There are several methods to detect the deflection of the cantilever. The most commonly used detection method for AFM is the beam deflection technique. In this technique the cantilever displacement is measured by detecting the deflec-tion of a laser beam reflected on the back side of the cantilever with a quadrant detector. This can be seen in figure 2.2 (a). Another technique is shown in fig-ure 2.2 (b). This is the capacitance technique in which a small plate is mounted above the tip so the plate and cantilever form a capacitor. The capacity is a measure for the deflection of the cantilever. A third method uses a piezoelectric cantilever. When the cantilever bends it will produce an electric field which can be measured by a current detector as is shown in figure 2.2 (c).
I
(a) (b) ( ) c
Figure 2.2: Three cantilever deflection detection methods. (a) The beam de-flection technique with the quadrant detector above the cantilever. (b) The capacitance technique. (c) The technique with the piezoelectric cantilever.
The AttoAFM-I uses the interferometric technique, see figure 2.3. In theory this technique gives an equal signal to noise ratio as the commonly used beam deflection technique [3] but the beam deflection technique is easier to use because it has less optical components. The main problem with the beam deflection technique is that the detector has to be close to the cantilever and thus to the sample. Because our setup is operational at low temperatures it is practical to use the interferometric technique. Here the detector can be placed far away from the cantilever and does not have to be cooled down and operated in a magnetic field.
The amount of light that reaches the detector depends on the distance be-tween the fiber and the cantilever because of the interference bebe-tween the light that reflects from the cantilever and from the fiber-air interface, see figure 2.3.
The total electric field of the detected light consists of two parts and is given by equation 2.1, the first part is the light that reflected at the air-fiber interface and the second part is reflected from the cantilever.
Et= rfE0+ rct2fE0cosθ (2.1) Here Et is the total electric field going to the detector, E0 is the electric field of the original laser signal, rf and tf are the reflection and transmission coefficient of the end of the fiber and rc is the reflection coefficient of the can-tilever. θ is the phase difference between the two different parts of the signal and is given by equation 2.2. This phase difference gives rise to the interference.
θ = 4πd
λ (2.2)
E0 r Ef 0
t Ef 0 r t Ec f 0
r t E cosc f2 0 q
Figure 2.3: Schematic side view of the fiber end on top of the cantilever. The incoming light gets reflected by the fiber-air interface as well as by the cantilever.
The reflections give rise to the interferometric signal.
Here d is the distance between the fiber and the cantilever and λ is the wave-length of the laser light. The detector measures the power, therefore we rewrite the equation in terms of the irradiance I, the radiant power density. With I = ǫ0chE2i this gives the following equation.
It= ǫ0chEt2
i = I1+ I2+ 2p
I1I2cos θ = ǫ0cE02(r2f+ r2ct4f+ 2rfrctf2cos θ) (2.3) here It is the total irradiance, I1 is the irradiance of the beam reflected by the fiber-air interface and I2is the irradiance of the beam reflected by the cantilever.
The sum of the reflection and transmission coefficient of the irradiance should be equal to one. Therefore we can introduce new irradiance reflection and transmission coefficients to simplify things. Rf = r2f = (1 − t2f) and Rc = r2c which leads to:
It= ǫ0cE02(Rf+ Rc(1 − Rf)2+ 2(1 − Rf)p
RfRccosθ) (2.4) The reflectivity of the fiber can be calculated if we know the indices of refraction of air and the fiber [4]. nair ≈ 1 and nf iber ≈ 1.5, which gives:
Rf = (nf iber− nair
nf iber+ nair) ≈ 0.04 (2.5)
.
So Rf << 1 and we can simplify 2.4 to:
It≈ ǫ0cE02(Rf+ Rc+ 2p
RfRccos θ) (2.6)
The visibility of an interference signal is determined by contrast between the maximal and minimal irradiance. If a cantilever starts in an equilibrium position it can deflect up or down depending on the force working on it. To be able to measure both up and down deflections the cantilever should be positioned in the middle between the maximum and minimum where cos θ = 0. This means d = nλ8 . This is also the position that has the highest sensitivity for deflections and thus changes in d. If the cantilever starts at this equilibrium position the visibility is given by:
v =Imax− Imin
Imax+ Imin
= 2p RfRc
Rf+ Rc
(2.7) Now we can call Ieqthe irradiance at equilibrium, Ieq =Imax+I2 min = ǫ0cE02(Rf+ Rc) and we can write Itin its simplest form:
It= Ieq(1 − v cos θ) (2.8)
This is a good approximation when the distance between the fiber and the cantilever is large. For small distances the interference is not simply build up out of the two reflected signals but also higher order reflections between cantilever and fiber need to be taken into account as is shown schematically in figure 2.4.
Figure 2.4: Side view of the cantilever and the fiber end. The arrows represent the path of the light. For small cantilever-fiber distances higher order reflections have to be taken into account.
The total contribution to the interference signal by light reflected from the cantilever and the fiber is not constant, but depends on the distance between the fiber and the cantilever. Equation 2.1 now changes into a more complicated form [5].
Et= rfE0+ X∞ n=1
t2frc(rfrc)n−1E0V cos(ϕ) (2.9) ϕ is the phase difference between all the interference signals and is given by equation 2.10.
ϕ = 4πdn
λ + π(2n − 1) (2.10)
V is the loss due to the divergence of the laser beam after leaving the fiber and is also dependents on d. It is the area where the light can enter the fiber again divided by the area of the diverged laser beam as shown in figure 2.5.
After a distance d, D′= D + 2dtanφ. This gives the following relation for V : V = D2
D′2 = D2
(D + 4dntanφ)2 (2.11)
D D’
d f
Figure 2.5: The divergence of the laser beam after leaving the fiber. D is the diameter of the fiber core, D′ is the diameter of the diverged laser beam, d is the distance between fiber and cantilever and φ is the angle of divergence.
To see the relation between the measured signal by the detector and the distance between fiber and cantilever we take the following laser/fiber constants.
D = 9µm, φ = 7o, λ=1550nm and the reflectivity of the cantilever with an aluminium coating Rc=0.85. It/I0 is plotted versus d in figure 2.6. For large distances the signal is dominated by the reflection of the fiber. Most of the light reflected from the cantilever is lost due to the divergence of the laser beam.
As the cantilever moves closer to the fiber more and more light gets reflected back into the fiber and the visibility increases until at d = 20µm the cantilever reflection starts to dominate. At very small distances the signal approaches the reflection coefficient of the cantilever and almost all light is reflected back into the fiber.
Figure 2.6: The calculated irradiance detected divided by the irradiance of the incoming laser of a cantilever fiber interference as a function of the distance, d, between the cantilever and the fiber.