Broadband spatiotemporal axicon fields

Rahul Dutta; Kimmo Saastamoinen; Jari Turunen; Ari T. Friberg

doi:10.1364/OE.22.025015

1. Introduction

Axicons are optical elements, which convert coherent plane waves into conical waves [1, 2], thus allowing efficient generation of approximately propagation-invariant fields such as Bessel beams [3–5]. These elements can be realized using reflective or refractive conical surfaces [1]. Alternatively, one may make use of diffractive techniques [6, 7] to generate the same kind of monochromatic beams.

When illuminated with polychromatic light, reflective and diffractive axicons behave differently because of their differing dispersive properties [5]. Reflective axicons produce conical waves with the same cone angle for each frequency ω; however, the transverse scale of the diffraction pattern depends on ω, which leads to approximations of X-waves [8] and to broadband propagation-invariant beams with reduced spatial coherence [9,10] if illuminated with stationary light. Diffractive axicons, on the other hand, produce fields with frequency-dependent cone angle but an invariant transverse scale, thus resulting in approximations of Bessel pulses with a well-defined central peak and side lobes [11, 12]. Hence both types of axicons share an achromatic property but in different meanings. Stationary X-waves and pulsed Bessel beams are investigated theoretically and experimentally in more detail in [13, 14].

While the general characteristics of polychromatic fields produced by axicons are understood qualitatively, as described above, their detailed properties have not been studied. Such axicon fields are important for a variety of applications in which simultaneous temporal and spatial localization is needed. Polychromatic fields allow, additionally, the utilization of partial temporal coherence. In this work we compare reflective and diffractive axicons that, for monochromatic light, generate approximations of J₀ Bessel fields. We consider illumination by both spectrally fully coherent pulses and by spectrally incoherent (statistically stationary) light, and illustrate the remarkable differences in the character of the beams and pulses created by the two types of axicons under broadband illumination.

2. Representations of the incident field

We assume that the axicon is illuminated with the waist of a broadband isodiffracting Gaussian field (the Rayleigh range is independent of frequency), which has a frequency-domain representation

U (ρ^{'}; ω) = {(\frac{2}{π w^{2}})}^{1 / 2} {(\frac{ω}{\bar{ω}})}^{1 / 2} \sqrt{S (ω)} exp (- \frac{ω}{\bar{ω}} \frac{{ρ^{'}}^{2}}{w^{2}}) .

Here ρ′ is the transverse radial coordinate at the plane z = 0 of the axicon, ω̄ = 2πc/λ̄ is some reference frequency, w is the beam width at ω = ω̄, and S(ω) is some fixed spectral distribution. In this work we assume that

S (ω) = \frac{S_{0}}{Γ (2 n) \bar{ω}} {(2 n \frac{ω}{\bar{ω}})}^{2 n} exp (- 2 n \frac{ω}{\bar{ω}}),

where n ≥ 1 is a real parameter that determines the width of the spectrum and Γ represents the Gamma function. This profile [15, 16] is employed frequently in the analysis of broadband localized waves as it has the advantage of containing no negative frequencies. Its peak frequency ω̄ is taken as the reference frequency in Eq. (1).

Combining Eqs. (1) and (2), we find that the spatial distribution of the spectral density of the incident field, S(ρ′; ω) = |U(ρ′; ω)|², takes on the form

S (ρ^{'}; ω) = \frac{2 {(2 n)}^{2 n} S_{0}}{π w^{2} Γ (2 n) \bar{ω}} {(\frac{ω}{\bar{ω}})}^{2 n + 1} exp [- 2 \frac{ω}{\bar{ω}} (n + \frac{{ρ^{'}}^{2}}{w^{2}})] .

This distribution reaches its maximum at

{\bar{ω}}_{ρ} = \bar{ω} \frac{n + 1 / 2}{n + {ρ^{'}}^{2} / w^{2}} .

