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R.J. Jones Optical Sciences
OPTI 511L Fall 2008
Experiment 2: Second Harmonic Generation (SHG) _____
In this experiment we produce 0.53 μm (green) light by frequency doubling of a 1.06 μm (infrared) diode-pumped YAG laser, using a KTP crystal as the nonlinear medium. We will:
A. Demonstrate frequency doubling of a YAG laser (1064 nm -> 532 nm).
B. Observe how the optical power at 532 nm varies as a function of phase matching.
C. Optimize and measure the efficiency of second harmonic generation.
SAFETY NOTE
For this experiment we use an Amoco Model ALC 1064-150P laser, which produces at least 150 mW of CW single mode power at 1.064 μm. The human eye is not sensitive to this wavelength, but the radiation is transmitted through the eye and focused near the retina, where can cause irreversible damage. The danger threshold for CW viewing is around 1 mW/cm^2 entering the pupil, so you need to be careful. Protective goggles will be available, and should be used at all times.
Second Harmonic Generation (SHG)
Set up the following:
YAG Laser
KTP
crystal prism (^) beam block
power meter
f f ′
- Start with a lens of focal length f ~ 5cm. Assuming a 3 mm beam diameter of the 1.064 micron YAG laser, calculate the beam waist and confocal parameter at the focus (ie confocal parameter = 2 x Rayleigh Range). How does this compare to the length of the 5 mm KTP crystal? What will be the beam waist and confocal parameter for the second harmonic beam?
- The polarization of the YAG laser is linear. Adjust the rotation angle of the KTP
crystal such that the polarization of the 1.064 μm beam forms an angle θ = 45 ° with
its square edges, and so the beam is focused at the center of the crystal. (If the angle of the linear polarization is not known, adjust the KTP to optimize the second harmonic light) Fine adjust the tip and tilt angles of the KTP to maximize the green (532 nm) light. Make sure you also adjust the longitudinal crystal position so that the beam waist coincides with the crystal position. A translation stage will be useful to optimize the alignment.
- Observe the power of the 532 nm light as a function of the phase matching angle. Try to accurately measure the maximum green light you can generate. This is quite challenging because of the intense beam at 1.064 μm can easily corrupt your measurement. You may need more than a single prism to separate the beams sufficiently. A color filter may also be used. Make sure to estimate the losses from each. Be careful of scattered light from the beam block. You will also need to continually check and correct the alignment of the beams through the crystal and onto the power meter & beam block as you change the phase matching angle.
- Try a few different focussing lenses (e.g. 7.5 cm and 10 cm) in order to optimize the SHG conversion efficiency. Which gives you the maximum amount of SHG? Compute again the corresponding confocal parameters. Note that the choice of focal length f involves a tradeoff: a short focal length increases the intensity at the beam waist needed for efficient SHG, but at the same time decreases the Rayleigh range for the beam. If f becomes too short the beam diffracts rapidly, and the intensity does not remain high along the entire 5 mm path through the crystal.
i. e. we allow the medium to respond to the driving field in a nonlinear fashion. We see immediately that the polarization density will contain components P (^2 )^ ∝ E^2 , P ( )^3 ∝ E^3 etc. For
a monochromatic driving field of frequency ω , these components will radiate at 0 ω , ω , 2 ω ,
3 ω etc. The nonlinear susceptibilities are small, χ (1)^ >>χ (2)^ >>χ (3)^ >> ..., and the conversion
into higher harmonics is correspondingly inefficient - except for extremely intense fields.
Note that the nonlinearity brings with it a breakdown of the superposition principle. If the
driving field contains two frequencies ω 1 and ω 2 then in addition to higher harmonics we also
get sum- and difference frequency generation, ω 1 + ω 2 and ω 1 − ω 2. Electromagnetic waves
passing through a nonlinear medium interact!
Second harmonic generation: We now examine the wave equation (1) with a nonlinear polarization (2) up to and including the 2nd order susceptibility. The driving and second harmonic fields have the form
E ω = 12 E ω ( ) ze − i ( ω t − k ω t )
- c. c ., (3) E 2 ω = 12 E 2 ω ( ) ze − i (^) ( 2 ω t − k (^) 2 ω t )
- c. c ..
To leading orders these waves induce a polarization density
P = P 0 ( NL^ )^ ( ) z + 12 P ω(^ L^ )^ ( z ) e − i (^ ω^ t −^ k^ ω^ t )^ + c. c.
( χ (2) E ⋅ E contains a DC term which gives rise to a DC polarization component P 0 (^ NL )^ ( ) z , which
we will ignore). The linear terms in the polarization can be accounted for in the usual way: at frequencies well away from any absorption resonances they simply add to the (real) index of refraction. We therefore have
k ω = n ω
c
c
k n
ω ω
2 =^2.
In general n 2 ω ≠ n ω and therefore k 2 ω ≠ 2 k ω. Light radiated by the nonlinear polarization at
points z and z + Δ z differs in phase by e − i^^2 k^ ω^ Δ z^ , while the 2 ω wave propagating from z to
z + Δ z picks up a phase e − ik^^2 ω^ Δ^ z^. Clearly then, there is the possibility of a phase mismatch between the propagating and generated light, which prevents the two from interfering constructively. This subtle point turns out to be an all-important consideration in determining the efficiency of any nonlinear conversion process.
