Physics

Geometric Optics

Reflection, refraction, Snell's law, total internal reflection, ray tracing and plane mirrors.

Reflection: when a wave strikes the separation surface between two media, it is returned to the first along with part of the energy, and a change in the direction of propagation.

Snell's Law of reflection:
  • The incident and reflected rays, as well as the normal, lie in the same plane.
  • The angle of incidence (i) and the angle of reflection (r) are equal.
Snell's law of reflection
Refraction: Consists of the change in direction of a wave at the separation surface between two different media through which it propagates with different speed. The Index of Refraction, nn, of a medium is defined as the ratio between the speed of light in a vacuum and the speed of light in the considered medium.
n=cvn = \frac{c}{v}
Snell's Law of refraction:
  • The incident and refracted rays, as well as the normal, lie in the same plane.
  • The ratio of the sines of the angles of incidence  heta1\ heta_1 and refraction  heta2\ heta_2 is equal to the ratio between the propagation speeds in media 1 and 2.
sin⁡θ1sin⁡θ2=v1v2\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2}
Snell's law of refraction
Snell's law can also take the following forms:
sin⁡θ1sin⁡θ2=λ1λ2;n1⋅sin⁡θ1=n2⋅sin⁡θ2\frac{\sin \theta_1}{\sin \theta_2} = \frac{\lambda_1}{\lambda_2} ; \quad n_1 \cdot \sin \theta_1 = n_2 \cdot \sin \theta_2
Total internal reflection: Let us consider the case where the ray starts from medium 2, where n2>n1n_2 > n_1, if we increase the angle of incidence, the angle of refraction will also increase, the value of the latter being greater. For a certain angle of incidence, called the critical angle  hetaL\ heta_L, the angle of refraction is 90º, occurring in this case the so-called grazing refraction. For angles of incidence greater than the critical angle  hetaL\ heta_L, the light is totally reflected, a phenomenon known as total internal reflection.
The critical angle is found with  heta1=90∘\ heta_1=90^\circ in Snell's 2nd law:
θL=arcsin⁡n1n2\theta_L = \arcsin \frac{n_1}{n_2}
Total internal reflection
DIN sign convention:
  • Light rays and the object come from the left.
  • The optical center is the intersection of the axis with the vertical of the lens or mirror.
  • Distances to the right and upwards from the optical center are positive. Otherwise they are negative.
DIN symbols nomenclature:
  • s: Object distance
  • s': Image distance
  • f: Object focus
  • f': Image focus
  • y: Object height
  • y': Image height
  • m: Magnification. Ratio between y and y'
  • n: Index of refraction of the lens or diopter
  • R1: Radius of curvature of the left surface of the lens
  • R2: Radius of curvature of the right surface of the lens
  • O: Optical center
Problem solving process.
  1. Drawing of the optical elements and positions. We place the characteristic elements; axis, object position, focus, lens, image focus, mirror or lens... etc., and their symbols. Mirrors and lenses can be represented as a segment perpendicular to the axis.
  2. Ray tracing suitable for the optical element in question, as described in the following points. Direct rays are drawn with a solid line. The extensions of the rays are drawn with a dashed line. In this way we find the approximate position of the image.
  3. Calculation of magnitudes applying appropriate formulas.
  4. Description of the result. The following characteristics of the obtained image must be indicated.
    1. Real / virtual image: If the image is formed at the intersection of the traced rays it will be real, it can be collected on a screen. If it is formed at the intersection of the extensions of the rays it will be virtual.
    2. Magnified / diminished image: If the ratio m=y′/ym = y'/y is greater than 1, the image is magnified. If it is less, it will be diminished.
    3. Upright / inverted: If y′y' is + it will be upright. If it is - it will be inverted.
Optical ray tracing
Ray tracing:
  • Ray parallel to the axis: Its reflection or extension passes over itself.
  • Ray to the center: It is reflected at the same angle of incidence.
  • Ray from the base (optional): from the base of the object to the point where the parallel ray strikes. Its extension intersects the axis at the base of the object's image.
Equations of plane mirrors:
s=−s′y=y′s = -s' \quad y = y'
Plane mirrors tracing
Ray tracing:
  • Ray parallel to the axis: Its reflection or extension passes through the focus.
  • Ray passing through the center: It reflects back on itself.
Spherical mirrors tracing
Equations of spherical mirrors:
1f=1s′+1sf=R2m=y′y=−s′s\frac{1}{f} = \frac{1}{s'} + \frac{1}{s} \quad f = \frac{R}{2} \quad m = \frac{y'}{y} = -\frac{s'}{s}
Concave mirror: R<0;f<0R < 0 ; f < 0
Convex mirror: R>0;f>0R > 0 ; f > 0
Ray tracing:
  • Ray parallel to the axis: The refracted ray or its extension passes through the image focus.
  • Ray passing through the center: It is not deviated.
  • Ray passing through the object focus: It refracts parallel to the axis.
Spherical diopter tracing
Equations of spherical diopters:
n′s′−ns=n′−nRGauss Eq. f′s′+fs=1\frac{n'}{s'} - \frac{n}{s} = \frac{n' - n}{R} \quad \text{Gauss Eq. } \frac{f'}{s'} + \frac{f}{s} = 1
f=−Rnn′−nf′=Rn′n′−nf+f′=Rf = -R \frac{n}{n' - n} \quad f' = R \frac{n'}{n' - n} \quad f + f' = R
ff′=−nn′m=y′y=n⋅s′n′⋅s\frac{f}{f'} = -\frac{n}{n'} \quad m = \frac{y'}{y} = \frac{n \cdot s'}{n' \cdot s}
Ray tracing:
  • Ray parallel to the axis: The refracted ray or its extension passes through the image focus.
  • Ray passing through the center: It is not deviated.
  • Ray passing through the object focus: It refracts parallel to the axis.
Thin converging lens tracing
Thin diverging lens tracing
Equations of lenses:
f=−f′f = -f'
Optical power: power of a lens in diopters (D=m⁻¹)
P=1f′P = \frac{1}{f'}
Gaussian equation of thin lenses:
1f′=1s′−1s\frac{1}{f'} = \frac{1}{s'} - \frac{1}{s}
Lensmaker's equation and focal length:
1f′=(n−1)(1R1−1R2)\frac{1}{f'} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)
Magnification:
m=y′y=s′sm = \frac{y'}{y} = \frac{s'}{s}
Converging lens: f′>0f' > 0, diverging lens: f′<0f' < 0