We propose and experimentally demonstrate an interferometer for femtosecond pulses with spectral bandwidth about 100 nm. The scheme is based on a Michelson interferometer with a dispersion compensating module. A diffractive lens serves the purpose of equalizing the optical-path-length difference for a wide range of frequencies. In this way, it is possible to register high-contrast interference fringes with micrometric resolution over the whole area of a commercial CCD sensor for broadband femtosecond pulses.
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Femtosecond light sources provide a valuable tool for three-dimensional ranging through coherence gating. Depth resolution is controlled by the spectral bandwidth of the pulse through the equation , with λo and Δλ the center wavelength and the bandwidth of the source respectively. Optical coherence tomography (OCT) takes advantage of low-coherence interferometry to provide cross-section images of biological tissue. OCT images can be recorded either point by point through a fast two-dimensional scanning system or using wide-field techniques, which provide en-face (XY) tomographic images without transverse scanning .
Short-coherence digital holographic microscopy was proposed by Cuche et al in 1997 as a new wide-field tomographic modality . Here, an off-axis hologram results from the interference between an object wave scattered by the sample and a reference wave in a Michelson-type interferometer. The hologram is recorded onto a CCD camera allowing digital reconstruction. When a low-coherence source is used to record the hologram it is possible to perform tomography of the sample .
An axial resolution of 11 μm has been demonstrated with a 80 fs pulsed laser source . As noted by the authors, no further improvement in axial resolution is expected by reducing the pulse width as the bandwidth of the source must be kept small enough to preserve the formation of fringes to record the hologram. The point is the so-called walk-off effect [6,7]. Using a temporal picture, the reduced coherence length restricts interference fringes to a diamond-shaped region with size proportional to the coherence time, roughly the inverse of the spectral bandwidth. From a spectral point of view, fringes are washed-out as each spectral component generates its own fringe pattern but with a different period. Anyway, the number of recordable fringes is given by λo /Δλ  which, for instance, for a femtosecond laser delivering 10 fs pulses at a carrier wavelength of 800 nm is only 9. This fact leads to a reduced field of view for holographic reconstruction.
Inspired by the achromatic grating interferometer , this problem was addressed by the use of a diffraction grating as a beam splitter . Pulse front tilt originated at the grating forces pulses to overlap over the whole diameter of the beam. This smart approach has been extensively employed for transient grating experiments  but it lacks the flexibility necessary for holographic recording and tomographic measurement. Pulse front tilt has also been performed by the use of specific-purpose prisms in the two arms of a Michelson-type interferometer leading to an increased number of fringes for the off-axis geometry .
Here, we propose a new pulse front tilt mechanism in a Michelson interferometer that is fully compatible with holographic applications. The key is an easy-implemented dispersion compensating module (DCM) constituted by a diffractive lens located outside the interferometer. The device is designed to equalize the optical path length with frequency just to get a nearly zero differential group delay between the object and the reference beam for all the spectral components when the two arms of the interferometer have an identical length. This link guarantees a flat visibility, nearly unity, for the interference fringes along a considerable region of the beam. Optical path equalization is performed through the dispersion introduced by diffractive lenses. For 10 fs pulses at a wavelength carrier of about 800 nm, results of computer simulations and laboratory experiments demonstrate an enhancement in the number of recordable fringes by a factor of about 36.
2. Frequency-domain analysis of interference of short pulses traveling through dispersive media
Let us consider a pulsed plane wave with carrier angular frequency ωo, , traveling along a dispersive medium in the direction indicated by the z-axis. The pulse carries a total intensity , with , and we denote by the baseband spectral energy distribution of the pulse. In mathematical terms,
A further insight into the problem can be gained if ∆Φ(ω + ωo ) is expanded around the carrier frequency up to second order as11]. Also, Taylor series expansion is not adequate for causally dispersive, absorptive dielectrics when the propagation distance exceeds a characteristic absorption depth setting the limit for the penetration depth in tomographic applications . Anyway, experimental results in Section 4 validate the goodness of the above assumptions. Inserting Eq. (3) into Eq. (2) we find
It is thus apparent that the ability of the pulse to interfere with a delayed replica of itself is governed by a generalized complex degree of temporal coherence that accounts for the presence of dispersion within the interferometric setup.
