Optics and Optical Design Chapter 6: Polarization Optics Lectures 11 13 Cord Arnold / Anne L Huillier
Polarization of Light
Arbitrary wave vs. paraxial wave
One component in x direction y x z
Components in x and y direction y x z
Components in x and y direction y x z
Circular polarization y x z
Elliptic polarization y x z
Importance of polarization for light matter interaction Reflection/transmission at boundaries Absorption (e.g. polarizers) Light scattering Refractive index in anisotropic materials Interaction with molecules => The polarization state can be manipulated by certain media, e.g. Anisotropic, optical active, liquid crystals, and magnetooptic materials.
Polarization helix
The polarization ellipse
Polarization quiz E t, z 3cos t - z/c eˆ x y x Linear polarization in x
E Polarization quiz t, z 3cos t - z/cˆ 3sin t - z/c ˆ y e x e y x Circular polarization
E Polarization quiz t, z 3cos t - z/cˆ 2cos t - z/c ˆ y e x e y x Linear polarization at angle -33.7
E Polarization quiz, 4 t z 3cos t - z/ceˆ x 3cost - z/c eˆ y y x 2R arctan 2 1 R 2 cos 4 45 Elliptic polarization
E Polarization quiz, 4 t z 3cos t - z/ceˆ x cost - z/c eˆ y y x 2R arctan 2 1 R 2 cos 14 Elliptic polarization
Orientation and ellipticity angles y Orientation angle 0 3 5, 4 2 4 3, 2 2 x Ellipticity angle
Poincaré sphere
Jones vectors
Polarizer
Wave retarders (wave plates)
Conclusions for wave retarders A λ/4 plate at 45 transforms linear polarization into circular polarization; elliptic otherwise. Circularly polarized light is transformed into linear polarization at 45 to the crystal axes, independent of the orientation of the plate. A λ/2 plate turns linear polarization by twice the angle θ to the plate, but maximum 90. Left circular polarization in transferred into right and vice versa, independent of the orientation of the plate.
Reflection and refraction
Reflection and refraction at a boundary TE Transverse Electric orthogonal (s)enkrecht TM Transverse Magnetic parallel (p)arallel
Boundary conditions
Fresnel equations
TE polarization external reflection Example n 1, n2 1 1.5 n n 1 2
TE polarization internal reflection Example n 1.5, n2 1 1 c Total internal reflection Critical angle for total internal reflection c sin 1 n n 2 1 n n 1 2
TM polarization external reflection Example n 1, n2 1 1.5 B Brewster angle Brewster angle (TM reflection vanishes) B tan 1 n n 2 1 n n 1 2
TM polarization internal reflection Example n 1.5, n2 1 1 Brewster TIR n n 1 2
External Overview Internal TE TM
Power reflectance
Application < > polarization by reflection
Application < > polarization filters in photography Source: http://www.bhphotovideo.com
Application < > polarization change by total internal reflection Fresnel-Rhomb If n 1 =1.5 and n 2 =1 and θ 1 =47.6 or 55.5, the Fresnel-Rhomb transfers incoming linear polarization at 45 to circular by two internal reflections.
Optics in anisotropic media
Isotropic < > anisotropic
The index ellipsoid, impermeability tensor, indicatrix Biaxial: Uniaxial: Isotropic: All refractive indices are different n 1 n 2 n 3 Two refractive indices are identical, n 1 = n 2 =n o (ordinary refractive index) and n 3 = n e (extraordinary refractive index) n e >n o => positive uniaxial n e <n o => negative uniaxial The z-axis of a uniaxial crystal is the optic axis. The index ellepsoid is a sphere.
Propagation along a principal axis Nothing happens to linear polarization, if the light travels along a principal axis of the crystal and the direction of polarization is along another principal axis.
Propagation along a principal axis A linearly polarized wave with angle of polarization not along a principal axis is not a normal mode of the system. The polarization state changes upon propagation.
Propagation along an arbitrary direction
Finding the optical axes for biaxial crystals z OA n 3 OA The optic axes of a crystal are defined as direction in which a travelling wave suffers no birefringence. n 2 n2 n2 Without loss of generality we assume n 1 < n 2 <n 3. n 1 x Biaxial crystals obviously have two optical axes. y n 2 Uniaxial crystals have one optic axis along the direction of the extraordinary refractive index n e.
Propagation along an arbitrary direction of an uniaxial crystal Special case uniaxial crystal: n 1 = n 2 =n o
Propagation in an arbitrary direction in a uniaxial crystal Special case uniaxial crystal: n 1 = n 2 =n o H k D E D E, and Ordinary wave: Extraordinary wave: H k D D E and
Propagation in an arbitrary direction in a uniaxial crystal H E S H E S E k H H E k H k D D H k 2 1 0
Double refraction
Optical activity < > Circular birefringence Image source: Wikipedia
Faraday effect
Optics of liquid crystals
Optics of liquid crystals Orientation of a twisted nematic liquid crystal Propagation of light in a twisted nematic liquid crystal
Common elements to manipulate polarization
Polarizer
Applications of anisotropic media < > polarization beamsplitter no ne n o n o n o n e n o n e n o n e n e n e
Twistec nematic liquid crystal switch
Applications of anisotropic materials: Wave retarders Optic axis n e n o Uniaxial crystals: n 1 =n 2 =n o, n 3 =n e n o Example: λ/2-plate 4 2 3 4
Applications of anisotropic materials: Intensity control
Applications of anisotropic materials: Polarization rotators and optical isolators