1. Introduction

Conversion of a Gaussian laser beam to a plane wave with uniform irradiance is termed as beam shaping, a concept that can be extended to any process involving a transformation at wish of the irradiance and eventually the phase of an optical beam. In particular, wave patterns with fine structures in the order of the wavelength require the use of high-numerical-aperture microscope objectives. This is of interest, for instance, in optical trapping and manipulation of particles [1], material processing and multifocal microscopy [2]. Dispersion effects become evident with femtosecond laser sources in virtue of its inherent broadband nature. Shapers employing diffractive optical elements (DOEs) are particularly reactive to angular dispersion,[3] producing off-axis redshift and subsequent pulse stretching. For this reason, up-to-date solutions involve scanning processes about an on-axis focus [4]. However, a few proposals may be found in the literature concerning angular dispersion compensation in order to achieve a parallel processing. Some of them consider chromatic compensation uniquely along a privileged transverse direction, having practical uses with 1D diffraction gratings [5, 6, 7]. Minguez-Vega et al. demonstrated achromatization of paraxial diffracted fields in the Fourier plane [8], a procedure that may be combined with high-numerical-aperture objectives. However, a residual longitudinal chromatic aberration is induced, which may be of importance with ultrabroadband laser sources. In this paper we propose an optical arrangement, conceptually simpler in comparison with that presented in Ref. [9], capable of compensating isotropically diffraction-induced angular dispersion of wave fields initiated by a DOE. Moreover, we investigate the behavior of an infinity-corrected microscope objective collecting the diffracted wave in order to achieve a beam shaping in the focal plane. As an example, we perform a multifocal pattern of equienergetic single-cycle pulses with extremely-high spatial resolution and free of longitudinal chromatic aberration.

2. High-resolution diffraction-driven beam shaping

Consider a uniform plane wave, which without lost of generality is here of unit amplitude, to be transformed into a given 2D transverse field, F(r). Our approach is based on inducing the prescribed pattern by means of a DOE with an amplitude transmittance

As depicted in Fig. 1(a), an infinity tube microscope objective collects the diffracted light, which is focused onto the focal plane with an amplitude U(r) = ∫F(r)h(r- r)dr, being an approximated replica of the function F. Previously, we applied the following general considerations: (1) the beam is apertured by the objective lens and (2) the diffracted focal field has a linear and shift-invariant response. For instance, by choosing an on-axis point-like distribution F = δ(r), where δ is the Dirac delta function, the optical system provides the so-called point spread function (PSF),

for the sine condition [10], where k = 2π/λ is the wavenumber, r = ∥r∥, J₀ is a Bessel function of order 0, and sin α is the numerical aperture of the objective lens (in air).

More generally, the function F in Eq. (1) gives the spatial spectrum of the DOE output field. A given spatial frequency q initiates a tilted plane wave exp(i qr ₀) of transverse wavenumber q = k sin θ. In the paraxial regime (sin θ ≈ θ), beam steering is performed with a deviation angle θ = q/k. The tilted beam entering into the objective lens (of focal length f) is focused at a point a distance r= θf from the optical axis. Therefore, we may establish a relationship between a given excited spatial frequency q and the spatial coordinate r=q f/k where off-axis focusing is accomplished. As an example, in Fig. 1(b) we plot the intensity distribution generated by a Dammann diffraction grating of period Λ = 0.5 mm exciting a discrete number of equienergetic spatial frequencies q _nm = 2π(n,m)/Λ, where n and m are integers (|n|, |m| ≤ 2). The incident, monochromatic, plane wave has a wavelength λ₀ = 600 nm, and the numerical aperture of the collecting lens is 0.75. Within this configuration, we generate a 3 μm-pitch multiple beam array of 404 nm FWHM, appropriate for applications such as multifocal microscopy.

 

Fig. 1. (a) Schematic focusing arrangement of a wave field diffracted by a Dammann grating. (b) Monochromatic response with λ₀ = 600 nm.

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3. Resolution modes

Beam aperturing through the microscope objective determines the limit of resolution allowing U to be similar to the signal F. From a fundamental point of view, two close spatial frequencies are distinguishable if the modulus of its difference satisfies the inequality ∥Δq∥D ≥ 2π, where D is the diameter of the entrance pupil. In the spatial domain, this uncertainty principle is written ∥Δr∥ ≥ λf/D, giving a minimum distance η = λf/D between foci driven by both spatial frequencies to be discriminated. In a rigorous analysis, we consider the spot size given by the PSF and, following the Rayleigh criterion, the limit of spatial resolution is η = 0.6lλ/sinα.

