Abstract

We propose a laser frequency splitting method to determine the relative stress-optic coefficient and internal stresses in Nd:YAG crystals with high resolution. In this method a mirrored Nd:YAG crystal in a circular disk shape is made into an all-internal cavity laser. Once diametral compression force is applied, due to intracavity photoelastic effect, the single laser frequency is split in two. Through measuring the beat frequency of the split frequencies, the relative stress-optic coefficient and internal stresses in the Nd:YAG crystal can be determined according to the proportional relationship. In our experiment the measured value of the stress-optic coefficient, C111=1.27×1012m2/N, is close to the theoretical value; the principal stress difference at the center of the Nd:YAG crystal was also determined. The optical path retardation resolution is approximately 1.5×108λ, and correspondingly the principal stress difference resolution is approximately 5  Pa. The resolution is approximately 5–8 orders of magnitude higher than those of conventional methods, therefore even very small residual stresses can be easily determined with this method. Although this method is a kind of point-by-point method instead of a full-field method, it is a promising novel photomechanics method with attractive advantages such as extremely high resolution.

© 2008 Optical Society of America

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References

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  1. T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, 1981).
  2. A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974), pp. 88 and 108-110.
  3. H. Aben and C. Guillemet, Photoelasiticity of Glass (Springer-Verlag, 1993).
  4. J. Lu, Handbook of Measurement of Residual Stresses (Fairmont, 1996), pp. 225-231.
  5. Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075 (2001).
    [CrossRef]
  6. X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
    [CrossRef]
  7. S. Zhang, Orthogonally Polarized Laser Principles (Tsinghua University, 2005).
  8. S. Zhang and T. Xu, “Orthogonally linear polarized lasers (I)--principle and devices,” Prog. Nat. Sci. 15, 586-595 (2005).
    [CrossRef]
  9. S. Zhang and G. Liu, “Orthogonal linear polarized lasers (II)--study on the physical phenomena,” Prog. Nat. Sci. 15, 865-876 (2005).
    [CrossRef]
  10. S. Zhang, W. Du, and G. Liu, “Orthogonal linear polarized lasers (III)--applications in self-sensing,” Prog. Nat. Sci. 15, 961-971 (2005).
    [CrossRef]
  11. S. Zhang and T. Bosch, “Orthogonally polarized lasers and their applications,” Opt. Photon. News 18, 38-43 (2007).
    [CrossRef]
  12. W. Holzapfel and U. Riss, “Computer-based high resolution transmission ellipsometry,” Appl. Opt. 26, 145-153 (1987).
  13. W. Holzapfel and M. Finnemann, “High-resolution force sensing by diode-pumped Nd:YAG laser,” Opt. Lett. 18, 2062-2064 (1993).
  14. W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurements by solid-state laser transducers,” Measurement 28, 277-291(2000).
    [CrossRef]
  15. S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28 (1995).
    [CrossRef]
  16. W. Holzapfel, L. Hou, and S. Neuschaefer-Rube, “Error effects in microlaser sensors,” Proceedings of XVI IMEKO World Congress, Vol. 3, Vienna, Austria (Australian Society for Measurement and Automation, 2000), pp. 85-90.

2007 (1)

S. Zhang and T. Bosch, “Orthogonally polarized lasers and their applications,” Opt. Photon. News 18, 38-43 (2007).
[CrossRef]

2006 (1)

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

2005 (3)

S. Zhang and T. Xu, “Orthogonally linear polarized lasers (I)--principle and devices,” Prog. Nat. Sci. 15, 586-595 (2005).
[CrossRef]

S. Zhang and G. Liu, “Orthogonal linear polarized lasers (II)--study on the physical phenomena,” Prog. Nat. Sci. 15, 865-876 (2005).
[CrossRef]

S. Zhang, W. Du, and G. Liu, “Orthogonal linear polarized lasers (III)--applications in self-sensing,” Prog. Nat. Sci. 15, 961-971 (2005).
[CrossRef]

2001 (1)

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075 (2001).
[CrossRef]

2000 (1)

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurements by solid-state laser transducers,” Measurement 28, 277-291(2000).
[CrossRef]

1995 (1)

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28 (1995).
[CrossRef]

1993 (1)

1987 (1)

Aben, H.