Hence, compared to S(ω), the spectrum S(ρ′; ω) is blueshifted (ω̄_ρ > ω̄) if ρ′ < w and red-shifted (ω̄_ρ < ω̄) if ρ′ > w. It also gets narrower as ρ′ increases. These features are illustrated in Fig. 1(a).

Fig. 1 (a) Normalized spectral and (b) temporal intensity profiles of the incident field at distances ρ′ = 0 (blue), $ρ^{'} = w / \sqrt{2}$ (black), ρ′ = w (magenta), and $ρ^{'} = \sqrt{2} w$ (red).

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Let us first assume that all frequency components U(ρ′; ω) are completely correlated. In this case the space–time representation of the incident field is obtained by the well-known Fourier-transform relationship

U (ρ^{'}; t) = \int_{0}^{\infty} U (ρ^{'}; ω) exp (- i ω t) d ω .

Inserting from Eq. (1) into Eq. (5) we get

U (ρ^{'}; t) = \frac{U_{0}}{{(n + {ρ^{'}}^{2} / w^{2} + i \bar{ω} t)}^{n + 3 / 2}}

with

U_{0} = {(\frac{2 S_{0} \bar{ω}}{π w^{2} Γ (2 n)})}^{1 / 2} {(2 n)}^{n} Γ (n + 3 / 2) .

The spatiotemporal intensity profile I(ρ′;t) = |U(ρ′;t)|² of the incident field is then

I (ρ^{'}; t) = \frac{I_{0}}{{[{(n + {ρ^{'}}^{2} / w^{2})}^{2} + {(\bar{ω} t)}^{2}]}^{n + 3 / 2}},

where I₀ = |U₀|². The half-width at half-maximum (HWHM) value of I(ρ′; t), denoted by T, is readily seen to be given by

\bar{ω} T = \sqrt{2^{1 / (n + 3 / 2)} - 1} (n + {ρ^{'}}^{2} / w^{2}) .

Hence the pulse width increases with ρ′ as illustrated in Fig. 1(b), the reason being that the spectral profile narrows down off-axis.

Secondly, we assume that the incident field is stationary in time but has the same spectrum S(ω) as the fully coherent pulse. We are now mainly interested in the spatial dependence of the temporal coherence, i.e., in the mutual coherence function Γ(ρ′; τ), which is determined by the Wiener–Khintchine theorem

Γ (ρ^{'}; τ) = \int_{0}^{\infty} S (ρ^{'}; ω) exp (- i ω τ) d ω,

where τ is the time difference. Inserting from Eq. (3) into Eq. (10), we find that the spatial distribution of the complex degree of temporal coherence, γ(ρ′; τ) = Γ(ρ′; τ)/Γ(ρ′; 0), is of the form

γ (ρ^{'}; τ) = {(\frac{n + {ρ^{'}}^{2} / w^{2}}{n + {ρ^{'}}^{2} / w^{2} + i \bar{ω} τ / 2})}^{2 (n + 1)} .

The HWHM of |γ(ρ′; τ)|, denoted by Θ_i, is obtained from

\bar{ω} Θ_{i} = 2 \sqrt{2^{1 / (n + 1)} - 1} (n + \frac{{ρ^{'}}^{2}}{w^{2}}) .

The quantity Θ_i is a measure of the coherence time of the incident field. In view of Eq. (12), the coherence time increases off-axis. The reason again is that the spectrum there gets narrower.

3. Space–frequency representation of axicon fields

Figure 2 illustrates the operation of a reflective axicon, which converts a plane wave (arriving from the right and shown by the blue rays) into a conical wave with cone angle θ (propagating to the right and shown by the red rays). The nominal propagation-invariant region extends to the distance L where the marginal rays cross the optical axis. Reflective axicons are considered in this work since they are dispersion-free and therefore strictly achromatic in the sense that the cone angle θ is independent of ω. Diffractive (reflection-type) axicons are quasi-planar counterparts of the reflective axicons, formed by quantizing the surface profile height within the region [0, λ̄/2], where λ̄ = 2πc/ω̄. The result is a radial grating of period d, with the dispersion relation

sin θ (ω) = \frac{2 π c}{d ω}

providing the frequency dependence of the cone angle θ.

