Abstract

We analyze a new method for determining the out-of-plane coefficient of thermal expansion for thin films of transparent materials. The method is based on the measurement of interference peaks recorded in transmission spectra as a function of sample temperature. The locations of interference peaks depend on the optical path. We show how a consideration of Lorentz-Lorenz equation, in addition to the transmission peak equation, can separate the different contributions of index and physical pathlength to the optical path. The analysis is generalized to include the effects of uniaxial material properties (such as anisotropic linear thermal expansion and birefringence). By applying this method of analysis to recent data, we demonstrate the importance of including the effect of the thermo-optic coefficient in interpreting observed data.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. Boese, H. Lee, D. Y. Yoon, J. D. Swalen, and J. F. Rabolt, “Chain orientation and anisotropies in optical and dielectric properties in thin films of stiff polyimides,” J. Polym. Sci.: Part B: Polym. Phys. 30, 1321–1327 (1992).
    [Crossref]
  2. S. Ando, K. Sekiguchi, M. Mizoroki, T. Okada, and R. Ishige, “Anisotropic linear and volumetric thermal-expansion behaviors of self-standing polyimide films analyzed by thermomechanical analysis (tma) and optical interferometry,” Macromol. Chem. Phys. 219, 1700354 (2017).
    [Crossref]
  3. S. Ando, M. Harada, T. Okada, and R. Ishige, “Effective reduction of volumetric thermal expansion of aromatic polyimide films by incorporating interchain crosslinking,” Polymers 10, 761 (2018).
    [Crossref]
  4. T. Okada, R. Ishige, and S. Ando, “Effects of chain packing and structural isomerism on the anisotropic linear and volumetric thermal expansion behaviors of polyimide films,” Polymer 146, 386–395 (2018).
    [Crossref]
  5. G. Beadie, M. Brindza, R. A. Flynn, A. Rosenberg, and J. S. Shirk, “Refractive index measurements of poly(methyl methacrylate) (PMMA) from 0.4–1.6 um,” Appl. Opt. 54, 139–143 (2015).
    [Crossref] [PubMed]
  6. M. Brindza, R. A. Flynn, J. S. Shirk, and G. Beadie, “Thin sample refractive index by transmission spectroscopy,” Opt. Express 22, 28537–28552 (2014).
    [Crossref] [PubMed]
  7. A. Rosenberg, S. H. Lee, J. S. Shirk, and G. Beadie, “Opto-thermal characteristics of amorphous polyimides for optical applications,” Opt. Mater. Express 8, 2159–2172 (2018).
    [Crossref]
  8. C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, Inc., ISBN 0-471-49024-5, 1976), 5th ed. (Chpater 13).
  9. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., ISBN 0-471-43132-X, 1975), 2nd ed. (Chapter 4).
  10. G. Elsner, J. Kempf, J. W. Bartha, and H. H. Wagner, “Anisotropy of thermal expansion of thin polyimide films,” Thin Solid Films 185, 189–197 (1990).
    [Crossref]
  11. R. F. Saraf, H.-M. Tong, T. W. Poon, B. D. Silverman, P. S. Ho, and A. R. Rossi, “Thickness-direction thermal-expansion measurements,” J. Appl. Polym. Sci. 46, 1329–1337 (1992).
    [Crossref]
  12. S. Kanagaraj and S. Pattanayak, “Measurement of the thermal expansion of metal and frps,” Cryogenics 43, 399–424 (2003).
    [Crossref]
  13. H. M. Tong, H. K. D. Hsuen, K. L. Saenger, and G. W. Su, “Thickness-direction coefficient of thermal expansion measurement of thin polymer films,” Rev. Sci. Instruments 62, 422–430 (1991).
    [Crossref]
  14. J. A. Rogers, L. Dhar, and K. A. Nelson, “Noncontact determination of transverse isotropic elastic moduli in polyimide thin films using a laser based ultrasonic method,” Appl. Phys. Lett. 65, 312–314 (1994).
    [Crossref]
  15. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, Inc., ISBN 0-471-82866-1, 1988). (Chapter 4).
  16. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, ISBN 0-521-63921-2, 1988), 6th ed. (Section 1.6).
  17. F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., 1937), 1st ed. (Chap. ’Interference Involving Multiple Reflections’).
  18. J. F. Offersgaard, “Waveguides formed by multiple layers of dielectric, semiconductor, or metallic media with optical loss and anisotropy,” J. Opt. Soc. Am. A 12, 2122–2128 (1995).
    [Crossref]
  19. J. D. Lytle, Handbook of Optics, Vol II (McGraw Hill, Inc., ISBN 0-07-047974-7, 1995), 2nd ed. (Chapter 34, Table II).
  20. H. Mueller, “Theory of photoelasticity in amorphous solids,” Physics 6, 179–184 (1935).
    [Crossref]
  21. R. M. Waxler, D. Horowitz, and A. Feldman, “Optical and physical parameters of plexiglas 55 and lexan,” Appl. Opt. 18, 101–124 (1979).
    [Crossref] [PubMed]
  22. Z. Zhang, P. Zhao, P. Lin, and F. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
    [Crossref]
  23. M. F. Vuks, “Determination of the optical anisotropy of aromatic molecules from the double refraction of crystals,” Opt. Spectrosc. 20, 361–364 (1966).
  24. S. Kim, P. M. Cotts, and W. Volksen, “On-line measurement of the rms radius of gyration and molecular weight of polyimide precursor fractions eluting from a size-exclusion chromatograph,” J. Polym. Sci. B: Polym. Phys. 30, 177–183 (1992).
    [Crossref]

