Interferometer

In order to transform the light reflected off the cantilever into a voltage before supplying it to the feedback input of our PLLPro2 amplifier, we use a variable gain photoreceiver, i.e. the Femto Photoreceiver OE–200–IN2.

This fast optical power meter consists of an InGaAs PIN photodiode, which is calibrated at λphoto = 1550 nm and can deliver conversion gain accuracy of ± 5 %. It has an FC/ANP fiber optic receptable on the input side and a BNC connector on the output side. It can be operated both in ac and dc mode and has a tunable gain range that can vary between 10³ to 10⁹ in Low Noise settings and between 10⁵ to 10¹¹ for High Speed settings. Moreover, it can deliver additional filtering by applying a low–pass filter of 10 Hz to the dc signal.

When performing the alignment procedure we operate the interferometer in dc mode at full bandwidth (FBW) with an optical gain factor of 10⁶(Low Noise). For normal operation however, the interferometer needs to be operated in ac mode (FBW LN), since the PLLPro2 amplifier will only accept ac voltages ≤ 1.0 V.

Modeling and calibration

The interferometer signal can be modeled by considering a two–reflection system, the first part is the light re-flected at the air–fiber interface and the second part is rere-flected from the cantilever–end. Using electrodynamic wave theory, we write

E(r, t) = rfE0+ rct²_fE0cos θ, (4.6) here E(r, t) is the total electric field going to the detector, E0is 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 cantilever. θ is the optical path difference between the fiber and cantilever reflections, and is given by Eq. 4.7

θ = 4πd

λ , (4.7)

Here d is the distance between the fiber and the cantilever and λ is the wavelength 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= ǫ0chE(r, t)²i = I1+ I2+ 2p

I1I2cos θ = ǫ0cE₀²(r_f²+ r²_ct⁴_f+ 2rfrctf2cos θ), (4.8) here Itis the total irradiance, I1is 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 = r_f²= (1 − t²f) and Rc= r_c² which leads to

It= ǫ0cE₀²(Rf+ Rc(1 − R^f)²+ 2(1 − R^f)p

RfRccos θ). (4.9)

The reflectivity of the fiber can be calculated from the indices of refraction of air and the fiber, nair ≈ 1 and nf iber≈ 1.5, which gives

Rf = (nf iber− n^air

nf iber+ nair) ≈ 0.04. (4.10)

Since Rf<< 1, we can simplify 4.9 to

It≈ ǫ0cE²₀(Rf+ Rc+ 2p

RfRccos θ). (4.11)

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 = ^nλphoto₈ . This is also the position that has the highest sensitivity for deflections (i.e. changes in d). If the cantilever starts at this equilibrium position the visibility is given by:

ν = Imax− Imin

Imax+ Imin

=2p RfRc

Rf+ Rc

. (4.12)

If we now call Ieqthe irradiance at equilibrium, or more simple the average power, defined as Ieq =^Imax+I₂ ^min = ǫ0cE₀²(Rf+ Rc), we can write Itin its simplest form

It= Ieq(1 − ν cos θ). (4.13)

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. Eq. 4.6 now changes into a more complicated form [70]

E(r, t) = rfE0+ X∞ n=1

t²_frc(rfrc)ⁿ⁻¹E0flosscos ϕ, (4.14) ϕ is the phase difference between all the interference signals and is given by

ϕ = n · 4πd

λphoto + π(2n − 1), (4.15)

floss is the loss factor 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 Fig. 4.7a. After a distance d, D^′ = D + 2d tan φ. This gives the following relation for floss Figure 4.7: (a) Divergence of the laser beam after leaving the fiber. The calculated detected irradiance of a cantilever divided by the irradiance of the laser as a function of the distance d between the cantilever and the ferrule. (b) For large separation between ferrule and cantilever. (c) For small separation between ferrule and cantilever.

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 = 9 µm, φ = 7^◦, λ = 1550 nm and the reflectivity of the cantilever with an aluminum coating Rc= 0.85. It/I0 is plotted versus d in Fig. 4.7. 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 = 20 µm the cantilever reflection starts to dominate (Fig. 4.7b). At very small distances the signal approaches the reflection coefficient of the cantilever and almost all light is reflected back into the fiber (Fig. 4.7c).

The interferometer response to a cantilever motion, i.e. the change in the fiber–cantilever separation d, is given by

where V0 is the average voltage from the interferometer signal and ∆V is the voltage change. ∆λ is ideally small, such that the second term can be neglected. As seen before, the interferometer is most sensitive to the cantilever motion ∆z when d = n · λ/8, with n odd. To calibrate the interferometer, the signal is measured as a function of fiber–cantilever distance. This is done by applying a so–called ‘dither spectroscopy’, in which the dc voltage applied to the dither piezo is increase linearly, while measuring the interference signal. The maximum and minimum voltages are averaged to determine Ieq, which is used as the setpoint for the interferometer. The interferometer calibration is given by

in which Vmax,min are the maximum and minimum interferometer voltages measured as a function of fiber–

cantilever separation, respectively.

Ferrule

Holder chip

Dither piezo Fibersledge

Cu plate

Fiber Spring relief

(a) Cantilever–fiber system

Si wafer

cantilever laser spot

(b) Holder chip

(c) Interference pattern measured with the attocube system

Figure 4.8: (a) Picture of the cantilever–fiber system. (b) Top view schematic of the holder chip (left) and the cantilever (right). The laser spot should be centered exactly in the middle of the cantilever at the tip end.

(c) Typical interference pattern for the alignment of the cantilever to the ferrule. The measured deflection of the cantilever [V] is shown as a function of time [s] (distance [m]), while decreasing the distance between ferrule and cantilever. In the blue dot, a peak is shown in the interference signal, this is caused by the screwdriver when it is placed on top of the screw. The blue arrow indicates the moment at which the screwing commences. The random spacing in the interference signal is caused by the nonuniform velocity of the screwing. The insets show magnifications of the resultant waveform. Also see paragraph 4.3.1.