ABSTRACT
The relation between the electric field and the spectral function will be as follows: For a monochromatic wave with the angular frequency w0, the electric field is Ez(z, t) = E0ei(w0t – kz) (23.6) Setting the position at z = 0, we get E(t) = E0eiw0t (23.7) Equation 23.7 corresponds to the spectral function S(w) as S E t e dti t( ) ( )w w= -
Ú (23.8)
Ú0 0 (23.10) Due to d w w
Ú Ú¢ ¢ ¢12 1
2 e dt e dti t i t (23.11)we get S(w) = 2pE0d(w – w0) (23.12) If S0(w) = 2pE0(w) corresponds to the amplitude of the spectral component with the angular frequency w, then
S(w) = S0d(w – w0) (23.13) For a multiple wavelength wave, the electric field isE t E ei t( ) ( )=Â 0 w w
E t S ei t( ) ( )= Â1 2
(23.15) In the case of a continuous-wavelength wave, the electric field is modified as
E t S e di t( ) ( )= -•
Ú12p w w w (23.16)
Equation 23.16 also corresponds to the spectral function S(w) as S E t e dti t( ) ( )w w= -
Ú (23.17)
S S e d e dti t i t( ) ( ) 'w p w ww w= -•
• -ÚÚ 12 ¢ ¢ (23.18)
S S e dt di t( ) ( ) ( )w p
ÚÚ12 ¢ ¢ ¢ (23.19)
Due to d w w
Ú Ú¢ ¢ ¢12 1
2 e dt e dti t i t (23.20)
we get S S d( ) ( ) ( )w w d w w w= -
Ú ¢ ¢ ¢ (23.21)
The interference signal can be experimentally performed by a Michelson interferometer, as shown in Fig. 23.1.
