Formulae
1
sin πΌ cos π½ = [sin(πΌ + π½) + sin(πΌ − π½)]
2
1
cos πΌ cos π½ = [cos(πΌ + π½) + cos(πΌ − π½)]
2
1
sin πΌ sin π½ = [−cos(πΌ + π½) + cos(πΌ − π½)]
2
πΌ±π½
πΌβπ½
cos
2
2
πΌ+π½
πΌ−π½
cos πΌ + cos π½ = 2 cos
cos
2
2
πΌ+π½
πΌ−π½
cos πΌ − cos π½ = −2 sin
sin
2
2
sin πΌ ± sin π½ = 2 sin
sin(πΌ ± π½) = sin πΌ cos π½ ± cos πΌ sin π½
cos(πΌ ± π½) = cos πΌ cos π½ β sin πΌ sin π½
Complex numbers
π§ = π + ππ = π΄π , π
π{π§} = π = π΄ cos π, πΌπ{π§} = π = π΄ sin π,
π§ ∗ = π − ππ, |π§| = π§π§ ∗ = π + π = π΄ , tan π =
ππ = −ππ₯ , πΈ = ππ₯ + ππ£ , π₯ = π΄ cos(π π‘ + πΌ) , π₯ = π
π{π§} , π§ = π΄π
π =
π/π , π =
1/πΏπΆ , π =
ππ = −ππ₯ − ππ£ , π₯ = π΄π
/
π/π
cos(π π‘ + π) , π = π −
, π = π /πΎ
πΎ = π/π , πΎ = π
/πΏ , πΎ = 1/π
πΆ
π§ = π΄π
,π=
π −
+ π , π΄(π‘) = π΄ π
/
, πΈ(π‘) = πΈ π
, max. A at π = π −
π΄(π) =
(
π(π) =
)
=
max. P at π
ππ π΄ ≈
N coupled oscillators
π¦ , = π΄ , cos(π π‘ + πΏ ) , π = 2π sin
(
, π΄ , = πΆ sin
)
=π
, π = π/ππ
n: mode index, p: position index, πΏ = 0 if we have only one mode.
Stressed string:
Solid rod:
=
=
Boundary conditions:
fixed at x=0, π(π₯) sin
Fourier: π¦(π₯) = ∑
, π£ = π/π, y(π₯, π‘) = π(π₯) cos ππ‘
, π£ = π/π; π(π₯, π‘) = π(π₯) cos ππ‘ , Air: π£ =
; free at x=0, π(π₯) cos . π determined by boundary condition at π₯ = πΏ.
π£
π£
1
π = ππ or
π− π
πΏ
πΏ
2
π΅ sin
, for π¦(π₯) = −π¦(−π₯) (odd function). π΅ = ∫ π¦(π₯) sin
ππ₯
Maxwell Equations for non-magnetic insulators: ππ¬ = 0, ππ― = 0, π × π¬ = −μ
π = πΎπ , π =
, phase velocity π£ = , π£ =
=
√
=
π―
,π×π―=Ο΅
π¬
n: index of refraction
For plane waves π¬ = π¬ π (ππ ) and
π― = π― π (ππ ) ππ¬ = 0, ππ― = 0, π × π¬ = μ ππ― , π × π― = Ο΅ππ¬
Group velocity π£
=
;π£
≠ π£ if π(π) = πππππππππ ππ’πππ‘πππ ππ π, or , in other words, if index of
refraction depends on π or k.
Poynting vector: πΊ = π¬ × π―, irradiance: πΌ =
ππ ππΈ , π =
πΈ =
Normalized Jones vectors (normalization is important for calculation of intensities only) :
1
0
1
1
x-polarized:
, y-polarized:
, right circular polarized:
, left circular polarized:
√ −π
√ π
0
1
Jones matrices:
1 ±1
1 0
0 0
x polarizer:
, y polarizer:
, linear polarizer with ± 45 to horizontal:
±1 1
0 0
0 1
cos π sin π
linear polarizer rotated by angle π from horizontal: cos π
cos π sin π
sin π
1 0
1 0
quarter wave plate, fast axis along x:
, fast axis along y:
0 π
0 −π
1 π
1 −π
right
and left
circular polarizers.
−π 1
π 1
Fabry-Perot interferometer: πΌ = πΌ (
,
=
Snell’s law:
)
/
,
Finesse: πΉ = (
)
, Reflecting F: β± =
√
,
, where π = 1 − π΄ − π
is transmission, A is absorption, R is reflection coeff.
=
=π
Transverse electric: s wave. Transverse magnetic: p wave. Fresnel equations:
π =−
(
(
)
,π
)
=−
(
(
)
, π‘
)
=
(
)
, π‘ =
(
)
(
)
Reflection coefficients, after eliminating π:
π =
√
√
,
π =
√
√ ^
Reflectance (fraction of power reflected): π
= |π|
Reflection matrix for a horizontal surface: Define TM as horizontal, TE as vertical polarization. Jones
−π 0
π‘
0
matrix for reflected light:
; for transmission:
.
0
π
0 π‘
Critical angle sin π = 1/π; Brewster angle (TM reflection is zero) tan π = π.
Spatial coherence estimation (π : source size, π: distance to screen, π: wavelength, β: transverse coherence
length); β = ππ/π
Fresnel Kirchoff integral:
πππ π
π
π
π(π) = −
ππ₯′ ππ¦′
(cos π − cos π )
4π
π
π
Fraunhofer approximation, valid if size of aperture ( πΏ ) small relative to the distance to the source (π′) and
screen (π),
πΏ βͺπ:
+
π(π) = πΆ
ππ₯ ππ¦ π΄(π₯ , π¦ ) π (
/ )(
)
where π = 2π/π, L is the distance between the aperture and the screen, π₯ , π¦ are at the aperture, π₯ , π¦ are
at the screen and π΄(π₯ , π¦ ) is the aperture function.
Single slit of width b, x direction only:
, with π½ = ππ sin π ≈
π(π₯) = πΆ
; intensity πΌ ∝ |π| , πΌ = πΌ
Double slit of width b, distance d (x direction only):
πΌ =πΌ
cos πΎ , where πΎ = ππ sin π ≈
π π π
π π π
1
0
1
1 π‘
Lense of focal length f:
, propagation by distance t:
, inside a dielectric:
−1/π 1
0 1
0
1
0
1
0
Snell’s law (small angle approx.!):
, curved interface:
− 1 π /π
0 π /π
Ray matrices:
π
π
=
Thin lens: = (π − 1)
−
, Thick lens: = (π − 1)
π = 8.854 × 10
π=
or
= 3 × 10
−
+
(
)
π = 4π × 10
, π =
= 344Ω
, π = ππ‘
ππ
π‘/π
1
, π = ππ‘