lecture 10

Lecture 10
COHERENCE
Another important consideration in the study of lasers is the interaction of two electromagnetic waves that have only slightly different frequencies, or that originate from points only slightly separated spatially - for example, two closely located but separate laser beams or a single beam illuminating two closely positioned apertures. In such instances the two distinct waves or beams will interfere with each other to produce, in some cases, very dramatic effects. Such effects include the development of longitudinal modes, mode-locking, and frequency multiplication and phase matching.
We will consider two separate waves, with k vectors of k_1 and k_2 and with angular frequencies ?_1 and ?_2, both traveling in the z direction. The electric field amplitude of such waves can be expressed as
E_1=E_1^o e^(-i(k_1 z-?_1 t)) …………. (1-69)
E_2=E_2^o e^(-i(k_2 z-?_2 t)) …………. (1-70)
where E_1^o and E_2^o represent the maximum value of the field amplitudes during their oscillating cycle. We will consider the two waves to be closely located in space, so that the combined wave E is given as
E_1=E_1^o e^(-i(k_1 z-?_1 t))+E_2^o e^(-i(k_2 z-?_2 t)) ……….. (1-71)
We now wish to consider the intensity resulting from these two distinctly separate waves as
I=|E|^2 ………… (1-72)
?I=(E_1^o )?^2+(E_2^o )^2+E_1^o ?E_2^o (e?^(-i([k_1-k_2 ]z-[?_1-?_2 ]t))+e^(-i([k_1-k_2 ]z-[?_1-?_2 ]t))) ……….. (1-73)
?I=(E_1^o )?^2+(E_2^o )^2+?2E?_1^o E_2^o cos?? ……….. (1-74)
I=I_1+I_2+?2E?_1^o E_2^o cos?? ……….. (1-75)
where
?=[k_1-k_2 ]z-[?_1-?_2 ]t ……….. (1-76)
Thus, if we examine (1-75) and assume (for simplicity) that E_1^o=E_2^o, we see for values of ? near or equal to zero (and also near or equal to 2?) that the intensity can be twice that of the sum of the intensities of the two individual waves. Also, if ?=? and (3?,5?,…) then the total intensity will be zero. Thus the two waves are interfering with each other to produce effects not associated with each of the waves separately. Looking at ? more closely, we see that if the two waves have nearly identical properties - if k_1 and k_2 are similar and if ?_2 and ?_2 are similar - then these interference effects are likely to occur and the two waves are said to be coherent. If the frequencies or the k vectors are drastically different, or if there are many waves of different frequencies or originating from very different locations, then the waves do not interfere and are said to be incoherent. Thus, for completely coherent waves the combined intensity would be