For a pulse beam of time-domain spectrum S(ω), where the carrier (mean) frequency ω ₀ = ∫ ω∣S(ω)∣²dω/ ∫|S(ω)|²dω, we may estimate the resolution limit is that given by Rayleigh for the average wavelength λ₀ = 2πc/ω ₀. In the case of an ultrabroadband pulse wave, angular dispersion originated by DOEs provokes a remarkable spatiotemporal uncoupling and, therefore, an additional loss of resolution power. Shortly, given a specific excited spatial frequency, the dependence of k(ω) upon the optical frequency provokes that the angular deviation, θ = q/k, also varies on ω. In free-space propagation (k=ω/c), beam steering is performed with a paraxial deviation angle θ(ω) = θ ₀ ω ₀/ω, where θ ₀ = qc/ω ₀. Considering nondispersive objectives where f is independent upon frequency, off-axis chromatic focusing is performed at r(ω) = θ(ω)f. Moreover, if the source bandwidth is denoted by Δω, the angular dispersion of the initiated tilted wave is approximately given by Δθ = ∣$\dot{\theta }$ ₀∣Δθ, where $\dot{\theta }$ ₀ = dθ/dω at ω = ω ₀, giving θ ₀ = -θ ₀/ω ₀. As a consequence, radial dispersion driven over the off-axis focal point is determined by

where r₀ = r(ω ₀) and σ = Δω/ω ₀ is the fractional bandwidth. Transverse focal delocalization, spatially variant since it increases linearly upon the distance r₀, may be higher than η, a means of lengthening the resolution limit.

 

Fig. 2. Polychromatic response of the system plotted in Fig. 1(a): Normalized power spectrum of (a) the incident wave and (b) off-axis foci of order (n,m). (c) Integrated intensity in the focal plane, and (d) field dynamics along the y-axis.

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From Eq. (3) we may establish two different modes of spatial resolution induced by diffraction. In the central region of the focal plane satisfying r < η/σ, aperturing dominates over chromatic dispersion; η > Δr and we speak of an aperturing mode of the spatial resolution. Otherwise (r > η/σ) radial dispersion features spatial resolution; in such a case we use the term dispersion mode. For quasi-monochromatic radiation (σ ≪ 1), the working area of the focal plane normally holds the aperturing mode. In particular, the spot array shown in Fig. 1(b) presents off-axis foci at maximum r₀ = 8.5 μm (for ω = 3.14 fs^-1) and, therefore, pulse sources of fractional bandwidth σ < η/r₀ = 5 10^-2 (approximately 10 fs pulse duration for transform-limited pulses) would not exhibit a resolution power reduction induced by angular dispersion.

In the single-cycle regime, σ is close to unity and the diameter of the central region featuring the aperturing mode is in the order of the (mean) wavelength. In this case, angular dispersion determines the spatiotemporal resolution of practically the entire focal plane. As illustration, Fig. 2(c) depicts the integrated intensity patterned by the Dammann grating described above when the illumination corresponds to a single-cycle plane wave of Poisson-type spectrum, shown in Fig. 2(a). In the figure inset, the temporal evolution of the incident wave field is represented in the local time tʹ. Fig. 2(b) shows the power spectrum at coordinates r_nm = q _nm f/k ₀ associated with off-axis focal points for ω ₀. The bandwidth is reduced significantly, which may be estimated from Eq. (3) by mere substitution of Δr by η, giving an off-axis fractional bandwidth σ_off = η/r₀. This considers that neighboring spectral components of ω ₀ are significant if the chromatic focal shift is lower than the spot radius η dictated by aperturing. As a consequence, a pulse stretching is produced with a linear dependence upon r. This effect is clearly observed in Fig. 2(d), where we represent the temporal dynamics of the diffracted field along the y-axis of the focal plane. In the numerical computation, phase delay of different spectral components causing group velocity dispersion may be omitted assuming the collecting lens is placed at a plane immediately behind the DOE.

4. Dispersion compensation

Compensation of angular dispersion in broadband diffracted waves results in a field widening of the aperturing resolution mode, increasing the resolution power of the extended area. Following Ref. [11], the here-proposed compensating element consists of a highly-dispersive thin lens, concretely a kinoform-type zone plate (ZP) of focal length Z(ω) = Z ₀ ω/ω ₀, being Z ₀ the focal length at ω ₀. When the DOE is located at a distance a in front of ZP, as shown in Fig. 3(a), the amplitude distribution at the conjugate plane is replicated with a lateral magnification M(ω) = Z(Z-a)^-1. If the DOE excites a given spatial frequency q, the diffracted field carries a frequency q/M in the image space and, therefore, the angular deviation of the tilted plane wave is θʹ = q/Mk. A product Mk independent of ω is associated with angular invariance; however, such a property cannot be found in free space propagation for positive values of the axial distance a. Our approach consists in minimizing the spectral dependence of Mk within a band around ω ₀. Thus, we impose ω ₀ to be a stationary point of Mk, i.e., d(Mk)/dω= 0 at ω= ω ₀. A solution is found if ZP is a positive lens and a = Z ₀/2, in which case

 

Fig. 3. (a) Diffractive image former of stationary diffraction-induced angular dispersion. (b) Angular dispersion for low orders of an 1D diffraction grating without (thin solid line) and with compensation (bold solid line) for λ₀ = 600 nm (0.5 magnification omitted). (c) Hybrid diffractive-refractive achromatic beam shaper.

