Abstract

This study reports both analytical and numerical thermal–structural models of polymer Bragg grating (PBG) waveguides illuminated by a light emitting diode (LED). A polymethyl methacrylate (PMMA) Bragg grating (BG) waveguide is chosen as an analysis vehicle to explore parametric effects of incident optical powers and substrate materials on the thermal–structural behavior of the BG. Analytical models are verified by comparing analytically predicted average excess temperatures, and thermally induced axial strains and stresses with numerical predictions. A parametric study demonstrates that the PMMA substrate induces more adverse effects, such as higher excess temperatures, complex axial temperature profiles, and greater and more complicated thermally induced strains in the BG compared with the Si substrate.

© 2012 Optical Society of America

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References

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  1. K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
    [CrossRef]
  2. L. Eldada and L. Shacklette, “Advances in polymer integrated optics,” IEEE J. Sel. Top. Quantum Electron. 6, 54–68 (2000).
    [CrossRef]
  3. W. C. Wang, M. Fisher, A. Yacoubian, and J. Menders, “Phase-shifted Bragg grating filters in polymer waveguides,” IEEE Photon. Technol. Lett. 15, 548–550 (2003).
    [CrossRef]
  4. J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
    [CrossRef]
  5. H. Zou, K. W. Beeson, and L. W. Shacklette, “Tunable planar polymer Bragg gratings having exceptionally low polarization sensitivity,” J. Lightwave Technol. 21, 1083–1088 (2003).
    [CrossRef]
  6. A. Sato, S. Atsushi, and R. K. Kostuk, “Holographic edge-illuminated polymer Bragg gratings for dense wavelength division optical filters at 1550 nm,” Appl. Opt. 42, 778–784 (2003).
    [CrossRef]
  7. R. Lausten, P. Rochon, M. Ivanov, P. Cheben, S. Janz, P. Desjardins, J. Ripmeester, T. Siebert, and A. Stolow, “Optically reconfigurable azobenzene polymer-based fiber Bragg filter,” Appl. Opt. 44, 7039–7042 (2005).
    [CrossRef]
  8. K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of an intrinsically heated polymer fiber Bragg grating,” Appl. Opt. 46, 4357–4370 (2007).
    [CrossRef]
  9. K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of polymer fiber Bragg grating illuminated by light emitting diode,” Int. J. Heat Mass Transfer 50, 5241–5248 (2007).
    [CrossRef]
  10. C.-L. Chen, Elements of Optoelectronics and Fiber Optics(Irwin, 1996).
  11. K. J. Kim, “Thermo-structural influences on optical characteristics of polymer Bragg gratings,” Ph.D. dissertation (University of Maryland, 2006).
  12. A. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment (Hemisphere, 1983).
  13. E. Suhir, “An approximate analysis of stresses in multilayered elastic thin films,” J. Appl. Mech. 55, 143–148 (1988).
    [CrossRef]
  14. S. Moaveni, Finite Element Analysis-Theory and Application with ANSYS (Prentice Hall, 2002).
  15. M. Ozisik, Heat Conduction (Wiley, 1993).
  16. ANSYS 7.0 Manual (ANSYS Inc., 2003).
  17. B. Boley and J. Weiner, Theory of Thermal Stresses (Wiley, 1960).
  18. M. Zhou, “Low-loss polymeric materials for passive waveguide components in fiber optical communication,” Opt. Eng. 41, 1631–1643 (2002).
    [CrossRef]
  19. M. Weber, CRC Handbook of Laser Science and Technology Supplement 2: Optical Materials (CRC, 1995).

2007

K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of an intrinsically heated polymer fiber Bragg grating,” Appl. Opt. 46, 4357–4370 (2007).
[CrossRef]

K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of polymer fiber Bragg grating illuminated by light emitting diode,” Int. J. Heat Mass Transfer 50, 5241–5248 (2007).
[CrossRef]

2005

2003

W. C. Wang, M. Fisher, A. Yacoubian, and J. Menders, “Phase-shifted Bragg grating filters in polymer waveguides,” IEEE Photon. Technol. Lett. 15, 548–550 (2003).
[CrossRef]

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
[CrossRef]

H. Zou, K. W. Beeson, and L. W. Shacklette, “Tunable planar polymer Bragg gratings having exceptionally low polarization sensitivity,” J. Lightwave Technol. 21, 1083–1088 (2003).
[CrossRef]

A. Sato, S. Atsushi, and R. K. Kostuk, “Holographic edge-illuminated polymer Bragg gratings for dense wavelength division optical filters at 1550 nm,” Appl. Opt. 42, 778–784 (2003).
[CrossRef]

2002

M. Zhou, “Low-loss polymeric materials for passive waveguide components in fiber optical communication,” Opt. Eng. 41, 1631–1643 (2002).
[CrossRef]

2000

L. Eldada and L. Shacklette, “Advances in polymer integrated optics,” IEEE J. Sel. Top. Quantum Electron. 6, 54–68 (2000).
[CrossRef]

1997

K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

1988

E. Suhir, “An approximate analysis of stresses in multilayered elastic thin films,” J. Appl. Mech. 55, 143–148 (1988).
[CrossRef]

Atsushi, S.