2018 (3)

S. Ando, M. Harada, T. Okada, and R. Ishige, “Effective reduction of volumetric thermal expansion of aromatic polyimide films by incorporating interchain crosslinking,” Polymers 10, 761 (2018).
[Crossref]

T. Okada, R. Ishige, and S. Ando, “Effects of chain packing and structural isomerism on the anisotropic linear and volumetric thermal expansion behaviors of polyimide films,” Polymer 146, 386–395 (2018).
[Crossref]

A. Rosenberg, S. H. Lee, J. S. Shirk, and G. Beadie, “Opto-thermal characteristics of amorphous polyimides for optical applications,” Opt. Mater. Express 8, 2159–2172 (2018).
[Crossref]

2017 (1)

S. Ando, K. Sekiguchi, M. Mizoroki, T. Okada, and R. Ishige, “Anisotropic linear and volumetric thermal-expansion behaviors of self-standing polyimide films analyzed by thermomechanical analysis (tma) and optical interferometry,” Macromol. Chem. Phys. 219, 1700354 (2017).
[Crossref]

2015 (1)

G. Beadie, M. Brindza, R. A. Flynn, A. Rosenberg, and J. S. Shirk, “Refractive index measurements of poly(methyl methacrylate) (PMMA) from 0.4–1.6 um,” Appl. Opt. 54, 139–143 (2015).
[Crossref] [PubMed]

2014 (1)

2006 (1)

Z. Zhang, P. Zhao, P. Lin, and F. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[Crossref]

2003 (1)

S. Kanagaraj and S. Pattanayak, “Measurement of the thermal expansion of metal and frps,” Cryogenics 43, 399–424 (2003).
[Crossref]

1995 (1)

1994 (1)

J. A. Rogers, L. Dhar, and K. A. Nelson, “Noncontact determination of transverse isotropic elastic moduli in polyimide thin films using a laser based ultrasonic method,” Appl. Phys. Lett. 65, 312–314 (1994).
[Crossref]

1992 (3)

R. F. Saraf, H.-M. Tong, T. W. Poon, B. D. Silverman, P. S. Ho, and A. R. Rossi, “Thickness-direction thermal-expansion measurements,” J. Appl. Polym. Sci. 46, 1329–1337 (1992).
[Crossref]

D. Boese, H. Lee, D. Y. Yoon, J. D. Swalen, and J. F. Rabolt, “Chain orientation and anisotropies in optical and dielectric properties in thin films of stiff polyimides,” J. Polym. Sci.: Part B: Polym. Phys. 30, 1321–1327 (1992).
[Crossref]

S. Kim, P. M. Cotts, and W. Volksen, “On-line measurement of the rms radius of gyration and molecular weight of polyimide precursor fractions eluting from a size-exclusion chromatograph,” J. Polym. Sci. B: Polym. Phys. 30, 177–183 (1992).
[Crossref]

1991 (1)

H. M. Tong, H. K. D. Hsuen, K. L. Saenger, and G. W. Su, “Thickness-direction coefficient of thermal expansion measurement of thin polymer films,” Rev. Sci. Instruments 62, 422–430 (1991).
[Crossref]

1990 (1)

G. Elsner, J. Kempf, J. W. Bartha, and H. H. Wagner, “Anisotropy of thermal expansion of thin polyimide films,” Thin Solid Films 185, 189–197 (1990).
[Crossref]

1979 (1)

1966 (1)

M. F. Vuks, “Determination of the optical anisotropy of aromatic molecules from the double refraction of crystals,” Opt. Spectrosc. 20, 361–364 (1966).