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where θʹ ₀ = qc/(2ω ₀). Therefore, θʹ ₀ = θ ₀/2 pointing out that angular compensation is performed with a magnification of one half. In Fig. 3(b) we plot θʹ of Eq. (4) corresponding to a discrete number of equidistant spatial frequencies. In terms of the coordinate λ, θʹ represents a convex parabola of maximum at the stationary wavelength λ₀. Normal dispersion is observed in the domain λ < λ₀, but redder components suffer from anomalous angular dispersion. Finally, a DOE displacement along the z-axis may be employed as a mechanism for tunning the stationary frequency up to ω_s = 2ω ₀ a/Z ₀.

An important point in question is the fact that, in spite of assuming plane wave illumination, the compensating setup shown in Fig. 3(a) produces an output spherical beam. Obviously, the DOE may drive itself an additional beam focusing, which is not considered in the following analysis. The field collected by the microscope objective has a ZP-induced wavefront sphericity that causes a strong longitudinal chromatic aberration. Collimation of the ZP emerging wave may be achieved by introducing an optical setup capable of converting the launched plane wave into a dispersive spherical beam of appropriate focus at a distance -Z from the zone plate. The illuminating system selects chromatically a phase curvature radius Z -a for the spherical wave impinging over the DOE; therefore, the amplitude distribution in a plane immediately in front of the DOE is proportional to exp ⌊ik(2Z-Z ₀)^-l r ² ₀⌋. Our proposal is based on an afocal doublet composed of nondispersive thin lenses, L ₁ and L ₂ with coincident focal lengths (f ₁ =f ₂), which includes a zone plate ZP₁. Fig. 3(c) depicts the illuminating system as it works together with the DOE-based beam shaper. Placed at the front focal plane of L₁, ZP₁ is in the conjugate plane of ZP through the inverter doublet and generates the required dispersive spherical wave when it has a focal length of opposite sign (-Z).

 

Fig. 4. Achromatic response of the system depicted in Fig. 3(c): (a) Normalized power spectrum and temporal evolution of the field (inset) for foci of order (n,m). (b) Integrated intensity in the focal plane, and (c) field dynamics along the y-axis.

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Figure 4 shows the compensating response of the proposed system illustrated in Fig. 3(c). Again, the DOE corresponds to a Dammann grating; however, doubled spatial frequencies have been employed in order to overcome an scaled-pattern shaping. As expected from the aperturing resolution mode, chromatic compensation of off-axis focal mismatch r(ω) =θʹ(ω)f seen in Fig. 4(b) induces a conservation, at least at first order, of time-domain magnitudes such as power spectra (Fig. 4(a)) and pulse field dynamics (Fig. 4(a,c)); this features characterizes the so-called in-plane isodiffracting beams [12]. High-order compensation procedures should be applied in order to minimize distortions of the power spectrum. Moreover, we place the objective lens at a distance l = 0 from ZP in order to minimize phase delay, though the distance a from the DOE to ZP leaves a residual alteration of the amplitude spectrum. For a given excited spatial frequency q, the spectral modifier yields exp(-iϕ), where ϕ = Z ₀ q ² c(4Mw)^-1. In the numerical simulations, where Z ₀ = 100 mm, we observe that such a pulse distortion corresponds fundamentally to a second-order effect since ω ₀ is a stationary point of Mω.

5. Summary

In conclusion, we have recognized that the double-diffraction problem found in a DOE-based shaper establishes two different resolution modes. Off-axis beam shaping is naturally performed within the angular-dispersion mode, with poor spatiotemporal resolution power in comparison with the on-axis aperturing mode. A compensation scheme employing diffractive lenses has been proposed to compensate angular dispersion and, therefore, optimizing the performance of the shaper. We point out that our proposal may be extended straightforwardly to account for other sort of dispersive thin lenses. Finally, examination of pulse broadening due to dispersion of glass of refractive lenses L₁, L₂ and the microscope objective, should be carried out in a more realistic model. Although not analyzed in this short communication, this effect is also of importance if oil-immersion objectives are used in order to achieve an increase of resolution power.

This research was funded by the Generalitat Valenciana under the project GV/2007/043.