Math 2.0

Optics, electromagnetic theory Definition

Total internal reflection

Builds on Refractive indexSnell's lawFresnel equations

Definition

Let a flat interface separate two lossless, isotropic media with real refractive indices n1>n2>0n_1 > n_2 > 0. Let a plane wave in medium 11 hit the interface at angle of incidence θi∈[0,π/2)\theta_i \in [0, \pi/2). The critical angle is θc=arcsin⁡n2n1.\theta_c = \arcsin\frac{n_2}{n_1}. Total internal reflection is the situation θc<θi<π/2\theta_c < \theta_i < \pi/2. In this situation Snell's law n1sin⁡θi=n2sin⁡θtn_1 \sin\theta_i = n_2 \sin\theta_t has no real solution θt\theta_t, and the reflection coefficients of the Fresnel equations satisfy ∣rs∣=∣rp∣=1\abs{r_s} = \abs{r_p} = 1.

Remarks

Check: for θi>θc\theta_i > \theta_c one has sin⁡θi>n2/n1\sin\theta_i > n_2/n_1, so the required sin⁡θt=(n1/n2)sin⁡θi\sin\theta_t = (n_1/n_2)\sin\theta_i exceeds 11. Then cos⁡θt=i(n1/n2)2sin⁡2θi−1\cos\theta_t = i\sqrt{(n_1/n_2)^2 \sin^2\theta_i - 1} is purely imaginary, and rsr_s, rpr_p become quotients of a complex number by its conjugate, of modulus 11. Example: glass with n1=1.5n_1 = 1.5 against air with n2=1n_2 = 1 has θc=arcsin⁡(2/3)≈41.8∘\theta_c = \arcsin(2/3) \approx 41.8^\circ, so a ray at 45∘45^\circ inside a right-angle prism is totally reflected. Non-example: light going from air into glass (n1<n2n_1 < n_2) is never totally reflected, since arcsin⁡(n2/n1)\arcsin(n_2/n_1) is not defined and a refracted ray exists for every angle. The reflected wave acquires a phase shift that differs for ss and pp polarization, and the field beyond the interface is an evanescent field.

Reading path

To understand this page from the ground up, read these 75 pages in order. The foundations come first.

  1. Calculus of variations concept
  2. Constitutive relations of electromagnetism concept
  3. Electric displacement field concept
  4. Euclidean space object
  5. Euler–Lagrange equation concept
  6. Function concept
  7. Function composition concept
  8. Geodesic concept
  9. Infimum concept
  10. Magnetization concept
  11. Maxwell's equations concept
  12. Normal vector concept
  13. Plane wave concept
  14. Polarization of light concept
  15. Refractive index concept
  16. Supremum concept
  17. Fréchet derivative definition
  18. Total derivative concept
  19. Fermat's principle statement
  20. Function of several variables concept
  21. Vector-valued function concept
  22. Jordan measure definition
  23. Differentiable function of several variables concept
  24. Iterated integral definition
  25. Limit of a function of several variables concept
  26. Partial derivative concept
  27. Parametrised curve concept
  28. Snell's law statement
  29. Gradient concept
  30. Continuity in R^n definition
  31. Differentiable function of several variables definition
  32. Partial derivative definition
  33. Differentiable on an open set definition
  34. Double integral definition
  35. Jacobian matrix definition
  36. Chain rule in several variables statement
  37. Gradient definition
  38. Higher partial derivatives definition
  39. Fubini's theorem statement
  40. Iterated derivative definition
  41. Triple integral definition
  42. Chain rule for a composition with a curve statement
  43. C^k function characterised by continuous derivatives statement
  44. Differential form definition
  45. Parametrised curve definition
  46. Vector field definition
  47. Wedge product of forms definition
  48. Closed curve definition
  49. Rectifiable curve definition
  50. Parametrised surface definition
  51. Conservative vector field definition
  52. Divergence definition
  53. Curl definition
  54. Tangent vector of a curve definition
  55. Line integral of a scalar field definition
  56. Line integral of a vector field definition
  57. Tangent plane and normal vector of a surface definition
  58. Regular curve definition
  59. Green's theorem statement
  60. One-form definition
  61. Surface area definition
  62. Arc length parametrisation definition
  63. Divergence theorem on a rectangle in the plane statement
  64. Surface integral definition
  65. Exterior derivative definition
  66. Line integral as an integral over the parameter interval statement
  67. Orientable surface definition
  68. Gradient theorem statement
  69. Flux through a surface definition
  70. Divergence theorem statement
  71. Generalised Stokes theorem statement
  72. Stokes' theorem statement
  73. Interface conditions for electromagnetic fields statement
  74. Fresnel equations statement
  75. Total internal reflection definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Total_internal_reflection
About this page
TagCV9R40QRHH
URIm2://CV9R40QRHH
Kinddefinition
WritingW0 stub
MSC78A40, 78A05
DefinesTotal internal reflection
UsesRefractive index, Snell's law, Fresnel equations
Changed3 November 2026

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