1935 (1)

H. Mueller, “Theory of photoelasticity in amorphous solids,” Physics 6, 179–184 (1935).
[Crossref]

Ando, S.

S. Ando, M. Harada, T. Okada, and R. Ishige, “Effective reduction of volumetric thermal expansion of aromatic polyimide films by incorporating interchain crosslinking,” Polymers 10, 761 (2018).
[Crossref]

T. Okada, R. Ishige, and S. Ando, “Effects of chain packing and structural isomerism on the anisotropic linear and volumetric thermal expansion behaviors of polyimide films,” Polymer 146, 386–395 (2018).
[Crossref]

S. Ando, K. Sekiguchi, M. Mizoroki, T. Okada, and R. Ishige, “Anisotropic linear and volumetric thermal-expansion behaviors of self-standing polyimide films analyzed by thermomechanical analysis (tma) and optical interferometry,” Macromol. Chem. Phys. 219, 1700354 (2017).
[Crossref]

Bartha, J. W.

G. Elsner, J. Kempf, J. W. Bartha, and H. H. Wagner, “Anisotropy of thermal expansion of thin polyimide films,” Thin Solid Films 185, 189–197 (1990).
[Crossref]

Beadie, G.

Boese, D.

D. Boese, H. Lee, D. Y. Yoon, J. D. Swalen, and J. F. Rabolt, “Chain orientation and anisotropies in optical and dielectric properties in thin films of stiff polyimides,” J. Polym. Sci.: Part B: Polym. Phys. 30, 1321–1327 (1992).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, ISBN 0-521-63921-2, 1988), 6th ed. (Section 1.6).

Brindza, M.

G. Beadie, M. Brindza, R. A. Flynn, A. Rosenberg, and J. S. Shirk, “Refractive index measurements of poly(methyl methacrylate) (PMMA) from 0.4–1.6 um,” Appl. Opt. 54, 139–143 (2015).
[Crossref] [PubMed]

M. Brindza, R. A. Flynn, J. S. Shirk, and G. Beadie, “Thin sample refractive index by transmission spectroscopy,” Opt. Express 22, 28537–28552 (2014).
[Crossref] [PubMed]

Cotts, P. M.

S. Kim, P. M. Cotts, and W. Volksen, “On-line measurement of the rms radius of gyration and molecular weight of polyimide precursor fractions eluting from a size-exclusion chromatograph,” J. Polym. Sci. B: Polym. Phys. 30, 177–183 (1992).
[Crossref]

Dhar, L.

J. A. Rogers, L. Dhar, and K. A. Nelson, “Noncontact determination of transverse isotropic elastic moduli in polyimide thin films using a laser based ultrasonic method,” Appl. Phys. Lett. 65, 312–314 (1994).
[Crossref]

Elsner, G.

G. Elsner, J. Kempf, J. W. Bartha, and H. H. Wagner, “Anisotropy of thermal expansion of thin polyimide films,” Thin Solid Films 185, 189–197 (1990).
[Crossref]

Feldman, A.

Flynn, R. A.

G. Beadie, M. Brindza, R. A. Flynn, A. Rosenberg, and J. S. Shirk, “Refractive index measurements of poly(methyl methacrylate) (PMMA) from 0.4–1.6 um,” Appl. Opt. 54, 139–143 (2015).
[Crossref] [PubMed]

M. Brindza, R. A. Flynn, J. S. Shirk, and G. Beadie, “Thin sample refractive index by transmission spectroscopy,” Opt. Express 22, 28537–28552 (2014).
[Crossref] [PubMed]

Harada, M.

S. Ando, M. Harada, T. Okada, and R. Ishige, “Effective reduction of volumetric thermal expansion of aromatic polyimide films by incorporating interchain crosslinking,” Polymers 10, 761 (2018).
[Crossref]

Ho, P. S.

R. F. Saraf, H.-M. Tong, T. W. Poon, B. D. Silverman, P. S. Ho, and A. R. Rossi, “Thickness-direction thermal-expansion measurements,” J. Appl. Polym. Sci. 46, 1329–1337 (1992).
[Crossref]

Horowitz, D.

