Nonlinear optical second harmonic generation

Nonlinear optical second harmonic
generation
I S Ruddock
Department of Physics and Applied Physics, University of Strathclyde, Glasgow, G4 ONC, UK
Received 22 February 1992, in final form 11 October 1993
Abstract. The generation of the second harmonic of the
632.8 MI light from ao He-Ne laser is described as the basis
of an undergraduate experiment. The dependence of the
signal on phase-matching angle and intensity of the
fundamental beam is discussed.
Rerumea. La gcnermbn del sepundo ~rmonicn dr In Iuz a
632 8 nm producio por un laser de Hz-h es descnto mmo
bru para un crpenmenio a nivel de hCZnCl31urd La
dependcnas de I3 slnd cambrando el anylo del cnsLaI y la
intesidad del ra,o fundamental ci dxuudo
1. Introduction
Second harmonic generation (SHG) is probably the
archetypal nonlinear optical process as it was the
first to be demonstrated 111 soon after the invention
of the laser, it is easy to perform and understand
and it contains within it the important elements of
nonlinear optics in general, e.g. intensity dependence
and phase-matching. SHG is commercially important
since frequency doubled infra-red lasers or dye lasers
excited by them are the usual sources of powerful
visible light pulses.
The design and operation of a wide range of existing
and potential devices involving light can be
understood in terms of the concepts of nonlinear
optics. However, illustrative examples are rarely
found as the basis of experiments in undergraduate
laboratories. When the Department of Physics and
Applied Physics at the University of Strathclyde
launched the BSc Honours course in Laser Physics
and Optoelectronics in 1985, SHG was one of the
experiments included in the new laboratory for the
third and penultimate year of this course. It
reinforces aspects of the lecture material given at
third year in nonlinear optics, laser physics and
instrumentation. Since 1985, the class has more than
doubled in size and the SHG experiment is now provided
in triplicate as it is recognized as a core topic all
students should perform at this stage.
In this paper, the experiment as performed in a
teaching laboratory is described and typical results
are presented to show the relevance of it in the curriculum
of a degree of this kind. The areas covered
are specifically: dependence of second harmonic
power on (a) angle in the vicinity of the phasematch-
ing direction, (b) fundamental power and (c) fundamental
spot size in the crystal; the last being an
exercise in the use of Gaussian beam optics.
2. Background and theory
The theoretical framework of SHG has been discussed
extensively in the literature during the past
thirty years and comprehensive surveys of the topic
may now be found in most student textbooks on
laser physics, modem optics and quantum or optoelectronics
[2-41. In addition, Dunn and Akerhoom
[5] give a good introduction to the theory and a
detailed account of the experimental realization of
SHG including the growth of a nonlinear crystal;
consequently only a resume of the basics of the
subject will he given here.
2.1 Nonlinear polarization
The second harmonic of a light wave of frequency w
is produced by a crystal if the induced electric polarization
has a component oscillating at 2w. This
happens when the polarization is a nonlinear function
of the electric field as occurs at or near the focus
when a laser beam is focused inside the crystal.
The relationship between polarization and electric
field can be represented by a polynomial of the form:
where xi,xz,x3,, . , are the first, second, third, . . .
order components of the susceptibility. The frequencies
of the terms in the expression for the polarization
P are the original frequency, w. and the higher

54 I S Ruddock
frequencies 2w, 3w, ... giving rise to harmonics of the
original frequency w. Because x2, x3,. .. are small
compared with xI, nonlinear optical effects were
not observed until after the invention of the laser in
1960.
The simplest way to obtain an expression linking
the second harmonic intensity with that of the fundamental
and the length of the crystal etc., is to Erst
assume that a wave El of amplitude O,, angular
frequency w, and wave number k,,
E, =Pl[exp i(k,r-qf)+exp-i(k,r-wlt)],
propagates in the z direction through the nonlinear
medium and generates a second harmonic wave E2
of amplitude 82, angular frequency zWl and wave
number k2,
Ea = P2[exp i(kp - zWl f) + exp -i(kzz - zWlr)],
(2)
Since the amplitude of the second harmonic grows
with distance z, &t2/az must be deduced and then
integrated over the thickness, 1, of the crystal.
In a nonmagnetic material of permitivity 6, the
wave equation is
2 a2E
V E = PO€-
az
az
= PO- [€DE + PI.
as
For the second harmonic wave equation, all polarization
wmponents oscillating at zWl must be
included in P and so (3) becomes:
(3)
gating a distance 1 through the crystal is given by
[ iAk ]
- ix2(2w1)8: expiAkl- I
4cn2
(7)
where Ak = k2 - Zk,, the mismatch parameter.