Fig. 2 Operation of a reflective axicon A, which converts a plane wave into a conical wave with cone angle θ. The geometrical propagation-invariant range L is defined by the crossing point of the marginal rays with the optical axis. A beam splitter B may be inserted in the system to separate the axicon field and the incident field spatially.

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We model the effect of the axicon onto the incident field by assuming that the thin-element approximation holds, and thereby describe the axicon response by a complex-amplitude transmission function t(ρ′; ω) of the form

t (ρ^{'}; ω) = \sqrt{η (ω)} exp [- i (ω / c) sin θ (ω) ρ^{'}],

where θ(ω) = θ for reflective axicons and it is given by Eq. (13) for diffractive axicons. Here η(ω) describes the (frequency-dependent) reflection coefficient for reflective axicons, and the diffraction efficiency for diffractive axicons.

The field immediately after reflection from the axicon is, according to the thin-element approximation, t(ρ′; ω)U(ρ′; ω). In the paraxial domain (small cone angle), its propagation is governed by the Fresnel integral for rotationally symmetric fields,

\begin{array}{l} U (ρ, z; ω) = & \frac{ω}{i c z} exp [\frac{i ω}{c} (z + \frac{ρ^{2}}{2 z})] \\ \times \int_{0}^{\infty} ρ^{'} t (ρ^{'}; ω) U (ρ^{'}; ω) J_{0} (\frac{ω}{c z} ρ ρ^{'}) exp (\frac{i ω}{2 c z} {ρ^{'}}^{2}) d ρ^{'}, \end{array}

where J₀ is the Bessel function of the first kind. We evaluate this integral using the method of stationary phase [17, 18], which leads to the result (see Appendix A for details)

\begin{array}{l} U (ρ, z; ω) = & - i \sqrt{\frac{η (ω)}{π}} \sqrt{\frac{z}{z_{R}}} \frac{ω}{c} sin θ (ω) \sqrt{S (ω)} exp [- \frac{ω}{\bar{ω}} \frac{z^{2}}{w^{2}} {sin}^{2} θ (ω)] \\ \times J_{0} [\frac{ω}{c} ρ sin θ (ω)] exp {i ω [\frac{z}{v (ω)} + \frac{ρ^{2}}{2 c z}]} . \end{array}

Here ρ is the radial coordinate at an arbitrary plane z, z_R = ω̄w²/2c is the Rayleigh range of the incident field, and v(ω) = c/[1 − sin² θ(ω)/2] is the phase velocity. For reflective axicons θ(ω) = θ = constant as defined above, and the spectral density readily takes on the form

S (ρ, z; ω) = - \frac{η (ω)}{π} \frac{z}{z_{R}} {(\frac{ω}{c})}^{2} {sin}^{2} θ S (ω) exp (- 2 \frac{ω}{\bar{ω}} \frac{z^{2}}{w^{2}} {sin}^{2} θ) J_{0}^{2} (\frac{ω}{c} ρ sin θ) .

For diffractive axicons we get, using Eq. (13),

\begin{array}{l} U (ρ, z; ω) = & - \frac{2 π i}{d} \sqrt{\frac{η (ω)}{π}} \sqrt{\frac{z}{z_{R}}} \sqrt{S (ω)} exp (- \frac{2 π^{2} c z^{2}}{d^{2} z_{R} ω}) \\ \times J_{0} (\frac{2 π ρ}{d}) exp [\frac{i ω}{c} (z + \frac{ρ^{2}}{2 z})] exp (- \frac{i 2 π^{2} c z}{d^{2} ω}), \end{array}

and the spectral density is

S (ρ, z; ω) = - \frac{4 π^{2}}{d^{2}} \frac{η (ω)}{π} \frac{z}{z_{R}} S (ω) exp (- \frac{4 π^{2} c z^{2}}{d^{2} z_{R} ω}) J_{0}^{2} (\frac{2 π ρ}{d}) .

