2.3 Femtosecond pulse shaping
2.3.2 Frequency domain pulse shaping with spatial light modulators
Following the analysis of Weiner [33,37], the resolution of a 4-f -pulse shaper in terms of frequency span per pixel can be calculated by starting with the grating equation [38]
sin θin+ sin θd=
mλ
Chapter 2. Theory and technology 41 where θinand θdare the angle of incidence and the diffraction angle, respectively. The
grating period is denoted with d, the wavelength with λ and the diffraction order with m. The angular dispersion D is found by differentiating equation (2.21).
D = dθd dλ =
m
d cos θd . (2.22)
With the FWHM pulse bandwidth ∆λ (≈ dλ), the approximation of small angles of θd
the lens focal length f and with m = 1, we can write the expression a = f · dθd=
f ∆λ
d cos θd , (2.23)
which gives the total spatial span of the pulse spectrum on the SLM masking element. The spatial dispersion follows as
α = a ∆ω =
f λ2
2πcd cos θd. (2.24)
∆ω is the pulse bandwidth expressed in angular frequency and c is the speed of light. A relation between the filter function H (ω) in equation (2.20) and the actual masking function M (x) describing the spatial distribution of amplitude and phase of the mask, can be found when we consider an electric field with a Gaussian profile directly after the mask
Emask(x, ω) ∝ Ein(ω) · e
−(x−αω)2w2
0 · M (x) . (2.25)
This equation describes the electric field as a function of space and frequency. The 1/e2-radius (of intensity) of each frequency component at the masking plane is
w0 =
cos θin
cos θd ·
f λ
πwin , (2.26)
where the 1/e2-radius of the collimated input beam is denoted with win. In order to
obtain the output electric field only as a function of frequency (or time), it is necessary to perform a spatial filtering operation. This can be achieved by expanding the masked field into a superposition of Hermite-Gaussian modes [38]. Higher-order modes, which describe spatial diffraction of the mask (inter-pixel gaps or pixel edges), are eliminated by spatial filtering, e.g. by coupling into a fibre or placing an iris after the pulse shaper. The shaped field is thus described by the lowest order Hermite-Gaussian mode (fun- damental Gaussian mode), whose coefficient is defined as the filter function
H (ω) = s 2 πw2 0 Z ∞ −∞M (x) · e −2(x−αω)2w2 0 dx . (2.27)
Chapter 2. Theory and technology 42 with the Gaussian (intensity) profile of the beam. The FWHM of this Gaussian profile determines the spectral resolution limit of the shaper as
δω ≈√ln 2 ·wα0 . (2.28)
Note that for pixellated devices, physical features smaller than w0 are smeared out
by the convolution and this ultimately limits the finest features that can be imprinted onto the filtered spectrum. The maximum achievable complexity η of the pulse shaper system can be defined as a function of the shaper resolution limit (equation (2.28)) and the pulse bandwidth ∆ω. With equations (2.24) and (2.26), we get
η = ∆ω δω = ∆λ λ0 π √ ln 2 win d cos θin . (2.29)
The complexity is a measure of the number of distinct spectral features with width δω that can be controlled within the pulse bandwidth ∆ω. Equivalently in the time domain, it means the number of temporal features with width τ that can be synthesised into a waveform within the time window T (see below). The complexity is typically between 100 and 400. Note that it is not only the optical components placed around the SLM that determine the complexity but also the applied SLM technology. An LC SLM with 128 pixels, for instance, limits the complexity to 128 and having a much higher complexity from the surrounding optical system is useless.
It is clear that the resolution limitation in the frequency domain infers a limitation in time as well due to the fundamental Fourier theory. The convolution of equation (2.27) yields the product h (t) = m (t) g (t) for the impulse response h (t), where m (t) is the inverse Fourier transform of the mask function. The function
g (t) = e−
w20 t2
8α2 (2.30)
describes a Gaussian envelope that restricts the time window of the pulse shaper. The (intensity) FWHM of this envelope is
T = 4α √ ln 2 w0 = 2 √ ln 2 winλ0 cd cos θin , (2.31)
Within this time window, the tailored output pulse can precisely reflect the response of the infinite-resolution mask. For a given grating, a larger time window can only be achieved with a larger input beam size. In a CPA system, the time window has no practical implications, since the stretched pulses are typically much longer than the calculated time window and it has been shown numerically that shaping of a linearly chirped and thus temporally stretched pulse beyond the time window is possible [39]. The time window is limited by the Fourier limit T δν = 0.441, where δν = δω/2π,
Chapter 2. Theory and technology 43 and is thus related to the spectral resolution of the pulse shaper in equation (2.28). The Fourier limit also applies to the full pulse bandwidth ∆ν = ∆ω/2π and the finest temporal feature with width τ , i.e. ∆ντ = 0.441. The complexity can be expressed as η = ∆ν/δν = T /τ . Hence, we get the maximum time-bandwidth product
T ∆ν = 0.441η . (2.32)
All of the above quantities are given as FWHM. In general, the pulse shaper optics must be chosen such that its complexity is equal or higher than the complexity given by the masking element (for LC SLM: number of pixels).
