Elements of Modern Physics
194
For high E values, a series expansion for P can be written, giving
P = α 1 E(t) + α 2 E(t)
2
+ ...
= α 1 E 0 cos ωt + α 2 E 0
2
cos
2
ωt + ...
(6.90)
Since cos
2
ωt = [1 + cos (2ωt)]/2, the second term builds a field component
with a frequency of 2ω. This is called frequency doubling. For example, when a
dielectric medium is irradiated with a powerful ruby laser beam with λ = 6943
Å, an ultraviolet component with λ ≈ 3472 Å is observed to emerge from the
medium. In general, higher harmonics with frequencies 3ω, 4ω, etc. also may
be present.
If two beams with different frequencies, ω 1 and ω 2 , at least one of them
being a laser beam, are incident on the medium, the non-linear term will have
terms with frequencies 2ω 1 , 2ω 2 , ω 1 + ω 2 and ω 1 – ω 2 . The emerging beam
therefore will contain components with these frequencies. Thus, the effect of a
low frequency beam (e.g., ω 2 in the infra-red region) may be observed in the
optical region by choosing ω 1 in the optical range.
In many substances, the refractive index of the substance increases as the
intensity increases. Thus, the effective refractive index of the material is larger
near the centre of the propagating laser beam so that the rays bend towards the
beam axis. This is known as self-focussing and is again a consequence of nonlinear optics. This property is utilized in fibre-optics communication.
Holography
An extremely interesting application of lasers is to holography, i.e., the
production of the whole or complete, 3-dimensional picture of an object. It is
based on the reconstruction of the electromagnetic fields reflected by the object.
Preparation of the photographic plate: Consider the electromagnetic field
of a laser beam reflected by an object. For simplicity, the reflected beam is
assumed to be a plane wave. This wave with amplitude R is allowed to interfere
with a reference laser beam of amplitude A, at an angle θ, on a photographic
plate [Fig. 6.5 (a)]. The exposed plate registers the interference fringes and is
processed to give a hologram. The intensity registered is [Fig. 6.5 (a)]
I = | A exp (i2π(x – ct)/λ) + R exp (i2π (n . r – ct)/λ)|
2
x = 0
= A
2
+ R
2
+ 2AR cos [2πy (sin θ)/λ]
(6.91)
where it is assumed that A and R are real. The separation between the interference
fringes is
d = sin
λ
θ
(6.92)
Reconstruction of the wavefront: A similar reference beam is allowed to
fall on the hologram which acts as a diffraction grating (grating separation
d = λ/sin θ) and produces diffraction images. The angular separation between
the central maximum and the first maximum on the two sides, is [see Fig. 6.5(b)]
194
For high E values, a series expansion for P can be written, giving
P = α 1 E(t) + α 2 E(t)
2
+ ...
= α 1 E 0 cos ωt + α 2 E 0
2
cos
2
ωt + ...
(6.90)
Since cos
2
ωt = [1 + cos (2ωt)]/2, the second term builds a field component
with a frequency of 2ω. This is called frequency doubling. For example, when a
dielectric medium is irradiated with a powerful ruby laser beam with λ = 6943
Å, an ultraviolet component with λ ≈ 3472 Å is observed to emerge from the
medium. In general, higher harmonics with frequencies 3ω, 4ω, etc. also may
be present.
If two beams with different frequencies, ω 1 and ω 2 , at least one of them
being a laser beam, are incident on the medium, the non-linear term will have
terms with frequencies 2ω 1 , 2ω 2 , ω 1 + ω 2 and ω 1 – ω 2 . The emerging beam
therefore will contain components with these frequencies. Thus, the effect of a
low frequency beam (e.g., ω 2 in the infra-red region) may be observed in the
optical region by choosing ω 1 in the optical range.
In many substances, the refractive index of the substance increases as the
intensity increases. Thus, the effective refractive index of the material is larger
near the centre of the propagating laser beam so that the rays bend towards the
beam axis. This is known as self-focussing and is again a consequence of nonlinear optics. This property is utilized in fibre-optics communication.
Holography
An extremely interesting application of lasers is to holography, i.e., the
production of the whole or complete, 3-dimensional picture of an object. It is
based on the reconstruction of the electromagnetic fields reflected by the object.
Preparation of the photographic plate: Consider the electromagnetic field
of a laser beam reflected by an object. For simplicity, the reflected beam is
assumed to be a plane wave. This wave with amplitude R is allowed to interfere
with a reference laser beam of amplitude A, at an angle θ, on a photographic
plate [Fig. 6.5 (a)]. The exposed plate registers the interference fringes and is
processed to give a hologram. The intensity registered is [Fig. 6.5 (a)]
I = | A exp (i2π(x – ct)/λ) + R exp (i2π (n . r – ct)/λ)|
2
x = 0
= A
2
+ R
2
+ 2AR cos [2πy (sin θ)/λ]
(6.91)
where it is assumed that A and R are real. The separation between the interference
fringes is
d = sin
λ
θ
(6.92)
Reconstruction of the wavefront: A similar reference beam is allowed to
fall on the hologram which acts as a diffraction grating (grating separation
d = λ/sin θ) and produces diffraction images. The angular separation between
the central maximum and the first maximum on the two sides, is [see Fig. 6.5(b)]