We note that the method of stationary phase, being an asymptotic short-wavelength technique, does not preserve energy owing to the approximate radial integration but leads to results, such as Eqs. (17) and (19), that in usual circumstances are highly accurate on and sufficiently near the axis [17].

In order to obtain analytical results also in the time domain, we shall assume from now on that η(ω) = η = constant, although this condition is difficult to satisfy even approximately for diffractive axicons over the broad wavelength regimes we are considering. Fortunately, the results presented below do not depend critically on the exact form of η(ω), and for brevity we simply write η = 1.

4. Spatiotemporal axicon fields with spectrally coherent illumination

Considering first space–time fields created by reflective axicons, we first insert from Eq. (16) into Eq. (5), which leads to

U (ρ, z; t_{r}) = - i {(2 n)}^{n} {[\frac{S_{0}}{π Γ (2 n)} \frac{z}{z_{R}} \frac{\bar{ω}}{c}]}^{1 / 2} sin θ \int_{0}^{\infty} x^{n + 1} J_{0} (\frac{\bar{ω}}{c} ρ sin θ x) exp (- a x) d x .

Here we have introduced a normalized variable x = ω/ω̄ and denoted

a = n + {(\frac{z}{L})}^{2} + i \bar{ω} (t_{r} - \frac{ρ^{2}}{2 c z}),

where t_r = t − z/v is the retarded time, v = c/(1 − sin² θ/2) is the (constant) phase velocity, and L = w/sinθ is defined as the characteristic geometrical propagation-invariant range (although the field remains propagation-invariant further on unless the edge of the axicon truncates the incident field at ρ′ = w). The integral in Eq. (20) does not have a general analytical solution, but on axis we have a simple expression for the intensity

I (0, z; t_{r}) = I (0, z; 0) \frac{{(n + z^{2} / L^{2})}^{2 (n + 2)}}{{[{(n + z^{2} / L^{2})}^{2} + {(\bar{ω} t_{r})}^{2}]}^{n + 2}},

where

I (0, z; 0) = \frac{{(2 n)}^{2 n} Γ^{2} (n + 2) {sin}^{2} θ}{{(n + z^{2} / L^{2})}^{2 (n + 2)}} \frac{S_{0}}{π Γ (2 n)} \frac{\bar{ω}}{c} \frac{z}{z_{R}} .

The HWHM value of I(0, z; t_r) at t_r = T_r is given by

\bar{ω} T_{r} = \sqrt{2^{1 / (n + 2)} - 1} [n + {(z / L)}^{2}] .

According to this result, the effective temporal width of the pulse increases with the propagation distance; for n = 1 it is doubled at z = L. This phenomenon has a simple intuitive explanation: the geometrical interpretation in Fig. 2 indicates that the pulse at an axial distance z originates from a ring of radius ρ′ = z tanθ at the plane z = 0. On the other hand, in view of Eq. (9) and Fig. 1(b), the pulse width increases with ρ′. Thereby it should also increase with z.

Figure 3 illustrates 2D cross-sections (in a meridional plane) of the field generated by the reflective axicon, obtained by numerical integration of Eq. (20). Towards the end of the propagation-invariant range, the field profile resembles the expected X-wave. However, at shorter distances, the field has a distorted X-wave-like shape. The distortion arises from the quadratic phase term in the expression of the axicon field in Eq. (16). As the phase varies inversely with z, in the beginning the phase term behaves like a spherical wave and becomes increasingly planar on propagation.

Fig. 3 Amplitude of the field $\sqrt{I (ρ; t)}$ after reflective axicon at distances (a) z = L/2, and (b) z = L. The parameters here are n = 1, w = 15 μm, λ̄ = 550 nm, and θ = 5°. The video animation Media 1 illustrates the temporal evolution of the pulse shape on propagation. The bar on the left shows the change of the axial peak intensity in relation to its maximum value.

