Download Beam Optics

MEFT / Quantum Optics and Lasers
Suggested problems – Set 2
Gonçalo Figueira, spring 2015
Note: some problems are taken from “Fundamentals of Photonics”, in which case the
corresponding number is indicated. At the end of the list you may find the solutions to
selected problems.
Beam Optics
(3.1-1) Parameters of a Gaussian Laser Beam
A 1 mW He-Ne laser produces a Gaussian beam at a wavelength of λ = 633 nm with a spot
size 2W0 = 0.1 mm.
(a) Determine:
• the angular divergence of the beam
• its depth of focus, and
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• its diameter at z = 3.5 × 10 km (approximately the distance to the moon).
(b) What is the radius of curvature of the wavefront at z = 0, z = Z0, and z = 2Zo?
2
(c) What is the optical intensity (in W/cm ) at the beam center (z = 0, ρ = 0) and at the axial
point z = z0? Compare this with the intensity at z = z0 of a 100 W spherical wave produced by
a small isotropically emitting light source located at z = 0.
(3.1-3) Determination of a Beam with Given Width and Curvature
Consider a Gaussian beam whose width W and radius of curvature R are known at a
particular point on the beam axis (see figure). Show that the beam waist is located to the left
at a distance
z=
R
1+ (λR / πW 2 )2
and that the waist radius is given by
W0 =
W
(
1+ πW / λR
2
)
2
.
(3.1-6) Beam Parameters
The light emitted from a Nd:YAG laser at a wavelength of 1.06 µm is a Gaussian beam of 1 W
optical power and beam divergence 2θ0 = 1 mrad. Determine
a) the beam waist radius W0
b) the depth of focus 2Z0
c) the maximum intensity
d) the intensity on the beam axis at a distance z = 100 cm from the beam waist.
(3.2-6) Beam Focusing
An argon-ion laser produces a Gaussian beam of wavelength λ = 488 nm with waist radius
W0 = 0.5 mm. Design a single-lens optical system for focusing the light to a spot of diameter
100 µm. What is the shortest focal-length lens that may be used?
Suggestion: use the following formulas that relate the Gaussian beam parameters before
(unprimed, e.g. W0) and after (primed, e.g. W’0) a lens of focal distance f.
(3.2-9) Transmission of a Gaussian Beam Through a Graded-Index Slab
The ABCD matrix of a SELFOC graded-index slab with quadratic refractive index
n(y ) ≈ n 0 (1− 21 α 2 y 2 ) and length d is
⎡A B ⎤ ⎡ cos αd
⎢C D ⎥ = ⎢−α sinαd
⎣
⎦ ⎣
(1/ α)sinαd ⎤
cos αd ⎥⎦
for paraxial rays along the z direction. A Gaussian beam of wavelength λ0, waist radius W0 in
free space, and axis in the z direction enters the slab at its waist. Use the ABCD law to
determine an expression for the beam width in the y direction as a function of d. Sketch the
shape of the beam as it travels through the medium.
Laser Drilling. A CO2 laser (wavelength 10.6 µm) is used to drill a hole into a copper plate
with a thickness of 1.0 mm. The beam waist is located in the middle of the plate.
a) What is the smallest beam diameter that can be obtained at the surface and what is
the corresponding diameter at the beam waist? (Suggestion: Write W(z) as a function
of W0 and the thickness of the plate. Calculate the derivative and make it equal to
zero.)
b) Sometimes it is better to make a hole with only a slight variation in the diameter. What
is the diameter at the surface and at the beam waist, if the difference between the two
must not be larger than 0.5%?
Electromagnetic Optics
(5.2-1) Dielectric Media
Identify the media described by the following equations, regarding linearity, dispersiveness,
spatial dispersiveness, and homogeneity. Assume that all media are isotropic.
P = ε 0 χE − a∇ × E
P + aP 2 = ε 0E
a1
∂2P
∂P
+ a2
+ P = ε 0 χE
2
∂t
∂t
{
[(
P = ε 0 a1 + a 2 exp − x 2 + y 2
)]}E
(5.4-1) Electric Field of Focused Light
2
a) 1 W of optical power is focused uniformly on a flat target of size 0.1 x 0.1 mm placed
in free space. Determine the peak value of the electric field E0 (V/m). Assume that the
optical wave is approximated as a TEM plane wave within the area of the target.
b) Determine the electric field at the center of a Gaussian beam (a point on the beam
axis at the beam waist) if the beam power is 1 W and the beam waist radius W0 = 0.1
mm.
Resonator Optics
(10.1-3) Resonance Frequencies of a Resonator with an Etalon
a) Determine the spacing between adjacent resonance frequencies in a resonator
constructed of two parallel planar mirrors separated by a distance d = 15 cm in air (n
= 1).
b) A transparent plate of thickness d1 = 2.5 cm and refractive index n = 1.5 is placed
inside the resonator and is tilted slightly to prevent light reflected from the plate from
reaching the mirrors. Determine the spacing between the resonance frequencies of
the resonator.