Hsuen, H. K. D.

H. M. Tong, H. K. D. Hsuen, K. L. Saenger, and G. W. Su, “Thickness-direction coefficient of thermal expansion measurement of thin polymer films,” Rev. Sci. Instruments 62, 422–430 (1991).
[Crossref]

Ishige, R.

T. Okada, R. Ishige, and S. Ando, “Effects of chain packing and structural isomerism on the anisotropic linear and volumetric thermal expansion behaviors of polyimide films,” Polymer 146, 386–395 (2018).
[Crossref]

S. Ando, M. Harada, T. Okada, and R. Ishige, “Effective reduction of volumetric thermal expansion of aromatic polyimide films by incorporating interchain crosslinking,” Polymers 10, 761 (2018).
[Crossref]

S. Ando, K. Sekiguchi, M. Mizoroki, T. Okada, and R. Ishige, “Anisotropic linear and volumetric thermal-expansion behaviors of self-standing polyimide films analyzed by thermomechanical analysis (tma) and optical interferometry,” Macromol. Chem. Phys. 219, 1700354 (2017).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., ISBN 0-471-43132-X, 1975), 2nd ed. (Chapter 4).

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., 1937), 1st ed. (Chap. ’Interference Involving Multiple Reflections’).

Kanagaraj, S.

S. Kanagaraj and S. Pattanayak, “Measurement of the thermal expansion of metal and frps,” Cryogenics 43, 399–424 (2003).
[Crossref]

Kempf, J.

G. Elsner, J. Kempf, J. W. Bartha, and H. H. Wagner, “Anisotropy of thermal expansion of thin polyimide films,” Thin Solid Films 185, 189–197 (1990).
[Crossref]

Kim, S.

S. Kim, P. M. Cotts, and W. Volksen, “On-line measurement of the rms radius of gyration and molecular weight of polyimide precursor fractions eluting from a size-exclusion chromatograph,” J. Polym. Sci. B: Polym. Phys. 30, 177–183 (1992).
[Crossref]

Kittel, C.

C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, Inc., ISBN 0-471-49024-5, 1976), 5th ed. (Chpater 13).

Lee, H.

D. Boese, H. Lee, D. Y. Yoon, J. D. Swalen, and J. F. Rabolt, “Chain orientation and anisotropies in optical and dielectric properties in thin films of stiff polyimides,” J. Polym. Sci.: Part B: Polym. Phys. 30, 1321–1327 (1992).
[Crossref]

Lee, S. H.

Lin, P.

Z. Zhang, P. Zhao, P. Lin, and F. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[Crossref]

Lytle, J. D.

J. D. Lytle, Handbook of Optics, Vol II (McGraw Hill, Inc., ISBN 0-07-047974-7, 1995), 2nd ed. (Chapter 34, Table II).

Mizoroki, M.

S. Ando, K. Sekiguchi, M. Mizoroki, T. Okada, and R. Ishige, “Anisotropic linear and volumetric thermal-expansion behaviors of self-standing polyimide films analyzed by thermomechanical analysis (tma) and optical interferometry,” Macromol. Chem. Phys. 219, 1700354 (2017).
[Crossref]

Mueller, H.

H. Mueller, “Theory of photoelasticity in amorphous solids,” Physics 6, 179–184 (1935).
[Crossref]

Nelson, K. A.

J. A. Rogers, L. Dhar, and K. A. Nelson, “Noncontact determination of transverse isotropic elastic moduli in polyimide thin films using a laser based ultrasonic method,” Appl. Phys. Lett. 65, 312–314 (1994).
[Crossref]

Offersgaard, J. F.

Okada, T.

T. Okada, R. Ishige, and S. Ando, “Effects of chain packing and structural isomerism on the anisotropic linear and volumetric thermal expansion behaviors of polyimide films,” Polymer 146, 386–395 (2018).
[Crossref]

S. Ando, M. Harada, T. Okada, and R. Ishige, “Effective reduction of volumetric thermal expansion of aromatic polyimide films by incorporating interchain crosslinking,” Polymers 10, 761 (2018).
[Crossref]

S. Ando, K. Sekiguchi, M. Mizoroki, T. Okada, and R. Ishige, “Anisotropic linear and volumetric thermal-expansion behaviors of self-standing polyimide films analyzed by thermomechanical analysis (tma) and optical interferometry,” Macromol. Chem. Phys. 219, 1700354 (2017).
[Crossref]

Pattanayak, S.