Bar-Cohen, A.

K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of an intrinsically heated polymer fiber Bragg grating,” Appl. Opt. 46, 4357–4370 (2007).
[CrossRef]

K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of polymer fiber Bragg grating illuminated by light emitting diode,” Int. J. Heat Mass Transfer 50, 5241–5248 (2007).
[CrossRef]

A. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment (Hemisphere, 1983).

Beeson, K. W.

Boley, B.

B. Boley and J. Weiner, Theory of Thermal Stresses (Wiley, 1960).

Cheben, P.

Chen, C.-L.

C.-L. Chen, Elements of Optoelectronics and Fiber Optics(Irwin, 1996).

Desjardins, P.

Eldada, L.

L. Eldada and L. Shacklette, “Advances in polymer integrated optics,” IEEE J. Sel. Top. Quantum Electron. 6, 54–68 (2000).
[CrossRef]

Fisher, M.

W. C. Wang, M. Fisher, A. Yacoubian, and J. Menders, “Phase-shifted Bragg grating filters in polymer waveguides,” IEEE Photon. Technol. Lett. 15, 548–550 (2003).
[CrossRef]

Han, B.

K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of polymer fiber Bragg grating illuminated by light emitting diode,” Int. J. Heat Mass Transfer 50, 5241–5248 (2007).
[CrossRef]

K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of an intrinsically heated polymer fiber Bragg grating,” Appl. Opt. 46, 4357–4370 (2007).
[CrossRef]

Hill, K. O.

K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Ivanov, M.

Janz, S.

Kang, J.-W.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
[CrossRef]

Kim, D. Y.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
[CrossRef]

Kim, J.-J.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
[CrossRef]

Kim, J.-P.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
[CrossRef]

Kim, K. J.

K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of polymer fiber Bragg grating illuminated by light emitting diode,” Int. J. Heat Mass Transfer 50, 5241–5248 (2007).
[CrossRef]

K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of an intrinsically heated polymer fiber Bragg grating,” Appl. Opt. 46, 4357–4370 (2007).
[CrossRef]

K. J. Kim, “Thermo-structural influences on optical characteristics of polymer Bragg gratings,” Ph.D. dissertation (University of Maryland, 2006).

Kim, M.-J.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
[CrossRef]

Kostuk, R. K.

Kraus, A.

A. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment (Hemisphere, 1983).

Lausten, R.

Lee, J.-S.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
[CrossRef]

Meltz, G.

K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Menders, J.

W. C. Wang, M. Fisher, A. Yacoubian, and J. Menders, “Phase-shifted Bragg grating filters in polymer waveguides,” IEEE Photon. Technol. Lett. 15, 548–550 (2003).
[CrossRef]

Moaveni, S.

S. Moaveni, Finite Element Analysis-Theory and Application with ANSYS (Prentice Hall, 2002).

Ozisik, M.

M. Ozisik, Heat Conduction (Wiley, 1993).

Ripmeester, J.

Rochon, P.

Sato, A.

Shacklette, L.

L. Eldada and L. Shacklette, “Advances in polymer integrated optics,” IEEE J. Sel. Top. Quantum Electron. 6, 54–68 (2000).
[CrossRef]

Shacklette, L. W.

Siebert, T.

Stolow, A.

Suhir, E.

E. Suhir, “An approximate analysis of stresses in multilayered elastic thin films,” J. Appl. Mech. 55, 143–148 (1988).
[CrossRef]

Wang, W. C.

W. C. Wang, M. Fisher, A. Yacoubian, and J. Menders, “Phase-shifted Bragg grating filters in polymer waveguides,” IEEE Photon. Technol. Lett. 15, 548–550 (2003).
[CrossRef]

Weber, M.

M. Weber, CRC Handbook of Laser Science and Technology Supplement 2: Optical Materials (CRC, 1995).

Weiner, J.

B. Boley and J. Weiner, Theory of Thermal Stresses (Wiley, 1960).

Yacoubian, A.

W. C. Wang, M. Fisher, A. Yacoubian, and J. Menders, “Phase-shifted Bragg grating filters in polymer waveguides,” IEEE Photon. Technol. Lett. 15, 548–550 (2003).
[CrossRef]

Yoo, S.-J.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
[CrossRef]

Zhou, M.

M. Zhou, “Low-loss polymeric materials for passive waveguide components in fiber optical communication,” Opt. Eng. 41, 1631–1643 (2002).
[CrossRef]

Zou, H.

Appl. Opt.

Appl. Phys. Lett.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, “Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,” Appl. Phys. Lett. 82, 3823–3825 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

L. Eldada and L. Shacklette, “Advances in polymer integrated optics,” IEEE J. Sel. Top. Quantum Electron. 6, 54–68 (2000).
[CrossRef]

IEEE Photon. Technol. Lett.