H. Aben and C. Guillemet, Photoelasiticity of Glass (Springer-Verlag, 1993).

Bosch, T.

S. Zhang and T. Bosch, “Orthogonally polarized lasers and their applications,” Opt. Photon. News 18, 38-43 (2007).
[CrossRef]

Du, W.

S. Zhang, W. Du, and G. Liu, “Orthogonal linear polarized lasers (III)--applications in self-sensing,” Prog. Nat. Sci. 15, 961-971 (2005).
[CrossRef]

Finnemann, M.

Guillemet, C.

H. Aben and C. Guillemet, Photoelasiticity of Glass (Springer-Verlag, 1993).

Guo, H.

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28 (1995).
[CrossRef]

Han, Y.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075 (2001).
[CrossRef]

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28 (1995).
[CrossRef]

Holzapfel, W.

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurements by solid-state laser transducers,” Measurement 28, 277-291(2000).
[CrossRef]

W. Holzapfel and M. Finnemann, “High-resolution force sensing by diode-pumped Nd:YAG laser,” Opt. Lett. 18, 2062-2064 (1993).

W. Holzapfel and U. Riss, “Computer-based high resolution transmission ellipsometry,” Appl. Opt. 26, 145-153 (1987).

W. Holzapfel, L. Hou, and S. Neuschaefer-Rube, “Error effects in microlaser sensors,” Proceedings of XVI IMEKO World Congress, Vol. 3, Vienna, Austria (Australian Society for Measurement and Automation, 2000), pp. 85-90.

Hou, L.

W. Holzapfel, L. Hou, and S. Neuschaefer-Rube, “Error effects in microlaser sensors,” Proceedings of XVI IMEKO World Congress, Vol. 3, Vienna, Austria (Australian Society for Measurement and Automation, 2000), pp. 85-90.

Kuske, A.

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974), pp. 88 and 108-110.

Li, K.

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28 (1995).
[CrossRef]

Li, Y.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075 (2001).
[CrossRef]

Liu, G.

S. Zhang and G. Liu, “Orthogonal linear polarized lasers (II)--study on the physical phenomena,” Prog. Nat. Sci. 15, 865-876 (2005).
[CrossRef]

S. Zhang, W. Du, and G. Liu, “Orthogonal linear polarized lasers (III)--applications in self-sensing,” Prog. Nat. Sci. 15, 961-971 (2005).
[CrossRef]

Liu, W.

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

Lu, J.

J. Lu, Handbook of Measurement of Residual Stresses (Fairmont, 1996), pp. 225-231.

Narasimhamurty, T. S.

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, 1981).

Neuschaefer-Rube, S.

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurements by solid-state laser transducers,” Measurement 28, 277-291(2000).
[CrossRef]

W. Holzapfel, L. Hou, and S. Neuschaefer-Rube, “Error effects in microlaser sensors,” Proceedings of XVI IMEKO World Congress, Vol. 3, Vienna, Austria (Australian Society for Measurement and Automation, 2000), pp. 85-90.

Riss, U.

Robertson, G.

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974), pp. 88 and 108-110.

Xu, T.

S. Zhang and T. Xu, “Orthogonally linear polarized lasers (I)--principle and devices,” Prog. Nat. Sci. 15, 586-595 (2005).
[CrossRef]

Xu, X.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075 (2001).
[CrossRef]

Zhang, S.