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Next, we proceed to examine the spatiotemporal properties of broadband fields generated by diffractive axicons, for which an analytical expression of the form

\begin{array}{l} U (ρ, z; t_{r}) = & - \frac{4 π i}{d} {(2 n)}^{n} {[\frac{S_{0} z}{π \bar{ω} Γ (2 n) z_{R}}]}^{1 / 2} J_{0} (\frac{2 π ρ}{d}) \\ \times {[\frac{b (z)}{a_{n} (ρ, z; t_{r})}]}^{(n + 1) / 2} K_{n + 1} [2 \sqrt{a_{n} (ρ, z; t_{r}) b (z)}] \end{array}

can be found after inserting Eq. (18) into Eq. (5). Here K_m denotes the modified Bessel function of the second kind and order m,

a_{n} (ρ, z; t_{r}) = n + i [\bar{ω} t_{r} - \frac{π}{w / d} \frac{{(ρ / d)}^{2}}{z / L}],

where t_r = t − z/c and

b (z) = {(\frac{z}{L})}^{2} + i π \frac{z}{L} \frac{w}{d} .

These expressions do not allow us to find an analytical expression for the HWHM of the pulse even on-axis.

Some numerically evaluated pulse profiles are shown in Fig. 4. Considering axial pulses, we see that the pulse gets longer as the propagation distance increases and its peak is slightly delayed. Moving off-axis, we observe a shift of the temporal origin of the pulse. Most notably, however, the pulses now have a long tail and therefore are quite unlike the pulses generated by reflective axicons. Mathematically, the temporal pulse distortion arises from the spectral phase term which is inversely proportional to frequency in Eq. (18). To examine the character of these pulses in more detail we plot in Fig. 5 2D meridional cross-sections of the pulse at two propagation distances. Well-defined zeros are seen in the transverse direction, as expected from qualitative considerations that predict Bessel-type pulses. Trailing of the temporal front end of the pulse in the radial direction and the tails are also clearly seen.

Fig. 4 Intensity profiles of the field after the diffractive axicon on the optical axis (ρ = 0) at distances z = L/2 (black), z = L (blue), and off-axis at point ρ/d = 2.5 (red) at the distance z = L (these curves are scaled to the same maximum value to aid comparisons). Here n = 1, w = 15 μm, λ̄ = 550 nm, d = 1.5 μm, and θ(ω̄) = 21.5°.

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Fig. 5 Amplitude of the field $\sqrt{I (ρ; t)}$ after diffractive axicon at distances (a) z = L/2, and (b) z = L. The parameters are same as in Fig. 4. The change in the intensity of the pulse (bar on the left) as well as the evolution of its shape on propagation are shown in Media 2.

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To understand the striking tailing effect in diffractive-axicon pulses, let us consider a simple geometrical picture illustrated in Fig. 6(a). Because of the dispersion relation in Eq. (13), spectral contributions of light that cross the optical axis at a certain position z originate from rings of different radii at the axicon plane, as exemplified by ρ′_R and ρ′_B for red and blue contributions. Specifically, red light has a longer distance (r_R) to travel to the crossing point than blue light (r_B), and therefore arrives later: the time it takes for a ray with frequency ω to travel from the axicon plane to the axial point is

t = \frac{z}{c \sqrt{1 - {(2 π c / ω d)}^{2}}} .

In view of this picture, we should expect the spectrum to be blue-shifted in the beginning of the pulse and red-shifted at the trailing end of the pulse. To investigate the validity of this geometrical picture, we plot in Fig. 6(b) the time-dependent physical spectrum (see Appendix B for details) of the pulse [19–21]. We see that, indeed, the low-frequency part of the pulse is long and delayed compared to the high-frequency part.

