We have developed a bimorph piezoceramic deformable mirror with 28 independently controlled vertical sectors. When used in a reflective 2 f setup the mirror enables phase compensation in the range of a few hundred radians. We have demonstrated that such a compressor is able to compress femtosecond laser pulses that had been initially stretched by a factor of 60 close to their initial shape.
©2005 Optical Society of America
The ability to control the shape of femtosecond light pulses has been gaining importance ever since 1993 when Warren and his coworkers  started a new field of ultrafast laser technology – pulse shaping. Although femtosecond pulse shaping has been developing for over a decade now (see the review article ), there is still need to construct new tools that can be used for this purpose.
So far the most common method for pulse shaping has relied on a 4 f setup which gives the experimentalist access to individual spectral components of the pulse. By changing their amplitudes and phases one can freely tailor the pulse shape within the limits set by the input pulse spectrum. The most popular way of modulating the complex pulse amplitude in the frequency domain is provided by Liquid Crystal Modulators (LCMs) which enable an independent control of the phase and amplitude of transmitted light . LCMs with 640 sectors and 12 bit voltage accuracy are commercially available, allowing precise phase and amplitude modulation. However, LCM have significant drawbacks. Firstly, liquid crystals have non-negligible absorption in the visible and near infrared, which limits the maximum power of incident light. Secondly, the devices are pixelated leading to sub-pulses which can have a detrimental impact on many applications. In addition, LCM cannot be used below 430 nm so, for example, the second harmonic of Ti:Sapphire laser or visible NOPA pulses cannot be shaped in this manner. An alternative method to control both the spectral phase and the amplitude relies on acustooptic modulators  which do not suffer from pixelization and have a high damage threshold. Nevertheless the efficiency of such modulators is limited (although AOM with 80% efficiency was presented ). Moreover, they cannot be used for high repetition rate pulse trains. Another nonpixelated device enabling high resolution proposed by Dorrer and coworkers  is based on optically-addressed liquid-crystal mask acting as phase modulator.
In some applications it suffices to shape only the phase of the ultrashort light pulse. This can be achieved with a deformable mirror placed in the focal plane of a 2f setup – a zero dispersion line. This approach, although limited to pure phase shaping, facilitates low loss, polarization independence, high power and broadband operation. There have been several proposals for such mirrors, including mirrors deformed by a small number of actuators driven by either electrostrictive effect , or thermal expansion . Finer control is possible with Piezo Actuated Deformable Reflector (PADRE) . It consist of a vector of piezoceramics stacks, acting as electro controlled actuators, which bend a thin glass mirror bonded to the top of the actuators. Another solution called micro machined deformable mirrors (MMDM) relies on Coulomb’s forces to deform a very thin silicon membrane suspended over an array of electrodes . A step towards miniaturization are Micro Electro Mechanical Systems (MEMS)  which use the same principle as MMDM but with independent control of many individual membranes. MEMS are fabricated as an array or 2D matrix. They are pixelated devices with the resulting drawbacks.
In this letter we present a novel design of a deformable mirror for femtosecond pulse shaping. It is a bimorph piezoceramic device with construction based on an old idea from adaptive optics originally proposed by Steinhaus and Lipson  over 25 years ago and developed into commercial products in the last decade. A standard bimorph mirror is made of a glass sheet bonded to a piezoceramics layer with an appropriate electrode pattern painted on the back. By controlling the voltage applied to the electrodes one can locally bend the mirror and use it, for example, to correct atmospheric distortion in a telescope or correct wave-front and some phase distortions in a high power laser system . The first rigorous mathematical treatment of such a mirror was done by Kokorowski .
Our deformable mirror (see Fig. 1(a)) is rectangular with dimensions 72.5×25 mm2. Two 0.5 mm thick sheets made of properly (parallel) polarized piezoceramics (T120-A4E-60, Piezo System Inc. ) each with a set of rectangular electrodes (25 mm long, 2.25 mm wide, 0.25 mm gap) on one side were bonded back-to-back using a high strength epoxy resin. The plates were off-set by 2.5 mm along the longer side to enable electrical connection to the common ground electrodes. To such a piezoceramic sandwich a thin (0.3 mm) BK7 glass plate was glued with the same epoxy. The glass plate was narrower (by 2 mm) than the piezoceramic plates to allow access to the sectored electrodes of the top piezoceramics plate. The front surface of the glass plate was then ground and polished. As the total thickness of the mirror was about 1.4 mm, it was quite flexible and the quality of the polished surface was rather poor, although locally the mirror was quite flat. However, the low surface figure is not crucial in a deformable mirror which, by definition, can be deformed to any shape desired as long as there are no significant deviations from the flatness in the area covering a single sector. In the case of our mirror its shape along the longer dimension was almost parabolic - a distortion that, as will be shown below, can be compensated with a constant bias voltage applied to all electrodes. The mirror was vacuum coated with silver and a protective dielectric layer. Finally, thin (50 µm) wires were soldered to the electrodes and the mirror was glued by its central back electrode to a supporting metal rib using a silicone adhesive. The principle of operation of our deformable mirror is illustrated in Fig. 1(b). When a voltage is applied between the outer electrodes of a given sector and the common ground electrode the sector bends resulting in a, locally, cylindrical shape that is either concave or convex depending on the sign of the applied voltage. Because of its construction, the mirror does not experience any significant deformations in the direction parallel to its shorter side. Thus one can safely assume that using the mirror for pulse shaping will not result in any space-time effects. If one neglects the glass plate and epoxy layers, the radius of the curvature R of the cylindrical surface is given by:
where V is the applied voltage, t is the thickness of the plate and d 31 is the piezoelectric tensor coefficient. In the case of a continuous voltage distribution V(x) along the mirror - a good approximation for a sectored mirror when the density of sectors is high - its shape y(x) is given by:
One can easily integrate (Eq. 2) with appropriate boundary conditions in our setup with x=0 corresponding to the center of the mirror). In particular with a polynomial-shaped voltage V(x)= ak xk the integration returns another polynomial:
Thus, one can easily program polynomial shapes of the mirror. For example, a constant voltage applied to all electrodes results in a pure parabolic shape, and a voltage that is proportional to the sector index gives a pure cubic mirror shape. This property might be quite useful in applications where smooth phase errors well described by polynomials are expected as it is the case in pulse compression in CPA laser amplifiers. The equation Eq. 3 does not account for either the glass plate or the epoxy layers which lower the amplitude of the mirror’s motion. When used with the manufacturer’s data for the piezoceramics it gives the maximum displacement of the mirror’s surface which is approximately twice larger than the value measured. To control the shape of the mirror we used a multichannel 12 bit D/A card in a PC computer followed by high voltage operational amplifiers; one for each sector. The output voltage range was ± 160V.