W. C. Wang, M. Fisher, A. Yacoubian, and J. Menders, “Phase-shifted Bragg grating filters in polymer waveguides,” IEEE Photon. Technol. Lett. 15, 548–550 (2003).
[CrossRef]

Int. J. Heat Mass Transfer

K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of polymer fiber Bragg grating illuminated by light emitting diode,” Int. J. Heat Mass Transfer 50, 5241–5248 (2007).
[CrossRef]

J. Appl. Mech.

E. Suhir, “An approximate analysis of stresses in multilayered elastic thin films,” J. Appl. Mech. 55, 143–148 (1988).
[CrossRef]

J. Lightwave Technol.

Opt. Eng.

M. Zhou, “Low-loss polymeric materials for passive waveguide components in fiber optical communication,” Opt. Eng. 41, 1631–1643 (2002).
[CrossRef]

Other

M. Weber, CRC Handbook of Laser Science and Technology Supplement 2: Optical Materials (CRC, 1995).

S. Moaveni, Finite Element Analysis-Theory and Application with ANSYS (Prentice Hall, 2002).

M. Ozisik, Heat Conduction (Wiley, 1993).

ANSYS 7.0 Manual (ANSYS Inc., 2003).

B. Boley and J. Weiner, Theory of Thermal Stresses (Wiley, 1960).

C.-L. Chen, Elements of Optoelectronics and Fiber Optics(Irwin, 1996).

K. J. Kim, “Thermo-structural influences on optical characteristics of polymer Bragg gratings,” Ph.D. dissertation (University of Maryland, 2006).

A. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment (Hemisphere, 1983).

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Figures (10)

Fig. 1.
Fig. 1.

Structure of a polymer Bragg grating waveguide.

Fig. 2.
Fig. 2.

Thermal resistance model of a polymer Bragg grating waveguide.

Fig. 3.
Fig. 3.

Schematic of the thermal–structural model of a polymer Bragg grating waveguide.

Fig. 4.
Fig. 4.

Flowchart of 3D FEA analysis.

Fig. 5.
Fig. 5.

(a) 3D FEA models (global and local) of a waveguide, (b) boundary conditions of 3D FEA local models.

Fig. 6.
Fig. 6.

Temperature fields of (a) the entire waveguide and (b) the Bragg grating in the waveguide core on a PMMA substrate at an incident optical power of 0.3 W and an ambient temperature of 25 °C.

Fig. 7.
Fig. 7.

Excess temperatures in the PMMA Bragg gratings, associated with PMMA and Si substrates, as well as waveguide cores for various incident optical powers ranging from 0.01 to 0.3 W (Tamb=25°C). The Bragg grating is inscribed from z=0.5cm to z=1.5cm along the waveguide core.

Fig. 8.
Fig. 8.

Average excess temperatures in PMMA Bragg gratings and waveguide substrates for various optical powers from 0.01 to 0.3 W (Tamb=25°C).

Fig. 9.
Fig. 9.

Axial strains in PMMA Bragg gratings and waveguide cores associated with (a) Si and (b) PMMA substrates. The Bragg grating is inscribed from z=0.5cm to z=1.5cm along the waveguide core.

Fig. 10.
Fig. 10.

Axial stresses in Bragg gratings and waveguide cores associated with (a) Si and (b) PMMA substrates. The Bragg grating is inscribed from z=0.5cm to z=1.5cm along the waveguide core.

Tables (1)

Tables Icon

Table 1. Properties and Geometry of a PMMA Bragg Grating Waveguide

Equations (27)

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I(r)=2Pincπw2e2r2/w2,
qG=I(r)ea^za^,
q=VqGdV=VI(r)ea^za^dV=0LcPincea^za^dz=Pinc(1ea^Lc),
Rcs=1πLsksln[1sin(πrc/2rs)],
Rsa=1/(hAs),
Rcla=Rcs+Rsa.
Rcua=RcTop+RTopa,
RcTop=ΔxcTkcAc,
RTopa=1hAc.
RT=(Rcua1+Rcla1)1.
ΔTc=qRT.
ΔTs=qsa·Rsa,
εf=dufdz=αfΔTf+[(γfψfβ2)cosh(βz)cosh(βLs/2)γf]×Eoftf[αfΔTfαsΔTs],
β=γ/ψ,
γ=γf+γs=1Eoftf+1Eosts,
ψ=ψf+ψs,
ψf=2/3(1+νf)/(1νf)tf/Eof,
ψs=2/3(1+νs)/(1νs)ts/Eos.
σf(z)=Eof[αfΔTfαsΔTs][1cosh(βz)cosh(βLs/2)].
x(kTx)+y(kTy)+z(kTz)+qG=0,
qG=2Pincπωo·a^·exp[a^·z2(x2+y2)ωo2],
σxx=μe+2Gεxx(3μ+2G)αΔT,
e=εxx+εyy+εzz,
μ=νE(1+ν)(12ν).
σyy=μe+2Gεyy(3μ+2G)αΔT,
σzz=μe+2Gεzz(3μ+2G)αΔT.
σxy=2Gεxy;σyz=2Gεyz;σzx=2Gεzx.

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