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OPTI 511R, Spring 2015
Due Thursday April 2, 2015
Problem Set 8
Prof. E. M. Wright
The problems of this assignment deal with optical resonators and laser light. You do not need to
know the physics of how a laser works, but you need to assume the following:
(i) Only a single longitudinal and transverse mode of the cavity will be occupied with light.
(ii) One of the mirrors is just slightly transmitting, so that some of the light contained in the cavity
will leak out through this mirror, which we’ll call the “output coupler.” The light leaks out with the
same transverse profile as the occupied mode of the cavity, so if the TEMm,n transverse cavity
mode is occupied, then the light exiting the output coupler is a traveling-wave beam with a TEMm,n
field profile; we would call this a TEMm,n beam. The other cavity mirror has 100% reflectivity.
(iii) The Gaussian parameters of the cavity mode, w(z) and R(z), will also be the Gaussian parameters of the exiting beam until a new optical element such as a lens is inserted into the beam path.
(Remember that the Hermite-Gaussian solutions are valid for free space as well as within a cavity.)
Thus the beam will have a waist w0 that is the same as the waist of the occupied cavity mode, and
that is located in the same position in space (at z = 0) as the cavity mode waist. The transverse
profile of the beam can thus be treated the same as the profile of the cavity mode extended to regions
beyond the cavity, and the spatially dependent beam radius w(z) and wavefront radius of curvature
Rf (z) for the field can be determined from w0 and z0 of the occupied cavity mode. (Note that this
is strictly true only if the output coupler does not act as a lens. We will assume this is true, but in
general the output coupler will indeed make the beam’s Gaussian parameters slightly different from
the parameters of the mode generating the beam.)
1. In this problem, you will examine a few aspects of optical resonator construction. Assume that you
are building a two-mirror optical cavity that will be used to make a 633-nm wavelength Helium-Neon
laser.
(a) The radius of curvature of mirror 1 is R1 = 30 cm. The radius of curvature of mirror 2 is
R2 = 60 cm. Let mirror 2 be the output coupler. The positive radius of curvature of each mirror
indicates that both mirrors are concave. The mirrors are to be placed a distance L apart. For each of
the following ranges of L, indicate whether the resonator will be stable or unstable:
(i)
(ii)
(iii)
(iv)
0 < L < 30 cm,
30 cm < L < 60 cm,
60 cm < L < 90 cm,
90 cm < L.
(b) Let L = 75 cm. What is the Free Spectral Range of the cavity? Give your answer in Hz.
(c) Assume that the optical axis lies along the z axis. The cavity modes will have a beam waist
within the cavity at a position that defines z = 0. Let z1 be the position of mirror 1, and z2 the
position of mirror 2, with z1 < 0 and z2 > 0. Find z1 , z2 , and the Rayleigh range z0 , given the
previous values of R1 , R2 , and L = 75 cm. You may work out the relevant expressions on your
own (a good exercise), or use the ”Laser Physics” textbook (or Class Notes) to look up expressions
for these values in terms of g1 and g2 or R1 , R2 and L.
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(d) From your calculated value of z0 , determine the mode waist (radius) w0 , which is also the Gaussian
waist for the exiting beam (again assuming that the output coupler does not act like a lens).
(e) Suppose that when you turn on the laser, you find that the exiting beam has a TEM0,3 profile, not TEM0,0 . Illustrate the spot pattern for this mode as it would appear on a card inserted into
the beam path.
(f) For the optical cavity parameters used above, give an approximate value for q, the axial mode
number, assuming laser operation at 633 nm.
(g) What is the difference in frequency (in Hz) between the TEM0,0,q mode and the TEM0,3,q mode for
this cavity? At about 633 nm, what is the difference in wavelength between these two modes? Why is it
not as convenient to discuss wavelength differences between modes as it is to use frequency differences?
(h) Immediately after passing through the output coupler, will the exiting beam be diverging away
from a focus, converging towards a focus, or will it be a plane wave?
(i) Suppose R1 = 50 cm and R2 = ∞ (the output coupler is flat). Where is the beam waist located? (Answer this without doing any calculations!)
(j) Suppose R1 = −50 cm and R2 = ∞ (again, the output coupler is flat, but the other mirror
is now convex ). Without doing any calculations, convince yourself that this cavity is unstable, and
speculate what single optical element other than a different mirror might be inserted into the cavity
to make the cavity stable.
2. (a) An optical resonator is to be constructed with two mirrors, one of which is concave, having
a radius of curvature of R1 (where R1 > 0), and the other is convex, having a radius of curvature of
R2 = −A · R1 (where R2 < 0 and A > 0). For what range of cavity lengths L will this resonator be stable? Answer this question by deriving an inequality with L, R1 , and A (all of which
are positive numbers) as your only variables. (HINT: As is often the case in homework and exam
problems, there’s an easy way to solve this problem, and there’s a hard way. The harder way here is
brute force mathematics from the beginning, multiplying everything all out. The easy way involves
first writing out the cavity stability condition 0 ≤ (1 − L/R1 ) · (1 − L/R2 ) ≤ 1, then substituting in
the given expression for R2 . Then before expanding out the middle portion, you should be able to
determine by inspection the condition placed on L by the left inequality. The right hand inequality
can then be easily solved once the middle portion is multiplied out.)