Fig. 6 (a) Geometrical explanation of the tailing effect in pulses generated by diffractive axicons. (b) The time-dependent physical spectrum of the field after diffractive axicon. We assume that the measurement instrument is a tunable Fabry–Perot filter with bandwidth B_f = 0.5 · 10¹⁵s⁻¹.

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5. Spatiotemporal axicon fields with spectrally incoherent illumination

Proceeding to consider stationary but fully spatially coherent fields, we may evaluate the axial complex degree of coherence after an reflective axicon in a closed form

γ (0, z; τ) = {[\frac{n + {(z / L)}^{2}}{n + {(z / L)}^{2} + i \bar{ω} τ / 2}]}^{2 n + 3} .

The HWHM value Θ_r of this profile is given by

\bar{ω} Θ_{r} = 2 \sqrt{2^{1 / (2 n + 3)} - 1} [n + {(z / L)}^{2}] .

Hence the axial degree of temporal coherence of stationary axicon field increases with propagation distance at the same rate as the axial width of a coherent pulse with the same spectrum does, as one can seen by comparing Eqs. (24) and (30). For diffractive axicons we obtain

γ (ρ, z; τ) = {(\frac{2 n}{2 n + i \bar{ω} τ})}^{n + 1 / 2} \frac{K_{2 n + 1} (2 \sqrt{a_{n} b})}{K_{2 n + 1} (2 \sqrt{2 n b})},

where a_n(τ) = 2n + iω̄τ and

b (z) = \frac{8 π^{2} c^{2} z^{2}}{d^{2} w^{2} {\bar{ω}}^{2}} .

The HWHM value cannot be evaluated analytically but, as illustrated in Fig. 7, the degree of axial temporal coherence decreases with propagation distance, though not as quickly as the temporal coherence for reflective axicons increases. The opposing behaviors of the temporal coherence profiles are obvious from the power spectra of the fields after the axicons, as shown in Fig. 8. The axial spectral density of the field after reflective axicon can be written as

S (0, z; ω) = - \frac{η (ω)}{π} \frac{z}{z_{R}} {(\frac{ω}{c})}^{2} {sin}^{2} θ S (ω) exp [- 2 \frac{ω}{\bar{ω}} \frac{z^{2}}{w^{2}} {sin}^{2} θ],

and for diffractive axicon it is of the form

S (0, z; ω) = - \frac{4 π^{2}}{d^{2}} \frac{η (ω)}{π} \frac{z}{z_{R}} S (ω) exp (- \frac{4 π^{2} c z^{2}}{d^{2} z_{R} ω}) .

This phenomenon is well explainable from the geometrical point of view. In the case of reflective axicons, the spectrum at an axial distance z arises from a ring with the corresponding radius in the incident plane. The spectrum of the incident field gets narrower as we move off-axis, as shown in Fig. 1(a), and this results in narrower axial spectrum with increasing z. Thus the temporal coherence profiles have to broaden with axial distance z. On the other hand, for diffractive axicons the spectrum gets wider slowly with axial distance z, but towards the end of the propagation-invariant distance L most of the long-wavelength contributions of the incident light have already crossed the axis. This causes a strong blueshift and widening of the spectrum. Hence the decrease of the temporal coherence of the field is rather fast.

Fig. 7 Temporal coherence profiles for the incident field and the fields after the axicons at (a) z = L/2 and at (b) z = L. Here the black curves indicate the incident field, blue curves for diffractive axicons, and the red curves for reflective axicons.

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Fig. 8 Axial power spectra (in arbitrary units) of the incident field and the fields after the axicons at distances (a) z = L/2, and (b) z = L. Here the black curves stand for the incident field, the blue curves for diffractive axicons, and the red curves for reflective axicons. The different power spectra are scaled for ease of comparison.