We now discuss resolution issues of an SLM with a pixellated LC mask and its disper- sion shaping capabilities. The discrete modulation of each frequency component in an LC SLM is limited by the sampling theorem after Nyquist-Shannon [40], which states that a signal must be sampled at least at twice the frequency of its highest frequency component. Applied to LC SLMs, two adjacent pixels must cover the total range of 0 ≤ φ ≤ 2π of the spectral phase. In other words, a maximum number of Npix/2 phase
modulations can be synthesised with an LC mask with Npixpixels. A sufficiently large
effective phase range beyond only 2π is obtained, if phase wrapping is applied (in- stead of 0-π-2π-3π-4π-5π, the phase is wrapped to 0-π-2π-0-π-2π, for instance). Thus, with a maximum phase shift between two adjacent pixels of π and with the frequency span ∆ωpixon one pixel, the maximum rate of change of phase per angular frequency
is dφ/dω = ±π/∆ωpix (plus or minus symbolises pos. or neg. slope). With the spec-
tral phase expanded to a Taylor series as in equation (2.7), one can deduce the maxi- mum values of each individual dispersion parameter before reaching the Nyquist limit [39,41]
dφ
dω = ± λ2
0
2c∆λpix , (Group delay) (2.33)
d2φ dω2 = ± λ40 2Npixπc2∆λ2pix , (GDD) (2.34) d3φ dω3 = ± λ60 N2 pixπ2c3∆λ3pix , (TOD) (2.35)
where ∆λpix = ∆ωpixλ20/2πc is the wavelength span per pixel and Npixthe number of
used pixels. Each equation gives the maximum dispersion possible from this term only without the influence of the other terms. If more than one dispersion term acts on the pulse, the Nyquist limit may be reached sooner. Furthermore, if the LC SLM is used in reflection mode, the spatially dispersed beam traverses the device twice and hence the maximum dispersion values are doubled.
Chapter 2. Theory and technology 44
2.3.3 Spatial light modulators
To control phase and amplitude of a pulse, SLMs in 4-f -configurations are applied as spectral masking elements. Modern SLMs offer flexible modulation of both phase and amplitude, which can be controlled by an external signal, as opposed to a fixed mask modulator, where the experimental usability is strongly limited. A comprehensive re- view of SLM-based pulse shaping can be found in reference [37].
Deformable mirrors are mounted in two or more symmetric positions and a force is applied on both ends of the mirror membrane. The force can be of mechanical or elec- trostatic nature. In combination with a grating and a lens in a 2-f -configuration, the function is equal to a 4-f -setup using a transmissive masking element. Deformable mirrors are capable of performing a continuous phase modulation. The phase profile varies smoothly and the modulation is very precise. However, the actuators provide only a limited bending radius and hence phase shift. Also, arbitrary pulse shapes are not easily programmable. Deformable mirrors can, however, operate in spectral ranges not covered by other types including the MIR. This is achieved by state-of-the-art high- reflectivity coatings for the specific range.
In acousto-optic modulators, a piezo-electric transducer driven by a radio-frequency voltage generates a travelling acoustic wave in a crystal. This results in a periodic vari- ation of the refractive index and thus a temporary diffraction grating. The optical pulse effectively sees a fixed diffraction grating as the acoustic wave velocity is much slower than the passing pulse light wave. The waveform of the radio-frequency signal deter- mines the grating shape and therefore the diffraction angle and the delay per frequency component. The input beam is incident at the Bragg angle (highest efficiency) and each spatially dispersed frequency component diffracts off of this grating. The amplitude of the output waveform is directly related to the diffraction efficiency of the modulator grating, which is determined by the acoustic wave amplitude. The resolution is given by the minimum programmable acoustic wave feature and can be considered as contin- uous compared to pixellated devices (see below). The refresh rate of the grating pattern depends on the acoustic aperture time and is typically in the low microsecond range. Due to the finite speed of the acoustic wave, AOMs can not be used for high repetition rate pulse trains in the MHz-range and the efficiency of the AOM is only between 10% and 15%.
Liquid crystal (LC) arrays consist of transparent cells, referred to as pixels, which are filled with long, thin, rod-like molecules. Each pixel can be individually programmed by applying a voltage via transparent electrodes. With no voltage applied, the align- ment direction of the molecules defines the extraordinary axis of the crystal. The molecules will tilt in the direction of the electric field when a voltage is applied. Thus,
Chapter 2. Theory and technology 45 light linearly polarised along the extraordinary axis, sees a voltage-dependent refrac- tive index and consequently experiences a variable phase shift. The maximum phase shift is achieved when the light is subject to the full extraordinary index (no voltage and thus tilt). For independent phase and amplitude modulation, a combination of two LC-masks with a molecule orientation of ±45◦ relative to the light polarisation
state is sandwiched between two polarisers. The retardance or optical path length of an LC-cell is nonlinearly dependent on the drive voltage. Amplitude modulation is expressed by the normalised transmission
T (V ) = cos2π
λ(R1(V ) − R2(V ))