S. Kanagaraj and S. Pattanayak, “Measurement of the thermal expansion of metal and frps,” Cryogenics 43, 399–424 (2003).
[Crossref]

Poon, T. W.

R. F. Saraf, H.-M. Tong, T. W. Poon, B. D. Silverman, P. S. Ho, and A. R. Rossi, “Thickness-direction thermal-expansion measurements,” J. Appl. Polym. Sci. 46, 1329–1337 (1992).
[Crossref]

Rabolt, J. F.

D. Boese, H. Lee, D. Y. Yoon, J. D. Swalen, and J. F. Rabolt, “Chain orientation and anisotropies in optical and dielectric properties in thin films of stiff polyimides,” J. Polym. Sci.: Part B: Polym. Phys. 30, 1321–1327 (1992).
[Crossref]

Rogers, J. A.

J. A. Rogers, L. Dhar, and K. A. Nelson, “Noncontact determination of transverse isotropic elastic moduli in polyimide thin films using a laser based ultrasonic method,” Appl. Phys. Lett. 65, 312–314 (1994).
[Crossref]

Rosenberg, A.

A. Rosenberg, S. H. Lee, J. S. Shirk, and G. Beadie, “Opto-thermal characteristics of amorphous polyimides for optical applications,” Opt. Mater. Express 8, 2159–2172 (2018).
[Crossref]

G. Beadie, M. Brindza, R. A. Flynn, A. Rosenberg, and J. S. Shirk, “Refractive index measurements of poly(methyl methacrylate) (PMMA) from 0.4–1.6 um,” Appl. Opt. 54, 139–143 (2015).
[Crossref] [PubMed]

Rossi, A. R.

R. F. Saraf, H.-M. Tong, T. W. Poon, B. D. Silverman, P. S. Ho, and A. R. Rossi, “Thickness-direction thermal-expansion measurements,” J. Appl. Polym. Sci. 46, 1329–1337 (1992).
[Crossref]

Saenger, K. L.

H. M. Tong, H. K. D. Hsuen, K. L. Saenger, and G. W. Su, “Thickness-direction coefficient of thermal expansion measurement of thin polymer films,” Rev. Sci. Instruments 62, 422–430 (1991).
[Crossref]

Saraf, R. F.

R. F. Saraf, H.-M. Tong, T. W. Poon, B. D. Silverman, P. S. Ho, and A. R. Rossi, “Thickness-direction thermal-expansion measurements,” J. Appl. Polym. Sci. 46, 1329–1337 (1992).
[Crossref]

Sekiguchi, K.

S. Ando, K. Sekiguchi, M. Mizoroki, T. Okada, and R. Ishige, “Anisotropic linear and volumetric thermal-expansion behaviors of self-standing polyimide films analyzed by thermomechanical analysis (tma) and optical interferometry,” Macromol. Chem. Phys. 219, 1700354 (2017).
[Crossref]

Shirk, J. S.

Silverman, B. D.

R. F. Saraf, H.-M. Tong, T. W. Poon, B. D. Silverman, P. S. Ho, and A. R. Rossi, “Thickness-direction thermal-expansion measurements,” J. Appl. Polym. Sci. 46, 1329–1337 (1992).
[Crossref]

Su, G. W.

H. M. Tong, H. K. D. Hsuen, K. L. Saenger, and G. W. Su, “Thickness-direction coefficient of thermal expansion measurement of thin polymer films,” Rev. Sci. Instruments 62, 422–430 (1991).
[Crossref]

Sun, F.

Z. Zhang, P. Zhao, P. Lin, and F. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[Crossref]

Swalen, J. D.

D. Boese, H. Lee, D. Y. Yoon, J. D. Swalen, and J. F. Rabolt, “Chain orientation and anisotropies in optical and dielectric properties in thin films of stiff polyimides,” J. Polym. Sci.: Part B: Polym. Phys. 30, 1321–1327 (1992).
[Crossref]

Tong, H. M.

H. M. Tong, H. K. D. Hsuen, K. L. Saenger, and G. W. Su, “Thickness-direction coefficient of thermal expansion measurement of thin polymer films,” Rev. Sci. Instruments 62, 422–430 (1991).
[Crossref]

Tong, H.-M.

