Consider two coherent sources of light, A and B
λ(ω /2π) = c
Source
Source
Point of
observation
= 2 π/λ
Each of these two relationships defines a family
of surfaces, each surface being a hyperboloid of
revolution.
Source

rA

rB
Source
hyperboloid
of revolution
Several Coherent Oscillators
E. Hecht, "Optics," 4th Edition, Addison Wesley. Chapter-10 (Section 10.1.3 Several
Coherent Oscillations. Section 10.2.8 The diffraction grating.)
R. Feynman, " The Feynman Lectures on Physics," Vol-I , Chapter 30 (Sections 30-1,
30-2, 30-3 Diffraction grating.
ET
Imaginary axis
Real axis
For the moment we do not worry about the value of φ
For the moment we do not
worry about the value of φ
E = n E0
Another way to picture this condition for minimum of intensity is
When we impose the condition for the length of this segment to be
equal to λ,
then the total intensity is zero.
(The top half of the oscillators interfere destructively, one to one,
with the bottom half of the oscillators.
On the sharpness of the peak of max intensity
of the point B
where the intensity
(see figure below)
1
2
12
When nδ/2 was equal to π we obtained a minimum.
Notice another minimum occurs when nδ/2 =2π.
Therefore, there must be MAXIMUM in between. Indeed this occurs
λ2
λ1
n m λ2
Key condition
n m λ2 = n m λ1 + λ1
(As Suggested
in the previous
figure)
The difference between λ1 and λ2
can not be smaller than this value of
∆λ (in order to satisfy the Raleigh
criterion, which ensures the two
wavelength can be distinguish by
using the grating.)
The expression above gives an indication of the resolving power of
the grating. The smaller the value of ∆λ the better the ability of the
grating to resolve components from the incident radiation that have
wavelengths very close to each other.
Since ∆λ is very small compared to λ, in the expression above λ1 can
be replaced by λ2. That is, more generally, that expression is written
as
∆λ = λ/nm