S. Zhang and T. Bosch, “Orthogonally polarized lasers and their applications,” Opt. Photon. News 18, 38-43 (2007).
[CrossRef]

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

S. Zhang, W. Du, and G. Liu, “Orthogonal linear polarized lasers (III)--applications in self-sensing,” Prog. Nat. Sci. 15, 961-971 (2005).
[CrossRef]

S. Zhang and G. Liu, “Orthogonal linear polarized lasers (II)--study on the physical phenomena,” Prog. Nat. Sci. 15, 865-876 (2005).
[CrossRef]

S. Zhang and T. Xu, “Orthogonally linear polarized lasers (I)--principle and devices,” Prog. Nat. Sci. 15, 586-595 (2005).
[CrossRef]

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075 (2001).
[CrossRef]

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28 (1995).
[CrossRef]

S. Zhang, Orthogonally Polarized Laser Principles (Tsinghua University, 2005).

Zhang, Y.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075 (2001).
[CrossRef]

Zong, X.

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

Appl. Opt. (1)

Measurement (1)

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurements by solid-state laser transducers,” Measurement 28, 277-291(2000).
[CrossRef]

Opt. Eng. (2)

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075 (2001).
[CrossRef]

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

Opt. Lasers Eng. (1)

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28 (1995).
[CrossRef]

Opt. Lett. (1)

Opt. Photon. News (1)

S. Zhang and T. Bosch, “Orthogonally polarized lasers and their applications,” Opt. Photon. News 18, 38-43 (2007).
[CrossRef]

Prog. Nat. Sci. (3)

S. Zhang and T. Xu, “Orthogonally linear polarized lasers (I)--principle and devices,” Prog. Nat. Sci. 15, 586-595 (2005).
[CrossRef]

S. Zhang and G. Liu, “Orthogonal linear polarized lasers (II)--study on the physical phenomena,” Prog. Nat. Sci. 15, 865-876 (2005).
[CrossRef]

S. Zhang, W. Du, and G. Liu, “Orthogonal linear polarized lasers (III)--applications in self-sensing,” Prog. Nat. Sci. 15, 961-971 (2005).
[CrossRef]

Other (6)

S. Zhang, Orthogonally Polarized Laser Principles (Tsinghua University, 2005).

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, 1981).

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974), pp. 88 and 108-110.

H. Aben and C. Guillemet, Photoelasiticity of Glass (Springer-Verlag, 1993).

J. Lu, Handbook of Measurement of Residual Stresses (Fairmont, 1996), pp. 225-231.

W. Holzapfel, L. Hou, and S. Neuschaefer-Rube, “Error effects in microlaser sensors,” Proceedings of XVI IMEKO World Congress, Vol. 3, Vienna, Austria (Australian Society for Measurement and Automation, 2000), pp. 85-90.

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Figures (5)

Fig. 1
Fig. 1

Circular disk in diametral compression: F is the applied diametral compression force; σ 1 and σ 2 are the orthogonal principal stresses; D is the diameter of the circular disk; d is the thickness of the circular disk; and O is the center of the circular disk.

Fig. 2
Fig. 2

Basic configuration of the experimental setup.

Fig. 3
Fig. 3

Observation of laser frequency splitting phenomenon by a Fabry–Perot scanning interferometer. The upper trace is the scanning voltage of Fabry–Perot scanning interferometer (attenuated by 20 dB ): 2 V / div for vertical axis and 10 ms / div for horizontal axis. The lower trace is the laser frequency splitting modes (Mode 1 and Mode 2): 5 mV / div for vertical axis and 10 mV / div for horizontal axis.

Fig. 4
Fig. 4

Relationship between frequency difference and applied diametral compression force.

Fig. 5
Fig. 5

Relationship between principal stress difference and applied diametral compression force.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

Δ ν = ν L · δ ,
δ = ( C 1 C 2 ) · ( σ 1 σ 2 ) · d = C · ( σ 1 σ 2 ) · d ,
Δ ν = ν L · C · ( σ 1 σ 2 ) · d .
Δ σ = σ 1 σ 2 = L ν C d · Δ ν .
σ 1 = 2 F π D d ,
σ 2 = 6 F π D d .
Δ ν = ν L · C · 8 F π D .
C = π D L 8 ν · Δ ν F = π D L 8 ν · S ,
δ = L c 0 · Δ ν · λ ,

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