Birefringence Imaging

Simulate quantitative birefringence imaging with polarizer/analyzer rotation and compensators. Visualize stress-induced birefringence patterns.

Model: Textbook approximation

polarizer P → sample (R_s, slow axis φ) → compensator → analyzer A; Γ = 2πR/λ

Stokes vector turned by each retarder; T = ½(1 + S₁ cos 2A + S₂ sin 2A)

one plate: T = cos²(A − P) − sin 2(φ − P) sin 2(φ − A) sin²(Γ/2)

Polarizer angle P0 °

0–180 °

Analyzer angle A90 °

0–180 °

Sample slow axis φ45 °

0–180 °

Sample retardance R_s (nm)Wavelength λ (nm)Compensator

Angles are counter-clockwise looking into the beam from the x axis (E–W). Every axis is a slow axis.

Transmitted intensity

0.2961

fraction of the light behind the polarizer

Sensitivity dI/dΓ

0.4565 /rad

0.00525 per nm of sample retardance

White-light colour of the sample

R_s = 100 nm, constant over the spectrum

Intensity against sample retardance

Retardance in waves of λ = 546 nm; the marker is the current sample.

Intensity against analyzer angle

The minimum sits where the analyzer is perpendicular to the long axis of the emerging polarization ellipse.

Photoelastic example: plate with a hole in tension (Kirsch)

Remote stress S (MPa)Hole radius a (mm)Plate thickness d (mm)Stress-optic coefficient C (TPa⁻¹)

C = 2.77 TPa⁻¹ is N-BK7 (Schott). The image uses the polarizer, analyzer, wavelength and compensator above; the field spans ±4a.

Light

Infinite plate (Kirsch 1898); slow axis along σ₁ for C > 0. Tension acts along x; the grey disc is the hole.

Largest retardance in the field: 3SCd = 831 nm = 1.52 waves at 546 nm, on the hole edge across the load.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Ideal polarizers and linear retarders chained in Stokes space; white-light colours from D65 and CIE 1931 at constant retardance. Sénarmont R = λΔθ/180°. Kirsch hole in an infinite plate, R = C·d·(σ₁ − σ₂), averaged over each pixel so fringes finer than a pixel fade instead of aliasing.

Goldstein D. H. (2011). Polarized Light, 3rd ed., ch. Mueller Matrices for Polarizing Components, pp. 93–115. CRC Press. doi:10.1201/b10436
Sørensen B. E. (2013). A revised Michel-Lévy interference colour chart based on first-principles calculations. Eur. J. Mineral. 25(1), 5–10. doi:10.1127/0935-1221/2013/0025-2252
CIE (2019). CIE 1931 colour-matching functions, 2 degree observer (CIE dataset); sRGB display colours per IEC 61966-2-1:1999. doi:10.25039/CIE.DS.xvudnb9b
Timoshenko S. P., Goodier J. N. (1970). Theory of Elasticity, 3rd ed., art. 35 Effect of circular holes on stress distributions in plates (Kirsch solution). McGraw-Hill.
Dally J. W., Riley W. F. (2005). Experimental Stress Analysis, 4th ed., chapter Theory of photoelasticity: the stress-optic law. College House Enterprises.
SCHOTT N-BK7 data sheet: stress optical coefficient K = 2.77 × 10⁻⁶ mm²/N. media.schott.com

Report a problem

Birefringence Imaging

Simulate quantitative birefringence imaging with polarizer/analyzer rotation and compensators. Visualize stress-induced birefringence patterns.

polarizer P → sample (R_s, slow axis φ) → compensator → analyzer A; Γ = 2πR/λ

Stokes vector turned by each retarder; T = ½(1 + S₁ cos 2A + S₂ sin 2A)

one plate: T = cos²(A − P) − sin 2(φ − P) sin 2(φ − A) sin²(Γ/2)

Polarizer angle P0 °

0–180 °

Analyzer angle A90 °

0–180 °

Sample slow axis φ45 °

0–180 °

Sample retardance R_s (nm)Wavelength λ (nm)Compensator

Angles are counter-clockwise looking into the beam from the x axis (E–W). Every axis is a slow axis.

Transmitted intensity

0.2961

fraction of the light behind the polarizer

Sensitivity dI/dΓ

0.4565 /rad

0.00525 per nm of sample retardance

White-light colour of the sample

R_s = 100 nm, constant over the spectrum

Intensity against sample retardance

Retardance in waves of λ = 546 nm; the marker is the current sample.

Intensity against analyzer angle

The minimum sits where the analyzer is perpendicular to the long axis of the emerging polarization ellipse.

Photoelastic example: plate with a hole in tension (Kirsch)

Remote stress S (MPa)Hole radius a (mm)Plate thickness d (mm)Stress-optic coefficient C (TPa⁻¹)

C = 2.77 TPa⁻¹ is N-BK7 (Schott). The image uses the polarizer, analyzer, wavelength and compensator above; the field spans ±4a.

Light

Infinite plate (Kirsch 1898); slow axis along σ₁ for C > 0. Tension acts along x; the grey disc is the hole.

Largest retardance in the field: 3SCd = 831 nm = 1.52 waves at 546 nm, on the hole edge across the load.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Goldstein D. H. (2011). Polarized Light, 3rd ed., ch. Mueller Matrices for Polarizing Components, pp. 93–115. CRC Press. doi:10.1201/b10436
Sørensen B. E. (2013). A revised Michel-Lévy interference colour chart based on first-principles calculations. Eur. J. Mineral. 25(1), 5–10. doi:10.1127/0935-1221/2013/0025-2252
CIE (2019). CIE 1931 colour-matching functions, 2 degree observer (CIE dataset); sRGB display colours per IEC 61966-2-1:1999. doi:10.25039/CIE.DS.xvudnb9b
Timoshenko S. P., Goodier J. N. (1970). Theory of Elasticity, 3rd ed., art. 35 Effect of circular holes on stress distributions in plates (Kirsch solution). McGraw-Hill.
Dally J. W., Riley W. F. (2005). Experimental Stress Analysis, 4th ed., chapter Theory of photoelasticity: the stress-optic law. College House Enterprises.
SCHOTT N-BK7 data sheet: stress optical coefficient K = 2.77 × 10⁻⁶ mm²/N. media.schott.com

Report a problem