When no dispersion is present, or the DGDD is set to zero, we recover the well-established Wiener-Khintchine theorem. This result states that the complex degree of temporal coherence and the spectral energy distribution are linked through a Fourier transform relationship. The most common situation corresponds to free-space propagation after the beam splitter (see Fig. 1 ). For this case, the value of the phase difference is determined by the length mistmatch between the interferometer arms (∆e) and the mirror tilts θ 1 and θ2 through the expression ∆Φ(ω) = [2∆e + xS (sinθ2-sinθ1)] ω/c. In the above expression c denotes the speed of light and xS the transverse coordinate at the sensor plane. Note that ∆Φ(ω) is linear with frequency so that τ´ = 0. The two usual cases θ1 = θ2 and ∆e = 0 can be obtained from the above result. For the interference of parallel plane waves, we recover the interferogram trace , with τ = 2Δe/c. Also, the same trace holds when θ1≠θ2 and ∆e = 0 but now the interference pattern is codified in the tranverse coordinate xS at the output plane through the relationship τ = xS (sinθ2-sinθ1)/c.
As a matter of fact, for a pulse with Gaussian spectrum the visibility, defined in the usual way , is also a Gaussian function. The number of observable fringes is controlled by the coherence time, the width of the complex degree of coherence, and, for the Gaussian situation, is of the order of ωo /Δω = λo /Δλ with Δω the spectral width . Also the transverse position of the central fringe with the length mismatch is given by xS = 2Δe/(sinθ2-sinθ1) which accounts for the walk-off effect.
The dispersion introduced by the optical materials employed in the setting of the interferometer increases the pulse duration at the interference region. We assume perfect antireflection coatings for the whole spectrum of the pulse and that no-absorption bands of the materials are present so frequency-dependent lossy effects are neglected . Effects of dispersion are modelled through a chirp filter in the spectral domain and, then, the baseband spectrum of the incoming stretched pulse reads as , where the parameter accounts for dispersion introduced by the glass in both the lens and the beam-splitter. Note that both reference and object beams suffer from the same time broadening so no relative additional phase difference is introduced, which guarantees validity of Eq. (2). The time-integrated interferometric trace computes the autocorrelation signal for the input field and, thus, it is only sensible to the spectral energy distribution. In other words, due to the phase-only filtering nature of the dispersive effect and the interferometric signal in Eq. (2) is independent of the value of the dispersion parameter. For the case of frequency-dependent absorption a further visibility reduction is expected due to the narrowing of the effective spectral energy density.
We emphasize, however, that pulse broadening reduces the temporal resolution of the interferometric signal in the case that a time-resolved detection scheme (FROG, SPIDER, …) is employed. In the case temporal resolution of the interferometric fringes is of interest, as for transient-grating spectrometry, a conventional pre-chirping unit can be used at the exit of the laser system so that the pulse at the output plane results in a transform-limited signal which ensures maximum temporal resolution.
Interestingly, our approach can be easily applied in a different context to explain the formation of broadband interference fringes with unit visibility in the achromatic grating interferometer . Usually, the symmetric configuration corresponding to the interference between diffraction orders + 1 and −1 is considered. Here a wavelength independent phase difference ∆Φ(ωo) = 4πxS /d, with d the grating period, is achieved. In this way, τ = 0 and τ´ = 0. This results in a pure cosenoidal interferogram with no fringe modulation and period p = d/2.
3. Optical setup design
We consider the use of a diffractive lens after beam recombination in the beam splitter to equalize the phase difference with frequency at the output plane (see Fig. 2 ). We assume paraxial approximation and Δe = 0. Here, we take advantage of the inherent dispersive nature of diffractive optical elements to perform frequency equalization of the optical path length difference (OPLD). Specifically we recall that the focal length Z(ω) of a diffractive lens is a linear function of the wave frequency; i.e., Z(ω) = Zo ω/ωo, where Zo denotes the focal length for the carrier ωo. For this case, the value of the phase difference is given by
Ideally, ΔΦ(ω) should be independent on the frequency. This situation cannot be satisfied for the optical setup in Fig. 2. However, geometrical parameters can be tuned in order to minimize the above functional dependence. To this end, we impose the first-order condition τ = 0. In a temporal picture, we require an identical arrival-time over the whole output plane for the wave packets traveling through the reference and the object arms of the interferometer. As a result, a high visibility for the interferential fringes at the output is expected. By virtue of Eqs. (6) and (7), OPLD equalization leads to the constraints13],  and .
