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Unformatted text preview: Gaussian Beams Chap. 3 80
e. Thus this tion takes place in the TEMggg'mod
ompared to the laser,* we attempt to ensure that oscilla
mode has the greatest intensity (power/area) for the minimum beam spread as c all other modes or ﬁeld distributions.
Note that the TEMM mode has a uniphase surface, albei in phase on this spherical surface. The term spatial coherence i that is, the ﬁeld has one common phase on this spherical surface. ﬁeld of the TEMLO mode reverses direction for negative x; and for the ﬁeld reverses direction many times. Within each dot, the equiph
spherical curvature as the TEMOJ} mode. _ This also explains why a ﬂashlight beam spread even with the parabolic reﬂector and large aperture on ﬁlament of the tungsten wire radiate an incoherent wave; th
of atoms bears little, if any, relationship to another group. Consequently, we cannot, by any stretch of imagination, identify the spot from a ﬂashlight as being a uniphase surface. The
characteristic dimension corresponding to we is much, much smaller than the physical size of the parabolic reﬂector; hence, its divergence is quite large compared to a laser. t curved but still the ﬁeld is
s used to describe this fact;
For contrast, note that the
the higherlorder modes,
ase surface has the same s so much faster than a laser beam,
the former. The atoms in the heated
at is, the phase from one group PROBLEMS s are intended as a review and
Gaussian beam modes.
atics, or a few sentences: 3.1. The following question to test your understanding and
appreciation of the Hermite Answer these questions with a
sketch, some simple mathem r, (a) What is the physical sign'ﬁcance of the distance Zo
(b) If z = zo and r2 = 2(20), by how much does the phase of the ﬁeld lag that at r = 0‘?
(c) Which factor expresses the idea that the beams are not plane waves and the
phase velocity is greater than c?
rtain commercial helium/neon laser is
‘ adian at 10 = lead or advertised to have a farﬁeld di
6323 nm. What is the spot size mo?
(b) The power emitted by this laser is per centimeter at r = z = 0‘?
(c) How many photons per second are emitted by this laser beam?
ly come in packages of hu. If one more photon (d) Electromagnetic energy can on
er, what is the new power speciﬁcation? _ per second were emitted by this las
art of the problem is to recognize that there is a time and a (The point of this p
place for making the distinction between a classical ﬁeld and a photon: Should we start here?)
3.3. Given a lW TEMM) beam of lo : 5 14.5 nm from an argon ion laser wi spot size of mo = 2 mm located at z z: 0 5 mW. What is the peak electric ﬁeld in volts th a minimum ar, and laser communications with iderations all apply to focusing.
spread is a factor in raw power sideration. For a laser transit, laser rad mum beam spread. But these same cons
lled by we. Thus, beam *The beam spread is always a con freespace transmission, we desire a mini
The smallest spot size achievable by a lens is also contro applications. Problems (a) How far will this beam propagate before the spot size is 1 cm?
(b) What is the radius of curvature of the phase front at this distance?
(c) What is the amplitude of the electric ﬁeld at r = 0 and z : 0?
3.4. A IOW argon ion laser oscillating at 4880 A has a minimum spot size of 2 mm.
(a) How far will this beam travel before the spot size is 4 mm?
(b) What fraction of the 10 W is contained in a hole of diameter 210(2)?
(c) Express the frequency/wavelength of this laser in eV, nm, urn, v(IIz), and
v(cm 1).
(d) What is the amplitude of the electric ﬁeld when w— — 1 cm? 3.5. Sketch the variation of the intensity with x (y— — 0) of a beam containing 1 W of
power in a TEMQO mode and 1/2W in the TEMLO mode (i.e., total power=l.5W).
There are two possibilities: (1) The frequencies and the phases of the two modes
are the same, in which case we should add the ﬁelds and then square to obtain the
intensity, or (2) the phase changes with respect to time. If this change is fast with
respect to the observation time, then we should add the intensiﬁes. 3.6. Consider a linear combination of two equal amplitude TEMW, modes given by:
E = E0 {(TEMLOmy :l: ﬁres/10.0%.} (3) Sketch the “dot” pattern or equal intensity contours for each component (i.e.,
a; or 3),). Indicate the direction of the electric ﬁeld. (b) Sketch the pattern for the linear combination. (c) Label the positions where the intensity is a maximum and a minimum. (This is
sometimes referred to as the “donut mode” or TEMSJ. 3.7. The intensity of a laser has the following visual appearance when projected on a
surface. (a) Name the mode (i.e., TEMmyp; m = ?; p = ‘2). (b) A plot of the relative intensity of another mode as a function of x (for y = 0) is
shown below at the right. The variation with respect to y is a simple bellshape
curve. What is the spot size w? this 0 1mm 3.8. Suppose that a TEMm, p mode impinged on a perfectly absorbing plate with a hole
of radius a centered on the axis of the beam. Plot the transmission coefﬁcient of this Illllwilllll ' ‘ “iv”V't‘iﬁhiar inl'irlrl'rirf'iﬁ “‘ Gaussian Beams Chap. 3 hole as a function of the ratio of a/w for the (0,0), (0,1), and (1,1) modes assuming
that the ﬁelds are not affected by the plate. . Show that the Hermite—Gaussian beam modes are orthogonal in the following sense: Re [f(E,,,,,, x H316.) as] = 0 . Repeat the analysis from (3.2.6) to (3.3.14) for the case where the index of refraction
is nonuniform and is given by n(r) = n0(1 — W212) The same arguments advanced for the derivation of (3.2.4) can be used for the mag
netic ﬁeld intensity H. Check the accuracy of this equation by considering a dominant
TE”, mode in a rectangular waveguide of width a and height 19 and computing the
ratio of Hz/Hx. . The news media has shown the astronauts placing laser retroreﬂectors on the moon.
Use the expansion law for Gaussian beams to predict the diameter of a laser beam
when it hits the moon. Use A0 = 6943 A. Consider two cases: (a) A laser rod of 2 cm diameter. (b) This same laser sent through a telescope backward so that the beam starts with
a diameter of 2 m. (c) Eye damage intensities are in the range of 10 uW/cmz. If the laser on earth
produced a pulse power of 10 MW, was there danger to the astronauts from the
optical radiation? . Verify the ABC D law for a continuous lens by starting with (3.6.l6b) and following
the analysis of Sec. 3.1 through 3.3. A convenient, if oversimpliﬁed, deﬁnition of a focal length of a lens is that it converges
a parallel beam of light to a point. But if the spot size at the focus were zero, as implied
by a point, the expansion of the beam would be inﬁnitely fast and by symmetry would
also correspond to its convergence, both statements being obvious contradictions. Use
a simple geometric argument based on the convergence (and expansion) to estimate
the minimum spot size in the focal region of a lens. Compare with the exact answer. . Suppose that a Gaussian beam with w :2 cm and a planar wave front impinges on
alens of focal length f = 4cm (A0 = 1.0,um).
(a) If z = 0 is the location of the lens, where does the output beam reach its
minimum spot size?
(b) What is the farﬁeld expansion angle? . Repeat the analysis of Sec. 3.1 and 3.3 for a medium in which the dielectric constant
is complex and depends on r in the following manner: 2
60') = 60 6’ — je” (1 —~ r7)
la ...
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 Fall '10
 Anderson

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