tan-1(P2/κ1)
(m=0, 1, 2,),
P2=κ2tan[tan-1(P3/κ2)-κ2d2],
Pj=(β2-k02nj2)1/2(j=0, 3),
κj=(k02nj2-β2)1/2(j=1, 2).
Φ2=tan-1(P2/κ2)
=mπ+tan-1(P3/κ2)-κ2d2
(m=0, 1, 2,);
tan-1(P2/κ1)=tan-1(κ2/κ1tan Φ2).
(κ1d1+κ2d2)+Φ(r)=mπ+tan-1(P0/κ1)+tan-1(P3/κ2),
Φ(r)=Φ2-tan-1(κ2/κ1tan Φ2).
Mi=cos(κih)-1/κisin(κih)κisin(κih)cos(κih)(i=1, 2,,l),
Mj=cosh(αjh)-1/αjsinh(αih)-αjsinh(αjh)cosh(αjh)
(j=l+1,l+2,,l+m),
κi=[k02n2(xi)-β2]1/2,
αj=[β2-k02n2(xj)]1/2.
Ey(0)Ey(0)=i=1lMi j=l+1l+mMj Ey(xc)Ey(xc).
Ey(x)=Acexp[-Pc(x-xc)](x>xc),
Pc=(β2-k02nc2)1/2.
[-P01]i=1lMi j=l+1l+mMj 1-Pc=0.
[-P01]i=1lMi 1-Pt=0,
Pt=Pl+1,
Pj=αjsinh(αjh)+Pj+1/αjcosh(αjh)cosh(αjh)+Pj+1/αjsinh(αjh),(j=l+1,l+2,,l+m),
Pl+m+1=Pc.
Ey(x)=Atexp[-Pt(x-xt)](x>xt).
Pt=(β2-k02neq2)1/2.
Pl+1>αl+1.
Pj=Pj+1cosh(αjh)+αj/Pj+1sinh(αjh)cosh(αjh)+Pj+1/αjsinh(αjh).
Pj<Pj+1,
Pt=Pl+1<Pl+m+1=Pc.
αl+1<Pt<Pc.
nc<neq<n(xl+1).
0xt κ(x)dx+Φ(r)=mπ+tan-1(P0/κ1)+tan-1(Pt/κl),
Φ(r)=i=1l-1[Φi+1-tan-1(κi+1/κitan Φi+1)],
Φi=tan-1(Pi/κi).
κl=[κ02n2(xl)-β2]1/2[k02n2(xt)-β2]1/2=0,
tan-1(Pt/κl)=tan-1β2-k02neq2k02n2(xl)-β21/2π2
(l).
tan-1β2-k02n2(xl+1)k02n2(xl)-β21/2π4(l).
0xtκ(x)dx+Φ(r)=mπ+tan-1P0κ1+π2.
0xt κ(x)dx=mπ+tan-1P0κ1+π4.

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