(10.1-7) Optical Energy Decay Time
How much time does it take for the optical energy stored in a resonator of finesse  = 100,
length d = 50 cm, and refractive index n = 1, to decay to one-half of its initial value?
(10.2-1) Maximum Resonator Length for Confined Rays
A resonator is constructed using concave mirrors of radii 50 cm and 100 cm. Determine the
maximum resonator length for which rays satisfy the confinement condition.
(10.2-3) Resonance Frequencies of a Confocal Resonator
A symmetric confocal resonator has a length d = 30 cm, and the medium has refractive index
n = 1. Determine the frequency spacing νF and the displacement frequency (Δζ/π)νF.
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9
Determine all resonance frequencies that lie within the band 5×10 ± 2×10 Hz.
(10.2-5) Stability of Spherical-Mirror Resonators
a) Can a resonator with two convex mirrors ever be stable?
b) Can a resonator with one convex and one concave mirror ever be stable?
(10.2-11) Gaussian Beam in a Symmetric Confocal Resonator
A symmetric confocal resonator with mirror spacing d =16 cm, mirror reflectances 0.995, and
n = 1 is used in a laser operating at λ0 = 1 µm.
a) Find the radii of curvature of the mirrors.
b) Find the waist of the Gaussian mode.
c) Assuming that losses arise only from imperfect mirror reflectances, determine the
distributed resonator loss coefficient αr.
Photon Optics
(12.1-1) Photons in a Gaussian Beam
a) Consider a single photon described by a Gaussian beam. What is the probability of
detecting the photon at a point within a circle whose radius is the waist radius of the
beam, W0? Recall that at the waist 𝑧 = 0 , 𝐼(𝜌, 𝑧 = 0) ∝ exp (−2𝜌 ! /𝑊!! ), where ρ is
the radial coordinate.
b) If the beam carries a large number n of independent photons, estimate the average
number of photons that lie within this circle.
(12.1-2) Photons-Momentum Recoil
198
Calculate the recoil velocity imparted to a Hg atom that has emitted a photon of energy
4.88 eV. Compare this with the root-mean-square thermal velocity v of the atom at a
temperature of T = 300 °K (obtained by setting the average kinetic energy equal to the
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average thermal energy, !!𝑚𝑣 ! = !!𝑘𝑇, where k = 1.38 × 10 J /K is Boltzmann's constant).
(12.1-6) Position of a Single Photon at a Screen
Consider a monochromatic light beam of wavelength λ0 falling on an infinite screen in the
plane z = 0, with an intensity𝐼 𝜌 = 𝐼! exp (−𝜌/𝜌! ), where 𝜌 = 𝑥 ! + 𝑦 ! . Assume that the
intensity of the source is reduced to a level at which only a single photon strikes the screen.
a) Find the probability that the photon strikes the screen within a radius ρ0 of the origin.
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b) If the beam contains exactly 10 photons, how many photons strike within a circle of
radius ρ0 on average?
(12.1-7) Momentum of a Free Photon
Compare the total momentum of the photons in a 10 J laser pulse with that of a 1 g mass
moving at a velocity of 1 cm/s and with an electron moving at a velocity c0/10.
(12.2-2) Photon Flux
Show that the power of a monochromatic optical beam that carries an average of one photon
per optical cycle is inversely proportional to the squared wavelength.
(12.2-4) Photon Statistics of a Coherent Gaussian Beam
Assume that a 100 pW He-Ne single-mode laser emits light at 633 nm in a Gaussian beam.
a) What is the mean number of photons crossing a circle of radius equal to the waist
radius of the beam W0 in a time T = 100 ns?
b) What is the variance and signal to noise ratio of the number of photon counts in (a)?
c) What is the probability that no photons are counted in (a)?
Selected Solutions
3.1-1
a) 8 mrad; 24.8 mm; 2800 km
b) ∞, 2z0, 5z0/2
2
2
2
c) I(0,0) ≈ 25.5 W/cm ; I(0,z0) ≈ 12.7 W/cm ; I ≈ 5.2 W/cm
3.1-6
a) W0 ≈ 675 µm
b) 2z0 ≈ 2.7 m
c) I0 ≈ 140 W/cm2
2
d) I(1 m) ≈ 90 W/cm
3.2-6
One possible solution is (it’s advisable to choose integer values for f, e.g. 40 cm, 50 cm, etc):
f = 40 cm; z ≈ 159 cm; z’ ≈ 44.7 cm
f ≥ z0M ≈ 32 cm
Laser drilling
a) W0 = 41 µm, W(d/2) = 58 µm
5.4-1
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a) 2.75×10 W/m
10.1-3
a) 1 GHz
b) W0 = 129.8 µm, W(d/2) = 130.45 µm
5
b) 2.2×10 W/m
b) 0.92 GHz
10.1-7
36.8 ns
10.2-1
1.5 m
10.2-3
14
9
5×10 + [–1.75, –1.25, –0.75, –0.25, +0.25, +0.75, +1.25, +1.75] ×10 Hz
12.1-1
a) 0.865
b) 0.865n
12.1-2
7.87 mm/s
12.2-4
a) 27.6
b) 27.6
c) ~1×10
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