The wave equation for the second harmonic component is
∂^2
∂ z^2
E 2 ω −
n 22 ω c^2
∂^2
∂ t^2
E 2 ω =μ 0
∂^2
∂ t^2
P 2 ( ω^ NL^ )^ (5)
Plugging in the field and nonlinear polarization at 2 ω from eqs. (3) and (4), using the slowly
varying envelope approximation for E 2 ω ( z ) and P 2 ( ω^ NL )^ ( z ) we obtain
d E 2 ω ( ) z dz
= i ω
μ 0 c
n 2 ω
P 2 ( ω^^ NL )^ ( ) ze i (^) ( 2 k ω − k (^) 2 ω) z
= i ω
μ 0 c
n 2 ω
d E ω ( ) z 2 e i Δ kz^ , (6)
where P 2 ( ω^ NL )^ ( ) z ≡ d E ω ( ) z 2 , Δ k ≡ 2 k ω − k 2 ω.
The quantity d is a material parameter that characterizes the effective nonlinearity of the medium. For KTP we have d ≈ 17 × 10 -9^ esu ≈ 6 × 10 -23^ C V 2.
When the power in the second harmonic wave remains much smaller than the power in the fundamental (no depletion approximation) we can set E ω ( z ) 2 = E ω ( 0 ) 2 and integrate eq. (6) to find
E 2 ω ( ) z = i ω
μ 0 c
n 2 ω
d E ω ( ) z 2 z
sin (Δ kz 2 ) Δ kz 2
e i Δ kz^^2.
Using I 2 ω ( ) z = 12 c ε 0 E 2 ω ( ) z
2 , and assuming an input intensity I ω and a doubling crystal of length L , we finally obtain the intensity of the second harmonic wave at the output facet,
I 2 ω ( ) L = κ I ω^2 L^2
sin (^) ( 12 Δ kL ) 1 2 Δ kL
2
. (7)
This allows us to estimate the power conversion efficiency,
η ≡
P 2 ω P ω
μ 0 ε 0
3 2 ω 2 d^2 L^2 n^3
P ω π w 2
sin^2 (Δ kL 2 ) (Δ kL^^2 )
2 ,^ (8)
where P ω , and P 2 ω is the optical power in the fundamental and second harmonic beam,
respectively. For simplicity we have approximated I ω ≈^ P ω 2 w^2 , where w is the beam radius of
the fundamental.
Equations (7) and (8) are our key result. Clearly, good conversion efficiency requires Δ k ~ 0 and large I ω.
ϕ z ′ k
n (^) z
n (^) x
n (^) y
x ′
n (^) e
n (^) o
y = y ′
Fig. 2. Indices of refraction in a uniaxial crystal.
In this situation we get extraordinary and ordinary indices of refraction for linear polarizations along x ′ and y ,
no = ny , ne =
nx^2 + nz^2
nx^2 cos 2 ϕ + nz^2 sin 2 ϕ
[9]
In angle tuned phase matching we adjust the angle ϕ (by rotating the crystal). There are two
ways of doing this:
Type I phase matching: n 2^ o ω^ ⋅ 2 ω = n ω^ e^ ⋅ ω + n ω^ e^ ⋅ ω ⇒ n 2^ o ω^ = n ω^ e
Type II phase matching: n 2^ o ω^ ⋅ 2 ω = n ω^ o^ ⋅ω + n ω^ e^ ⋅ω ⇒ n 2^ o ω^ = 12 ( n ω^ o^ + n ω^ e )
In Type I phase matching the fundamental is polarized along x ′ and the second harmonic is
polarized along y ; two extraordinary photons at ω combine to generate one ordinary photon at
In Type II phase matching the fundamental is polarized at 45° to the x ′ , y axes and the second
harmonic is polarized along y ; an ordinary and an extraordinary photon at ω combine to
generate an extraordinary photon at 2 ω.
Phase matching in our experiment is complicated by the fact that KTP is a biaxial crystal, i. e. nx ≠ ny ≠ nz. We will not discuss biaxial phase matching here, but simply note that our crystal
has already been cut to achieve Type II phase matching when the laser beam is normal to the entrance facet, and linearly polarized along the diagonal.
ne
no
θ
G
ε L
Fig. 3. Phase matching occurs for our precut KTP crystal when the laser polarization
G
ε L forms an angle θ = 45 ° with the ordinary/extraordinary axes.
Since only Type II phase matching can occur in our crystal, the second harmonic field E 2 ω ∝E ω^ o E ω^ e^ , where E ω^ o^ ∝E ω cos( ) θ and E ω^ e^ ∝E ω sin( θ ). This immediately tells us that
I 2 ω ∝ (^) [ I (^) ω cos( ) θ sin( θ )]^2 ∝ I ω^2 sin 2 ( 2 θ).
As our crystal is rotated around the axis of the laser beam, the second harmonic intensity will
therefore change as a function of θ , with a period of 9 0 °.
References: “Lasers”, P. W. Milonni and J. H. Eberly (Wiley 1988). “Quantum Electronics”, 2nd Ed., A. Yariv (Wiley 1975). “Nonlinear Optics”, R. W. Boyd (Academic Press 1992).