The average second harmonic intensity generated
at the exit of the crystal, I,, is thus
where use has been made of the general expression
I= L”,”
Only media which lack inversion symmetry, for
example anisotropic crystals, possess non-zero x2.
That this is so can be seen by inspection of equation
(1) since in a centrosymmetric crystal, reversal of the
electric field must leave the magnitude of the polarization
unchanged. A consequence of a crystal being
anisotropic is that it exhibits birefringence, but this
in turn can be exploited to produce ‘phasematching’
resulting in the efficient generation of the nonlinear
signal.
2.2 Phasematching
If Ak # 0, as is normally the case due to dispersion,
then the average second harmonic intensity, I, is proportional
to sin2(Akl/2) and it oscillates with distance
through the crystal. The second harmonic
-1
ir (4) waves generated at different points along the beam’s
V2E2 = I.L~~~~~~[E~+xIE~+x~~I.
The quantities within the brackets represent
respectively, the linear pola,+zation contrim
bution due to the second harmonic Wave itself
and the nonlinear driving term produced by the
fundamental.
Recognizing that the refractive index at the
second harmonic, n2 is related to the susceptibility
evaluated at zW,, by n: = 1 + x,, (4) may be written
as
a2
V2E2 =poeo-[n2E +xzEI]. 2
a
At the intensity levels encountered in these experiments,
the conversion efficiency is low and thus the
second harmonic amplitude grows slowly with
distance through the capital. In this case it may be
assumed that #8,/az2 < k2.a82/az, then substitution
of (2) as the solution of the second harmonic
wave equation (5) yields
path do not reinforce each other as they are not
travelling at the same speed as the fundamental wave.
Chaotic interference occurs and the second harmonic
wave does not succeed in growing in intensity. If
however, Ak=o,
(9)
and I, is proportional to l2 giving rapid growth of the
second harmonic intensity with distance. Phasematching
is said to occur under these conditions as
the refractive indices and phase velocities at the fundamental
and second harmonic Frequencies are equal
and thus X2 = XI 12.
The crystal used in this experiment, Ammonium
Dihydrogen Phosphate (ADP), is negatively uniaxial
with no > n. and its dispersion curves are shown
schematically in figure 1. It is clear that there exists
in this crystal a direction 9 relative to the optic axis
such that the ordinary index at XI is equal to the
extraordinary at A,, given by

Nonlinear optical second harmonic generation 55
.t \
Figure 1. The wavelength dependence of refractive index
for a negatively uniaxial crystal such as ADP. At M angle
B to the optic axis, the ordinary index at the fundamental
wavelength, A, is equal w the extraordinary at the second
harmonic wavelength, A,.
The opposite scheme applies in a positive uniaxial
crystal although both cases are referred to as Type
1 phasematching. Type 2 occurs when both ordinary
and extraordinary fundamental photons combine to
produce the second harmonic.
3. Experimental arrangement
Figure 2 shows schematically the basic set-up
required for the observation of SHG by a 632.8 nm
He-Ne laser. The plane polarized 2mW laser beam
is focused by a x 10 microscope objective into an
ADP crystal positioned on a turntable. The objective
is in a threaded flange with a rack and pinion
adjnstment so that the focus of the laser beam can
be tracked through the crystal. The position of the
lens relative to the crystal is measurable to a precision
of 5 pm by means of a standard engineer's dial
gauge indicator.
The UV light generated is detected by a
Hamamatsu IP28 photomultiplier tube through a
Schott Glass UGll fdter to block the red laser
light. A 632.8nm interference filter is also necessary
to reject the intense blue and UV radiation emitted
by the front mirror of the He-Ne laser which would
otherwise obscure the sewnd harmonic. Although
the signal is sufficiently strong for an oscilloscope
to be more than adequate for locating it and making
quantitative measurements, it is also convenient to
chop the laser beam and use a lock-in amplifier or
gated integrator. We have found the Bentham
Model 223 amplifier and Delta Developments
chopped signal integrator to be suitable for this
purpose.
The entire experiment is assembled on a single
triangular optical bench using simple saddles and
components as available from Precision Tool and
Instrument CO Ltd, except for the crystal turntable
which was supplied by Ealing Electro-optics and
has a Vernier angle scale. The x 10 microscope ohjective
(Ealing Cat. No 11-8265) is a low-cost flat
mounted lens with a large working distance enabling
the crystal to be rotated at the focus of the laser
beam.
