Formulae
1
sin 𝛼 cos 𝛽 = [sin(𝛼 + 𝛽) + sin(𝛼 − 𝛽)]
2
1
cos 𝛼 cos 𝛽 = [cos(𝛼 + 𝛽) + cos(𝛼 − 𝛽)]
2
1
sin 𝛼 sin 𝛽 = [−cos(𝛼 + 𝛽) + cos(𝛼 − 𝛽)]
2
𝛼±𝛽
𝛼∓𝛽
cos
2
2
𝛼+𝛽
𝛼−𝛽
cos 𝛼 + cos 𝛽 = 2 cos
cos
2
2
𝛼+𝛽
𝛼−𝛽
cos 𝛼 − cos 𝛽 = −2 sin
sin
2
2
sin 𝛼 ± sin 𝛽 = 2 sin
sin(𝛼 ± 𝛽) = sin 𝛼 cos 𝛽 ± cos 𝛼 sin 𝛽
cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽
Complex numbers
𝑧 = 𝑎 + 𝑖𝑏 = 𝐴𝑒 , 𝑅𝑒{𝑧} = 𝑎 = 𝐴 cos 𝜙, 𝐼𝑚{𝑧} = 𝑏 = 𝐴 sin 𝜙,
𝑧 ∗ = 𝑎 − 𝑖𝑏, |𝑧| = 𝑧𝑧 ∗ = 𝑎 + 𝑏 = 𝐴 , tan 𝜙 =
𝑚𝑎 = −𝑘𝑥 , 𝐸 = 𝑘𝑥 + 𝑚𝑣 , 𝑥 = 𝐴 cos(𝜔 𝑡 + 𝛼) , 𝑥 = 𝑅𝑒{𝑧} , 𝑧 = 𝐴𝑒
𝜔 =
𝑘/𝑚 , 𝜔 =
1/𝐿𝐶 , 𝜔 =
𝑚𝑎 = −𝑘𝑥 − 𝑏𝑣 , 𝑥 = 𝐴𝑒
/
𝑔/𝑙
cos(𝜔 𝑡 + 𝜙) , 𝜔 = 𝜔 −
, 𝑄 = 𝜔 /𝛾
𝛾 = 𝑏/𝑚 , 𝛾 = 𝑅/𝐿 , 𝛾 = 1/𝑅𝐶
𝑧 = 𝐴𝑒
,𝑝=
𝜔 −
+ 𝑖 , 𝐴(𝑡) = 𝐴 𝑒
/
, 𝐸(𝑡) = 𝐸 𝑒
, max. A at 𝜔 = 𝜔 −
𝐴(𝜔) =
(
𝑃(𝜔) =
)
=
max. P at 𝜔
𝑏𝜔 𝐴 ≈
N coupled oscillators
𝑦 , = 𝐴 , cos(𝜔 𝑡 + 𝛿 ) , 𝜔 = 2𝜔 sin
(
, 𝐴 , = 𝐶 sin
)
=𝜔
, 𝜔 = 𝑇/𝑚𝑙
n: mode index, p: position index, 𝛿 = 0 if we have only one mode.
Stressed string:
Solid rod:
=
=
Boundary conditions:
fixed at x=0, 𝑓(𝑥) sin
Fourier: 𝑦(𝑥) = ∑
, 𝑣 = 𝑇/𝜇, y(𝑥, 𝑡) = 𝑓(𝑥) cos 𝜔𝑡
, 𝑣 = 𝑌/𝜌; 𝜉(𝑥, 𝑡) = 𝑓(𝑥) cos 𝜔𝑡 , Air: 𝑣 =
; free at x=0, 𝑓(𝑥) cos . 𝜔 determined by boundary condition at 𝑥 = 𝐿.
𝑣
𝑣
1
𝜔 = 𝑛𝜋 or
𝑛− 𝜋
𝐿
𝐿
2
𝐵 sin
, for 𝑦(𝑥) = −𝑦(−𝑥) (odd function). 𝐵 = ∫ 𝑦(𝑥) sin
𝑑𝑥
Maxwell Equations for non-magnetic insulators: 𝛁𝑬 = 0, 𝛁𝑯 = 0, 𝛁 × 𝑬 = −μ
𝜖 = 𝐾𝜖 , 𝑐 =
, phase velocity 𝑣 = , 𝑣 =
=
√
=
𝑯
,𝛁×𝑯=ϵ
𝑬
n: index of refraction
For plane waves 𝑬 = 𝑬 𝑒 (𝒌𝒓 ) and
𝑯 = 𝑯 𝑒 (𝒌𝒓 ) 𝒌𝑬 = 0, 𝒌𝑯 = 0, 𝒌 × 𝑬 = μ 𝜔𝑯 , 𝒌 × 𝑯 = ϵ𝜔𝑬
Group velocity 𝑣
=
;𝑣
≠ 𝑣 if 𝜔(𝑘) = 𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑘, or , in other words, if index of
refraction depends on 𝜔 or k.