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6. Discussion and conclusions

In this work we have analyzed the nature of polychromatic optical fields as they emerge from axicons, the element of choice to generate extended, spatially localized fields (axicon line images) in monochromatic light. We have considered two types of axicons, reflective and diffractive, both of which exhibit certain achromatic features in polychromatic light: reflective axicons produce a wavelength-independent cone angle, whereas diffractive axicons create fields with transverse patterns that do not depend on wavelength. We also considered two types of illumination: coherent pulses and stationary fields. The former allows one to study the influence of the axicons’ different dispersion properties on spatiotemporal field localization, while the latter enables examining spatiotemporal coherence characteristics.

We showed, in particular, that in the coherent-field case reflective axicons generate temporally extended, X-wave-type pulses, while diffractive axicons produce elongated and slightly delayed, Bessel-type pulses with clear zeros in transverse plane. The latter fields also experience spectral rearrangement such that the pulse’s leading edge is blue-shifted and the trailing edge red-shifted. In the stationary-field case the on-axis temporal coherence width increases with propagation distance for reflective axicons, but it decreases (at a slower rate) for diffractive axicons. Physical explanations for all these phenomena were provided in terms of intuitive axicon pictures and the spatial variations of the fields’ power spectra. Some of the results obtained by our theoretical model are in agreement with experiments performed earlier [22, 23].

Our results are important in the rapidly progressing area of spatially and temporally localized optical waves [24, 25]. In various applications they will aid in the design of illumination and axicon conditions for optimal spatiotemporal distributions and temporal coherence properties of the ensuing fields. We have only considered the customary limiting circumstances of fully coherent and statistically stationary (spectrally uncorrelated) fields. However, allowing the spectral components of the illumination to be partially correlated one could likewise assess (temporally and spatially) partially coherent axicon fields [26, 27], thereby also bridging in a continuous manner the two extreme cases.

Appendix A

In this appendix we outline the stationary-phase calculation of the axicon field. To this end, let us first assume that f(ρ′) and g(ρ′) are by comparison slowly and rapidly varying functions in an integral in the form of

I (ω) = \int_{0}^{\infty} f (ρ^{'}) exp [i ω g (ρ^{'})] d ρ^{'} .

In the asymptotic limit ω → ∞ the general solution of this integral is [18]

I (ω) = \frac{f (ρ_{c}) exp [i ω g (ρ_{c})]}{\sqrt{ω g^{″} (ρ_{c})}},

where ρ_c is known as the critical point which is obtained when the derivative of g(ρ′) is zero, i.e., g′(ρ′) = 0. We may utilize this method in the evaluation of the field after the axicon by first inserting Eqs. (1) and (14), i.e., the incoming field in space–frequency domain and the transmission function of the axicon, into Eq. (15). This leads to a field in the form of

U (ρ, z; ω) = C_{0} F (ω) \int_{0}^{\infty} ρ^{'} J_{0} (\frac{ω}{c z} ρ ρ^{'}) exp (- \frac{ω {ρ^{'}}^{2}}{\bar{ω} w^{2}}) exp {i ω [\frac{{ρ^{'}}^{2}}{2 c z} - \frac{sin θ (ω) ρ^{'}}{c}]} d ρ^{'},

where

C_{0} = \sqrt{2 S_{0} / [π Γ (2 n) w^{2} \bar{ω}]}

and

F (ω) = \frac{ω}{i c z} exp [\frac{i ω}{c} (z + \frac{ρ^{2}}{2 z})] \sqrt{η (ω)} {(2 n)}^{n} {(\frac{ω}{\bar{ω}})}^{n + 1 / 2} exp (- n \frac{ω}{\bar{ω}}) .

Now, in view of Eq. (35), it is clear that

f (ρ^{'}) = ρ^{'} J_{0} (\frac{ω}{c z} ρ ρ^{'}) exp (- \frac{ω {ρ^{'}}^{2}}{\bar{ω} w^{2}})

and

g (ρ^{'}) = \frac{{ρ^{'}}^{2}}{2 c z} - \frac{sin θ (ω) ρ^{'}}{c} .