We have developed a first-order theory. So a residual modulation for the interference fringes is expected. The fringe envelope is evaluated through Eq. (5) with τ = 0 and τ´ = ΔΦo/ where ΔΦo is given by Eq. (6) for ω = ωo after the conditions derived in Eq. (8) are substituted into matrix elements A and B. Also, the value for τ´ is derived from the second derivative of Eq. (6) with achromatic compensation. Note that Eq. (5) indicates that the Fourier transform relation between the complex degree of coherence and the spectral energy is substituted by a Fresnel transformation. It is one of the main results of our work as it is the only way the optical design in Fig. 2 can be carried out. In Fig. 3 we plot the visibility function V(xS ) for the interferograms at the sensor for the optical setups in Fig. 1 (solid curve) and 2 (dashed curves) as a function of the normalized sensor coordinate xS /p. We assume a Gaussian spectrum centered at ωo and full-width-at-half-maximum (FWHM) width Δω. For the plot, we choose ωo = 2.40 1014 Hz and Δω = 2.44 1013 Hz (i.e., λo = 789 nm and Δλ = 81.6 nm). It appears that the diffractive lens enhances the sensor region over which interference fringes are observable by at least an order of magnitude. In fact, it can be demonstrated that the gain factor, defined as the ratio between the FWHM width of the function V(xS ) for the interference fringes obtained with the Michelson interferometer with and without DCM, respectively, is given by . Note that this is a theoretical value that provides the increment in the number of recordable fringes, which is entirely determined by the characteristics (carrier frequency and spectral width) of the pulse. For the present numerical example the gain factor is of about 36. From the point of view of laboratory results, this theoretical number is slightly reduced due to the effect of higher-order terms in the expansion of ∆Φ(ω + ωo ) and the presence of spatial variations in the input femtosecond beam. To take into account this effect, we test the goodness of the quadratic approximation in Eq. (3) in the same plot. To this end, visibility is evaluated directly through Eq. (2). Now, the full ω–dependence of the OPLD is retained for the calculation. Results are plotted in short-dashed line. Only a slight deviation is observed for a sensor area covering a huge amount of fringe periods.
4. Experimental results
We proved the feasibility of our diffractive-lens based DCM through laboratory experiments. A picture of the optical setup is shown in Fig. 4.a). In the right lower corner of the picture we see the spatial filter and the collimating lens to create the pulsed plane-wave. The achromatic lens (AL), beam splitter (BS), two-mirrors (M1 and M2), the diffractive lens (DL), and the CCD sensor are labeled in the picture. We used a nearly Gaussian pulse from a commercial Ti:sapphire laser oscillator. The laser system delivers 9 nJ pulses at 75 MHz repetition rate. In Fig. 4.b) the spectrum of the pulse (solid curve) together with its Gaussian fit (dashed curve) are plotted. Spectrum parameters ωo and Δω are those used for the plot in Fig. 3. Normally, the pulses are transform-limited to roughly 10 fs. However, the pulse can be pre-chirped so that at the sensor results in a transform-limited signal to increase temporal resolution of the interference as explained in Section 2. Fringes were captured by a CCD camera with a pixel size of 6 μm and a sensor area of 7.68 × 6.14 mm2. The focal lengths for the refractive achromatic objective f and the diffractive lens Zo were chosen as 200 mm and 143 mm, respectively. The diffractive lens was located at the image focal plane of the objective and the sensor was positioned 71.5 mm from the dispersive component. All the above parameters were chosen just to meet achromatic requirements specified through Eq. (8).
Figure 5 shows the interference fringes and the corresponding line scans showing the depth of fringe modulation recorded as a function of the sensor coordinate without (a) and with the DCM (b). For both pictures, fringe spacing is identical (p = 42 μm). This is accomplished by changing slightly the relative angle between the mirrors for the optical setups in Figs. 1 and 2. Interference visibility nicely fits the theoretical predictions shown in Fig. 3 for both situations. Note the significant improvement in the number of recordable fringes (by 240 in the picture). For the DCM situation they cover the whole sensor area keeping a visibility higher than 0.5. As noted previously, the slightly reduction with respect to the theoretical value of 36 is mainly due to the effect of higher-order terms in the expansion of ∆Φ(ω + ωo ) and the presence of spatial variations in the input femtosecond beam.
High-visibility interference fringes with broad bandwidth femtosecond pulses are recorded by means of a modified Michelson interferometer. The device incorporates a diffractive lens to equalize the optical path length difference between the object and the reference wave with frequency. In this way, the width of the visibility function, measured as its FWHM, in terms of the normalized coordinate xS /p is increased by a factor with respect to the non-equalized situation. We have experimentally recorded interference fringes with 42 μm spacing over the whole sensor area of a commercial CCD camera. A net gain of at least 30 for the the number of recordable fringes (visibility higher than 0.5) is achieved for a commercial Ti:sapphire femtosecond laser oscillator delivering 10 fs pulses at 800 nm.
This research was funded by the Spanish Ministry of Education and Science under projects FIS2007-62217 and Consolider Program Science and Applications of Ultrafast Ultraintense Lasers, CSD2007-00013. The authors are grateful to the Servei Central d’Instrumentació Científica of the University Jaume I for the use of the Ti:sapphire ultrafast laser.
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