The ADP crystals used here are 2.15 and 0.36mm in
thickness and are cut approximately for Type 1
phase-matching at 633 nm. Since ADP is negatively
uniaxial, the crystal and laser must he orientated relative
to each other such that the light is always incident
in the ordinary polarization plane as the
crystal is rotated; this is accomplished by rotating
the crystal about an axis parallel to the plane of
polarization of the laser beam but perpendicular to
the plane containing the crystal's optic axis (see
figure 2). If suitable nonlinear crystals are not available,
samples of urea may be easily grown in a couple
of days in a beaker and used straightaway without
polishing; at room temperature and a wavelength of
632.8 MI, the phasematching scheme is Type 2 (see
[SI for experimental details).
4. Experiments
SHG is immediately apparent if the laser is focused
on the crystal, which is being rotated while the photomultiplier
outpnt is observed on an oscilloscope. If
for any reason the effect is difficult to observe, the
lock-in amplifier phase can be correctly adjusted by
first removing the interference filter and then detecting
the chopped spontaneous emission from the laser
discharge.
,I
,I
U
Figure 2. Experimental arrangement for
generating and detecting the second harmonic

56 I S Ruddock
4.1 Phasematching angle
Intense second harmonic is only generated when the
fundamental and second harmonic waves are travelling
at the same phase velocity within the crystal.
As outlined in the theory this can be arranged by
exploiting the birefringence of the crystal to find a
direction in which the refractive indices are the
same. Data points (a) in figure 3 show the variation
in second harmonic power with angle of incidence
for the thin crystal. Depending on the accuracy of
the cut of a crystal, the angle of incidence may be
quite small; ideally it should be zero.
The angular tolerance for a particular set-up is
more properly demonstrated by converting the
measured angles of incidence to the corresponding
angles of refraction by Snell's law; the refractive
index of the crystal at 632.8~11 being obtained from
the Sellmeier equation [6]. Data points (b) show the
second harmonic power as a function of internal
angle which as expected is considerably narrower
than that of the external, due solely to refraction.
The residual width of the peak is a measure of the
laser beam's convergence and divergence as it passes
through the crystal.
If the phasematching angle, i.e. the angle between
the crystal's optic axis and the direction of light propagation,
is calculated using (IO), it is also instructive
for students to deduce, from their measurement of
the angle of incidence, the possible directions of the
optic axis within their crystal samples.
4.2. lntensiiy dependence
42.1. Laser power. Since SHG is mediated by the
second component, xz, in the expansion of the
susceptibility, the harmonic intensity generated is
proportional to the square of the fundamental
intensity. If, as is usually the case, the beam crosssectional
area is kept constant, then the signal is
Flgure 3. Dependence of the second harmonic signal on
(a) the fundamental beam's angle of incidence on the
crystal and @) the angle of refraction for an ADP crystal
of thickness 0.36mm. The angles of incidence and
refraction are related by Snell's law evaluated using the
ordinary refractive index at 632.8 om.
1
+I t .
proportional to the square of the laser power.
Although laboratory He-Ne lasers are normally of
fmed power outputs, they can be easily controlled by
means of neutral density (ND) filters. With NDs of
0.1, 0.2, 0.4 and I used singly and in combination,
transmissions of0.80,0.63,0.50,0.40,0.32,0.25,0.20
and 0.10 can be selected to conveniently control the
laser power incident on the crystal. (See [5] for the use
of polarizers for this purpose.) The second harmonic
power as a function of the incident laser power is
shown plotted in figure 4. The parabolic nature of the
curve is apparent and is easily confirmed by
replotting as an In-ln graph (figure 5). Most of the
points fit well to a line of slope 2.
4.2.2. Beam cross-sectional area. The intensity
dependence can also be demonstrated by varying
the size of the focused laser spot. This is achieved by
moving the microscope objective relative to the
noalinear crystal. The subsequent analysis uses the
propagation equations for Gaussian beams, theory
normally dealt with in a Laser Physics lecture course
but not often applied in an undergraduate
laboratory. Typical results for when the lens is
moved by k2.5mm on either side of the position of
maximum second harmonic signal are shown in
figure 6, data points (a) and @) for the thick and thin
crystals respectively. The dramatic increase in the
efficiency of the second harmonic process as the
tightest Focus passes through the crystal is clearly
illustrated.
From (9) and section 4.2.1, the second harmonic
intensity, la, is proportional to the square of the fundamental
intensity, If. Now, if the laser spot in the
crystal is approximated to be a disc of radius IV of
uniform intensity, then I2 is clearly inversely proportional
to w4. However, Iz as described by (8) is the
intensity of the second harmonic beam as it exits
the crystal and not that at the aperture of the photomultiplier.