Thus, the derivatives of g(ρ′) are

g^{'} (ρ^{'}) = \frac{ρ^{'}}{c z} - \frac{sin θ (ω)}{c},

g^{″} (ρ^{'}) = \frac{1}{c z} .

A straightforward calculation shows that the critical point exists at ρ_c = z sin θ(ω). Inserting Eqs. (39), (40), and (42) into Eq. (36) at the critical point gives the final result of the field after the axicon as stated in Eq. (16).

Appendix B

The time dependent physical spectrum is observable with a photon counting detector set behind a filter which fulfills causality and linearity conditions. The physical spectrum, as defined by Eberly and Wódkiewicz [19], can be written as

G (t, ω_{f}, B_{f}) = \int \int_{- \infty}^{\infty} H^{*} (t - t_{1}, ω_{f}, B_{f}) H (t - t_{2}, ω_{f}, B_{f}) Γ (t_{1}, t_{2}) d t_{1} d t_{2},

where Γ(t₁, t₂) = 〈U^*(t₁)U(t₂)〉 is the mutual coherence function of the source (with bracket angles denoting ensemble averaging), specifically in this case the field after the axicon. Additionally, H(t, ω_f, B_f) is the temporal response function of the filter at hand, where ω_f and B_f are the center frequency and the bandwidth of the filter, respectively. The time-domain representation in Eq. (43) may be converted to frequency-domain by utilizing the cross-correlation function

W (ω_{1}, ω_{2}) = \int \int_{- \infty}^{\infty} Γ (t_{1}, t_{2}) exp [- i (ω_{1} t_{1} - ω_{2} t_{2})] d t_{1} d t_{2}

and the frequency response of the filter function (obtained via Fourier transform)

\tilde{H} (ω, ω_{f}, B_{f}) = \int_{- \infty}^{\infty} H (t, ω_{f}, B_{f}) exp (i ω t) d t .

It is obvious that Eqs. (44) and (45) lead to the physical spectrum of the form

G (ω, ω_{f}, B_{f}) = \frac{1}{{(2 π)}^{2}} \int \int_{- \infty}^{\infty} {\tilde{H}}^{*} (ω_{1}, ω_{f}, B_{f}) \tilde{H} (ω_{2}, ω_{f}, B_{f}) W (ω_{1}, ω_{2}) exp [i (ω_{1} - ω_{2}) t] d ω_{1} d ω_{2},

which, at least in our case, is more convenient considering numerical computations. In the calculation of the physical spectrum it is useful to make use of a tunable Fabry–Perot filter with frequency response function [19]

\tilde{H} (ω, ω_{f}, B_{f}) = \frac{B_{f}}{B_{f} - i (ω - ω_{f})},

which is inserted to Eq. (46), giving the frequency-domain representation [21] as

G (t, ω_{f}, B_{f}) = \frac{B_{f}^{2}}{{(2 π)}^{2}} \int \int_{- \infty}^{\infty} \frac{exp [i (ω_{1} - ω_{f}) t]}{B_{f} + i (ω_{1} - ω_{f})} \frac{exp [- i (ω_{2} - ω_{f}) t]}{B_{f} - i (ω_{2} - ω_{f})} W (ω_{1}, ω_{2}) d ω_{1} d ω_{2} .

Assuming that the pulses after the axicon are fully coherent, i.e., W(ω₁, ω₂) = U^*(ω₁)U(ω₂), the field representation of Eq. (18) can be employed in Eq. (48). The ensuing expression is then numerically evaluated as presented in Fig. 6(b).

Acknowledgments

This work was partly funded by the Academy of Finland (projects 252910 and 268480).

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Broadband spatiotemporal axicon fields

Abstract

1. Introduction

2. Representations of the incident field

3. Space–frequency representation of axicon fields

4. Spatiotemporal axicon fields with spectrally coherent illumination

5. Spatiotemporal axicon fields with spectrally incoherent illumination

6. Discussion and conclusions

Appendix A

Appendix B

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (8)

Equations (48)

Optics Express