Suppose we desire the optical forces to reproduce a theoretical force Fth(x), which is some function of the mirror’s positionx. The aim is to find a power spectral distribution (PSD) for the input laser, p( ), that will produce an overall radiation pressure force
Frp(tot)(x) as close toFth(x) as possible.
Under the assumption that no interference e↵ects occur between the di↵erent fre- quency components of the input field, we have that the total optical force due to the input p( ) is Frp(tot)(x) = Z +1 1 d Frp(x, )p( ) = Z +1 1 d F0( +G0x)p( ) = (F0⇤p)( G0x), (11.3)
whereF0( ) ..=Frp(0, ) is the force obtained from a single-mode input when the mirror is in its rest position, and F0⇤p is its convolution with the PSD. The idea is then to choosep( ) to have the function (F0⇤p)( G0x) coincide withFth(x). For convenience, the convolution can be rewritten in the equivalent form
Frp(tot)(x) = (F0/ ⇤ p)( )| = G0x, (11.4)
combination of a smoothing by the normalized Lorentzian F0/ , a rescaling of the input field by , and a change of variable ! x = /G0. The smoothing action has a role analogous to that of a Gaussian blur, levelling out any feature finer than the linewidth of the cavity while preserving larger features. With this we identify one of the constraints of the approximation method: the fidelity of the approximation of the theoretical force function by the optical forces depends on the finesse of the cavity. Any theoretical force function whose features are larger than the cavity linewidth can be reliably approximated. For this reason we limit the analysis to the reproduction of functions that are not a↵ected significantly by the smoothing, i.e. Fth satisfying the condition
Fth( /G0)⇡(F0/ ⇤Fth|x= /G0)( ). (11.5)
With this assumption, the approximation of an arbitrary force function by the optical forces is satisfied by the choice
p( ) =Fth( /G0)/ , (11.6)
for which we have
Frp(tot)(x) = (F0/ ⇤ Fth|x= /G0)( ) = G 0x ⇡ Fth( /G0)| = G0x
=Fth(x), (11.7)
as desired. Choosing the input according to Eq. 11.6 will cause the mirror to experience an optical force which is modelled around the required theoretical force profile.
This result hinges on the linear superposition of the optical forces, as indicated by the integration in Eq. 11.3. Such superposition is only possible in the lack of interference e↵ects between di↵erent frequency components of the input field. For an input with a continuous PSD this assumption is very speculative. However, it is feasible in the case of a discrete frequency comb, as discussed in the last section. To confirm the validity of the result we need to prove its compatibility when the continuous PSDp( ) is replaced by a frequency comb of discrete modes of spacing✏. Applying a rectangular
approximation to Eq. 11.3, we have Frp(tot)(x) = Z d Frp(x, )p( ) ⇡X n ✏Frp(x, n✏)p(n✏). (11.8)
Then, if the PSD is chosen according to Eq. 11.6, the optical force resulting from the action of the cavity on the mirror is given by
Frp(tot)(x) =
X n
Frp(x, n✏)·Fth( n✏/G0)✏/ . (11.9) The right-hand side corresponds to the force obtained by a frequency comb input such that the component detuned by n✏ has power Fth( n✏/G0)✏/ , assuming that interference e↵ects are removed by appropriate shifting of each mode.
The required frequency comb could be generated in several ways. For many types of force functions, the modulation of a normal single-mode input might be enough to induce sidebands to the central frequency acting as the di↵erent components of the comb. The strength of each component is determined by the strength of the modulation. Potential asymmetries required in the comb may be enforced with a combination of amplitude and phase modulation. As the size of the comb would be determined by the maximum modulation frequency allowed, it could be possible to use a sequence of modulations to allow the generation of wider combs, at the expense of simplicity and flexibility. Alternatively, the di↵erent modes of the comb might be generated by commercial multi-channel laser systems, which are capable of independently tuning the frequency of each channel by up to a few tens of terahertz.
In summary, the optomechanical system can be engineered to let the oscillator expe- rience any theoretical force functionFth(x), as long as the profile of such function does not involve features finer than the linewidth of the cavity. The arbitrary force profile is resolved by an approximation which is mediated by the optical forces,Frp(tot)(x), and which is determined by the appropriate choice for the spectral distribution of the input. The realization of this technique relies on the absence of interference e↵ects. These can be suppressed by separating the frequency components of the input to separate free spectral ranges of the cavity to let the oscillator experience only the average e↵ect of the beating. For inputs with a continuous spectral density, Frp(tot)(x) can itself be approximated by an equivalent frequency comb input to allow the required separation of the modes.