R. F. Saraf, H.-M. Tong, T. W. Poon, B. D. Silverman, P. S. Ho, and A. R. Rossi, “Thickness-direction thermal-expansion measurements,” J. Appl. Polym. Sci. 46, 1329–1337 (1992).
[Crossref]

Volksen, W.

S. Kim, P. M. Cotts, and W. Volksen, “On-line measurement of the rms radius of gyration and molecular weight of polyimide precursor fractions eluting from a size-exclusion chromatograph,” J. Polym. Sci. B: Polym. Phys. 30, 177–183 (1992).
[Crossref]

Vuks, M. F.

M. F. Vuks, “Determination of the optical anisotropy of aromatic molecules from the double refraction of crystals,” Opt. Spectrosc. 20, 361–364 (1966).

Wagner, H. H.

G. Elsner, J. Kempf, J. W. Bartha, and H. H. Wagner, “Anisotropy of thermal expansion of thin polyimide films,” Thin Solid Films 185, 189–197 (1990).
[Crossref]

Waxler, R. M.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., 1937), 1st ed. (Chap. ’Interference Involving Multiple Reflections’).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, ISBN 0-521-63921-2, 1988), 6th ed. (Section 1.6).

Yeh, P.

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, Inc., ISBN 0-471-82866-1, 1988). (Chapter 4).

Yoon, D. Y.

D. Boese, H. Lee, D. Y. Yoon, J. D. Swalen, and J. F. Rabolt, “Chain orientation and anisotropies in optical and dielectric properties in thin films of stiff polyimides,” J. Polym. Sci.: Part B: Polym. Phys. 30, 1321–1327 (1992).
[Crossref]

Zhang, Z.

Z. Zhang, P. Zhao, P. Lin, and F. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[Crossref]

Zhao, P.

Z. Zhang, P. Zhao, P. Lin, and F. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[Crossref]

Appl. Opt. (2)

G. Beadie, M. Brindza, R. A. Flynn, A. Rosenberg, and J. S. Shirk, “Refractive index measurements of poly(methyl methacrylate) (PMMA) from 0.4–1.6 um,” Appl. Opt. 54, 139–143 (2015).
[Crossref] [PubMed]

R. M. Waxler, D. Horowitz, and A. Feldman, “Optical and physical parameters of plexiglas 55 and lexan,” Appl. Opt. 18, 101–124 (1979).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

J. A. Rogers, L. Dhar, and K. A. Nelson, “Noncontact determination of transverse isotropic elastic moduli in polyimide thin films using a laser based ultrasonic method,” Appl. Phys. Lett. 65, 312–314 (1994).
[Crossref]

Cryogenics (1)

S. Kanagaraj and S. Pattanayak, “Measurement of the thermal expansion of metal and frps,” Cryogenics 43, 399–424 (2003).
[Crossref]

J. Appl. Polym. Sci. (1)

R. F. Saraf, H.-M. Tong, T. W. Poon, B. D. Silverman, P. S. Ho, and A. R. Rossi, “Thickness-direction thermal-expansion measurements,” J. Appl. Polym. Sci. 46, 1329–1337 (1992).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Polym. Sci. B: Polym. Phys. (1)

S. Kim, P. M. Cotts, and W. Volksen, “On-line measurement of the rms radius of gyration and molecular weight of polyimide precursor fractions eluting from a size-exclusion chromatograph,” J. Polym. Sci. B: Polym. Phys. 30, 177–183 (1992).
[Crossref]

J. Polym. Sci.: Part B: Polym. Phys. (1)

D. Boese, H. Lee, D. Y. Yoon, J. D. Swalen, and J. F. Rabolt, “Chain orientation and anisotropies in optical and dielectric properties in thin films of stiff polyimides,” J. Polym. Sci.: Part B: Polym. Phys. 30, 1321–1327 (1992).
[Crossref]

Macromol. Chem. Phys. (1)

S. Ando, K. Sekiguchi, M. Mizoroki, T. Okada, and R. Ishige, “Anisotropic linear and volumetric thermal-expansion behaviors of self-standing polyimide films analyzed by thermomechanical analysis (tma) and optical interferometry,” Macromol. Chem. Phys. 219, 1700354 (2017).
[Crossref]

Opt. Express (1)

Opt. Mater. Express (1)

Opt. Spectrosc. (1)

M. F. Vuks, “Determination of the optical anisotropy of aromatic molecules from the double refraction of crystals,” Opt. Spectrosc. 20, 361–364 (1966).