Poynting vector: 𝑺 = 𝑬 × 𝑯, irradiance: 𝐼 =
𝑛𝜖 𝑐𝐸 , 𝑍 =
𝐸 =
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:
√ −𝑖
√ 𝑖
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 𝜃 sin 𝜃
linear polarizer rotated by angle 𝜃 from horizontal: cos 𝜃
cos 𝜃 sin 𝜃
sin 𝜃
1 0
1 0
quarter wave plate, fast axis along x:
, fast axis along y:
0 𝑖
0 −𝑖
1 𝑖
1 −𝑖
right
and left
circular polarizers.
−𝑖 1
𝑖 1
Fabry-Perot interferometer: 𝐼 = 𝐼 (
,
=
Snell’s law:
)
/
,
Finesse: 𝐹 = (
)
, Reflecting F: ℱ =
√
,
, where 𝑇 = 1 − 𝐴 − 𝑅 is transmission, A is absorption, R is reflection coeff.
=
=𝑛
Transverse electric: s wave. Transverse magnetic: p wave. Fresnel equations:
𝑟 =−
(
(
)
,𝑟
)
=−
(
(
)
, 𝑡
)
=
(
)
, 𝑡 =
(
)
(
)
Reflection coefficients, after eliminating 𝜙:
𝑟 =
√
√
,
𝑟 =
√
√ ^
Reflectance (fraction of power reflected): 𝑅 = |𝑟|
Reflection matrix for a horizontal surface: Define TM as horizontal, TE as vertical polarization. Jones
−𝑟 0
𝑡
0
matrix for reflected light:
; for transmission:
.
0
𝑟
0 𝑡
Critical angle sin 𝜃 = 1/𝑛; Brewster angle (TM reflection is zero) tan 𝜃 = 𝑛.
Spatial coherence estimation (𝑠: source size, 𝑟: distance to screen, 𝜆: wavelength, ℓ: transverse coherence
length); ℓ = 𝑟𝜆/𝑠
Fresnel Kirchoff integral:
𝑖𝑘𝑈 𝑒
𝑒
𝑒
𝑈(𝒓) = −
𝑑𝑥′ 𝑑𝑦′
(cos 𝜃 − cos 𝜃 )
4𝜋
𝑟
𝑟
Fraunhofer approximation, valid if size of aperture ( 𝛿 ) small relative to the distance to the source (𝑑′) and
screen (𝑑),
𝛿 ≪𝜆:
+
𝑈(𝒓) = 𝐶
𝑑𝑥 𝑑𝑦 𝐴(𝑥 , 𝑦 ) 𝑒 (
/ )(
)
where 𝑘 = 2𝜋/𝜆, L is the distance between the aperture and the screen, 𝑥 , 𝑦 are at the aperture, 𝑥 , 𝑦 are
at the screen and 𝐴(𝑥 , 𝑦 ) is the aperture function.
Single slit of width b, x direction only:
, with 𝛽 = 𝑘𝑏 sin 𝜃 ≈
𝑈(𝑥) = 𝐶
; intensity 𝐼 ∝ |𝑈| , 𝐼 = 𝐼
Double slit of width b, distance d (x direction only):
𝐼 =𝐼
cos 𝛾 , where 𝛾 = 𝑘𝑑 sin 𝜃 ≈
𝑎 𝑏 𝜃
𝑐 𝑑 𝜌
1
0
1
1 𝑡
Lense of focal length f:
, propagation by distance t:
, inside a dielectric:
−1/𝑓 1
0 1
0
1
0
1
0
Snell’s law (small angle approx.!):
, curved interface:
− 1 𝑛 /𝑛
0 𝑛 /𝑛
Ray matrices:
𝜃
𝜌
=
Thin lens: = (𝑛 − 1)
−
, Thick lens: = (𝑛 − 1)
𝜖 = 8.854 × 10
𝑐=
or
= 3 × 10
−
+
(
)
𝜇 = 4𝜋 × 10
, 𝑍 =
= 344Ω
, 𝑑 = 𝑓𝑡
𝑜𝑟
𝑡/𝑛
1
, 𝑑 = 𝑓𝑡