(b) If R1 = 200 cm, and R2 = −150 cm, what specific range of cavity lengths will allow the cavity to be stable?
(c) Assuming a value of L within the region of stability, make a sketch of the cavity geometry of
part (b). Include a qualitative picture of the mirrors’ curvatures, and a representation of the curved
wavefronts of a cavity mode field.
(d) Suppose these mirrors are to be used to make a laser (and assume L is in the stability range).
If R2 is the output coupler, where will the beam waist be located? (Just specify a region in space
relative to the cavity. You do not need to calculate a number).
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3. Suppose that there is a direct line-of-sight between the Optical Sciences building and the observatories on Mt. Lemmon and Mt. Bigelow, approximately 30 km away from OSC. Suppose also that a
TEM0,0 laser beam with a wavelength of λ = 500 nm is propagating from one of these observatories
to the roof of the OSC building. For the following questions, neglect all atmospheric effects.
(a) If the beam has a waist w0 of 50 µm at the observatory, what will be the beam radius at OSC?
Because of the large distance scale of this question, the precise location of w0 at the observatory is
irrelevant. All you really need to know is w0 , z0 (which is derived from w0 ), and the distance beyond
the waist that the beam has propagated.
(b) If the beam has a waist w0 = 0.5 mm, what is the beam radius at OSC?
(c) Telescopes may be used in reverse to turn a small-diameter laser beam into a large-diameter
beam. Suppose that a telescope on Mt. Bigelow is used in this way to create a beam with a waist of
w0 = 0.25 m located at the observatory. What would be the beam radius after it has propagated to
OSC?
(d) If this same beam from part (c) (w0 = 0.25 m) is projected to the moon (≈ 4 × 108 m away)
instead of the roof of OSC, what is the beam radius at the moon?
(e) If the beam with w0 = 0.5 mm (note the units!) is pointed at the moon, what is the beam
radius at the moon?
(f) Suppose that you wanted to completely cover the surface of the moon with the light from a
Gaussian laser beam with a wavelength of 500 nm, so that the radius of the beam w(z) at the surface
of the moon is equal to the radius R of the moon (R ≈ 1.7 × 106 m). The beam originates from the
Earth’s surface. What realistic beam waist w0 at the surface of the earth would produce a
Gaussian radius of w(z) = R at the moon? (Unless your calculator is capable of keeping track of
over 20 digits, you will need to make an approximation to the Gaussian beam waist formula that will
allow you to find a realistic answer.)
4. (a) Suppose you want to use a lens focus a Gaussian laser beam of wavelength λ in order to
obtain a beam waist radius w0 . You have a range of lenses available, so you can effectively choose
whatever w0 you desire. After focus, the beam freely propagates a given distance z beyond the focus.
The beam radius at this position, w(z), is given by the usual Gaussian beam radius formula. As a
function of z and λ, determine the initial beam waist w0 that will be needed in order to
achieve the smallest possible radius w(z) at the specified position z.
(b) As a function of z, what is the Rayleigh range z0 of the beam from part (a)?
(c) If a 633-nm TEM0,0 laser beam is projected from the earth to the moon, what beam waist w0
would you need at the surface of the earth to achieve the smallest possible illuminated spot on the
surface of the moon?
(d) What is the Gaussian radius of the beam from part (c) at the surface of the moon?
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5. Suppose that a two-mirror optical cavity has a frequency difference between any two adjacent
transverse modes (same value of q, but the sum m + n differs by 1) that is equal to one-third of the
cavity free spectral range. If the two mirrors have the same radius of curvature R, what is the length
L of the cavity in terms of R?
6. Assuming that an optical field occupies a single cavity mode specified by mode parameters m,
n, and q, the part of the expression describing the field’s phase variation along the cavity axis (z) is
written as
exp {i[kz − (1 + m + n) tan−1 (z/z0 )]}.
Define the longitudinal phase φ(z) as the real part of the argument of this term, so that
φ(z) ≡ kz − (1 + m + n) tan−1 (z/z0 ).
Assume that we’re analyzing the modes of a cavity with 2 concave mirrors. With the sign conventions
described in class and in the notes, we thus have
R1 > 0, R2 > 0, and z1 < 0, z2 > 0.
(Note that since the mode waist is within the cavity, and z = 0 is defined to occur at the waist
position, z1 is a negative number.)
In propagating from mirror 1 to mirror 2, an optical field will acquire a longitudinal phase shift
∆φ = φ(z2 ) − φ(z1 ),
where φ(zj ) represents the phase of the field at mirror j. In order for the standing wave condition to
be satisfied, we must have ∆φ = qπ, where the positive integer q is the longitudinal mode number.
(a) Beginning with ∆φ = φ(z2 ) − φ(z1 ) = qπ, write out an expression for the mode frequency νmnq
c
in terms of m, n, q (the mode numbers); z1 , z2 , and z0 ; and the cavity free spectral range 2L
(Your
−1
answer will have tan in the expression).
(b) Extra Credit (10 pts) Making use of the expressions for z1 , z2 , and z0 written in terms of
g1 and g2 (these expressions are given in the ”Laser Physics” textbook ), derive the equation for the
resonant frequencies of the cavity. (Hint: you may need to look up the law of cosines.)
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