Physics (1)

H. Mueller, “Theory of photoelasticity in amorphous solids,” Physics 6, 179–184 (1935).
[Crossref]

Polymer (2)

Z. Zhang, P. Zhao, P. Lin, and F. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[Crossref]

T. Okada, R. Ishige, and S. Ando, “Effects of chain packing and structural isomerism on the anisotropic linear and volumetric thermal expansion behaviors of polyimide films,” Polymer 146, 386–395 (2018).
[Crossref]

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S. Ando, M. Harada, T. Okada, and R. Ishige, “Effective reduction of volumetric thermal expansion of aromatic polyimide films by incorporating interchain crosslinking,” Polymers 10, 761 (2018).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1 Simulated transmission fringes for a thin film, with properties as described in the text. The horizontal axis is given in inverse wavelength. The solid curve corresponds to the film at low temperature. At elevated temperatures, fringes from the same film shift to the right (longer wavelengths). Also shown are the peak wavelengths corresponding to Eq. (8), if the analysis were performed for the M = 10 fringe peak.
Fig. 2
Fig. 2 Illustration of uniaxial birefringence. The refractive index along x or y is denoted nTE, while the refractive index along z is denoted nTM. For light, the refractive index is determined by the direction of its electric field, E. As illustrated on the right, E points perpendicular to the propagation direction. As diagrammed, this light would see the nTM index. Any light propagating normal to the film, along z, will see index nTE, regardless of its polarization. (Illustration on right modified from https://socratic.org/questions/58f0fd7e7c0149308e221579)
Fig. 3
Fig. 3 Schematic of the anisotropic thermal expansion of a uniaxial film, where a film at low temperature expands to a new outline given by the dashed lines. The direction-dependent rate of expansion is given by the linear coefficients of thermal expansion (CTE). The variables used, for convenience, to represent CTE in the equations are β and β . The volume coefficient of thermal expansion (VCTE) is defined by 2 β + β .

Tables (1)

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Table 1 Values in the first 5 columns are reported in [4]. Values in the last two columns are inferred from the same data, but based on the analysis presented in this paper.

Equations (18)

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M = 2   n ( λ )   d λ
d ( T ) = d ( T o ) [ 1 + Δ T β ]
n T E ( λ , T ) = n T E ( λ , T o ) + Δ T n ˙ T E ( λ )
M = 2   n T E ( λ o , T o ) d ( T o ) λ o
M = 2   [ n T E ( λ T , T o ) + Δ T   n ˙ T E ( λ T ) ] d ( T o ) [ 1 + Δ T β ] λ T
M = ( 2   n T E ( λ T , T o ) d ( T o ) λ T ) { 1 + [ β + n ˙ T E ( λ T ) n T E ( λ T , T o ) ] Δ T + [ β   n ˙ T E ( λ T ) n T E ( λ T , T o ) ] Δ T 2 }
| ( λ T n T E ( λ T ) ) ( d d λ   n T E ( λ T ) ) | < < 1
λ T λ o = 1 + [ β + n ˙ T E ( λ o ) n T E ( λ o , T o ) ] Δ T + [ β   n ˙ T E ( λ o ) n T E ( λ o , T o ) Δ T 2 ]
λ T λ o 1 + [ β + n ˙ T E ( λ o ) n T E ( λ o , T o ) ] Δ T
n 2 1 n 2 + 2 = 1 3   ε o N α = ρ   α 3   ε o ( N A M W )
d n d T = ( n 2 1 ) ( n 2 + 2 ) 6 n ( 1 ρ d ρ d T + 1 α d α d T )
d n d T = ( n 2 1 ) ( n 2 + 2 ) 6 n ( 1 L ) ( 1 ρ d ρ d T ) = ( n 2 1 ) ( n 2 + 2 ) 6 n ( 1 L ) ( 1 V d V d T )
n i 2 1 n 2 + 2 = 1 3   ε o N α i = ρ   α i 3   ε o ( N A M W )
n = ε = ( 2   ε T E + ε T M ) / 3 = ( 2   n T E 2 + n T M 2 ) / 3
α = ( α + 2   α ) / 3
d n d T = ( n 2 1 ) ( n 2 + 2 ) 6 n ( 1 L ) ( 1 V d V d T )
d n T E d T = ( n 2 1 ) ( n 2 + 2 ) 6 ( n T E Δ n / 3 ) ( 1 L ) [ ( β + 2 β ) ]
λ T λ o = 1 + β , r e p Δ T

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