The deformable mirror was tested by placing it in the Fourier plane of a 2 f zero dispersion line consisting of a 600-groove/mm diffraction grating and a 1 m focal length, 50 mm diameter, concave, gold coated mirror (see the experimental setup in Fig. 2). In such a set-up individual spectral components of the pulse are focused in the focal plane of the concave mirror and, at the same time, different frequencies are spread along that plane. When a flat mirror is placed in focal plane all the spectral components are retro-reflected and travel back to arrive at the output with the same delay, hence the mane “zero dispersion line”. Any change of the mirror shape causes, in such a shaper, different frequency components of the femtosecond light pulse to be advanced or retarded, so the electric field of the pulse after the compressor can be written as:
with ϕ(ω)=2ωd(ω)/c, where Ein is the input pulse field, ω is the frequency of the optical field, c is the speed of light and d – local (corresponding to the position of a given frequency component) mirror displacement.
To test the performance of our mirror we used the spectral interference method. A beam of pulses from a home-built Ti:Sapphire femtosecond oscillator was sent into a Michelson interferometer, with the shaper in one arm and a flat mirror in the other (reference) arm (see Fig. 2). The spectrum of the beam leaving the interferometer is given by:
where I 0(ω) is the spectral intensity of the input pulse,ω0 is the central frequency, τ is the delay between the reference and shaped pulses, and ϕ(ω) is the phase introduced by the shaper. The method is very convenient as it measures only the spectral phase modulation introduced by the shaper while being totally insensitive to the spectral phase of the input pulses. Phase ϕ(ω) was reconstructed from the measured spectrogram with an algorithm known from SPIDER  using a LabView code running on a PC.
Examples of the spectral phase patterns obtained in our experiment and recovered with the spectral interference method are shown in Fig. 3. Two upper graphs present a polynomial phase responses of the mirror. The first one shows a quadratic phase shape and was achieved by applying the same voltage to all actuators. The second graph shows the third order phase which results from the voltage that varies linearly with the actuator number. The dots mark the second and third order polynomials fitted to the experimental data. The range of the measured quadratic phase modulation exceeds 250 radians. Given that only 70% of the mirror’s aperture was used, one can expect phase modulation as large as 500 radians with full aperture operation still giving smooth phase modulation. Such range is not available in other designs. The resolution of the mirror is limited by its construction - given a finite number of electrodes the highest possible resolution is achieved by applying alternating positive and negative voltages to consecutive electrodes. As one can see in Fig.4(d) the modulation depth is, in this case, approx. 1 radian. The imperfections in the measured mirror response are probably due to non uniform epoxy layers thickens.
To demonstrate the performance of our mirror in pulse shaping we used the 2 f setup to compress long stretched pulses. Pulses from the Ti:Sapphire oscillator (approx. 30 fs long) were propagated through a 10 cm long block of SF11 glass, which introduced the second order dispersion of approximately 19000 f s 2, the third order dispersion of approx. 12000 f s 3 and a significant amount of higher order dispersion which is difficult to compensate using compressors based on passive components such as prisms and diffraction gratings. For example, the third order dispersion itself leads to a cubic phase of approximately 4 radians within the bandwidth of our laser. Propagation through the glass stretched the pulses about 65 times. The beam of stretched pulses was sent into the compressor and an evolutionary algorithm was applied to control the mirror shape with a signal from a two-photon photodiode as feedback. In such a setup we were able to compress chirped pulses close to their Fourier limit in a few minutes’ time. The spectral phase introduced by the SF11 glass block and the optimized pulse phase are shown in Fig. 4(a). Fig. 4(b) compares the interferometric autocorrelation function of the laser pulse with that of the stretched-compressed pulse.
In conclusion, we have demonstrated a novel design of a deformable mirror for phase shaping of femtosecond pulses. The mirror based on bimorph piezoceramics is compact, inexpensive and can be easily scaled to conform to the aperture of the zero dispersion delay line. We have demonstrated experimentally that the mirror provides very high-range smooth phase modulation as well as good resolution and accuracy. It is well suited for applications in which smooth high-range phase modulation is required. Due to the mirror’s construction polynomials shapes are easy to reproduce. It can be coated with dielectric layers and thus have very high damage threshold and operate in a broad spectral range.
This work was supported by KBN grant number 2P03B 029 26. Authors are indebted for partial support of this project by ONR grant number N00014-01-1-0661.
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