As long as this is greater than the beam
diameter, the quantity detected is the spatially
integrated second harmonic intensity, i.e. the average
second harmonic power, P2. At the crystal, the two
Figure 4. Dependence of the second harmonic signal on
the fundamental power: linear plot.
i +
t
I

~
Nonlinear optical second harmonic generation
57
Figure 5. Dependence of the second harmonic signal on
the fundamental power: In-In plot. The stmight line is of
slope 2.
quantities are related by PZ = I 2 . d and hence the
second harmonic signal is inversely proportional to
wz for constant laser power.
Figure 7 shows schematically the Gaussian
beam expanding from the output of the laser and
being focused by a lens of focal length f to a spot
of radius w0. The beam spot size, w(z), a distance z
symmetrically on either side of the focus given by
[2-41
w(z) = WO [ I +
(3'1 -
The spot sue 'w' denotes the I/e radius of
the electric field distribution and the l/ez radius
of the light intensity in the cross-section of a
fundamental mode Gaussian beam. The size of
the light spot at the focus, WO, is determined by
the focal length of the lens and the spot radius, w2,
of the light incident on it. By application of
the ABCD ray tracing matrices for the lens and
the displacement between it and the focal point,
wo may be shown to be given by [7]
For the dimensions encountered in most practical
situations, (12) can be simplified to
Figure 6. Dependence of the second harmonic signal on
the position of the crystal relative to the focus of the lens
at z = 0 for crystals of thickness (a) 2.16- and (b)
0.36mm. The solid curve is W(Z)-~ evaluated using only
the measured parameters of the experiment.
Since wz is in turn due to the divergence and diffraction
of the beam expanding from the beam
waist, w,, located either on one of the laser mirrors
or at some point between them, it may be calculated
by using (1 I ) again. The manufacturer's specification
for an He-Ne laser normally quotes wI or at
least the far field beam divergence and so the spot
size in the vicinity of the focus can be traced back
to the laser itself.
The laser used in the experiment Melles-Griot 05
LHP 121, bas a l/e2 beam radius (wl) of 0.2951~1.
Application of (1 1) yields a spot radius of 0.505 mm
(wz) at the input of the lens when it is 60cm from
the laser. The Ealing x 10 objective has an equivalent
focal length of 16" which results in a focus of
radius (wo) of 6.39 pm. Thus the variation in the spot
size on either side of the focus, ~ (z), is given by ( I 1)
to be
w(z) = 6.39pm [ 1 + 24.3 (14)
To fit theoretical curves to the experimental points,
w(z)-' is evaluated from (14) over the same range of
lens adjustment and is shown plotted as the solid
cume in figure 6.
The agreement of the computed points is exceptionally
good for the thin sample hut poor for the
thick one. The model effectively assumed that the
second harmonic was only generated in an infinitesimally
thin slice of crystal. In practice, the conver-
Laser
1
2Wl
Figure 7. Schematic view of the fundamental
beam diverging from the aperture of the laser
and being focused by the microscope
objective to a beam waist of spot diameter
2w". At a distance z from the focus, the
beam spot diameter is 2w(+ 2~7, 2w2, and
2w3(= 2w2) are the spot diameten at the
laser and input and output planes of the lens
respectively.

~~
58 I S Ruddock
sion effciency is high provided the focus is still
located within the crystal and so in the latter case,
the curve is a measure of the crystal's thickness
(- 2"). When the sample is thin, the variation in
beam spot radius through it is negligible and hence
the simple model applies.
5. Conclusions
An undergraduate experiment featuring the phenomenon
of second harmonic generation of a laser has
been described. The experiment illustrates the hasic
principles of the process, phasematching and power
dependence. In addition, it includes an exercise
requiring the application of Gaussian beam optics
and illustrating well some calculations which a
graduate scientist working in this field would be
expected to perform from time to time.
References
111 Franken PA. Hill A E, Peters C W and Weinreich
G 1961 Phys. Rev. Left. 7 118
[2] Guenther R 1990 Modem Optics(New York Wiley)
[3] Yariv A 1985 Opticul Elecrronics 3rd cdn (Holt-
Saunders)
[4] Chatak A K and Thyagarajan K 1989 Optical
Electronics (Cambridge: Cambridge University
Press)
[5l Dunn M H and Akerboom F 1985 Opricalsecond
hnrmonic generation. Physics Experiments and
Projecb for Studenfs ed C Isenberg and S Chomet
(Newman Hemisphere)
[a Kirby K W and DeShazer L G 1987 J. Opi. Soc.
Am. B 4 1072
[I Verdeyen J T 1981 her Electronics (Englewood
Cliffs, NJ: PrentiwHall). This undergraduate
textbook gives a good introduction to Gaussian
beam optics and ray tracing matrices, and deals
with the focusing of a laser beam as a worked
example