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Polychromatic stationary propagation-invariant fields with complete spatial coherence in the space-frequency domain are considered. In general, the field is shown to be spatially partially coherent in the space-time domain, apart from transversely achromatic fields with complete transverse coherence. Particular attention is paid to fields that possess the same cone angle at each frequency; these are stationary counterparts of pulsed conical fields known as X waves. It is shown that, for such fields, the radius of the space-time-domain transverse coherence area depends critically on the bandwidth of the power spectrum and can be comparable to the central-lobe radius of the monochromatic field component at the peak frequency of the spectrum.
©2008 Optical Society of America
1. Introduction
Following the introduction of a class of coherent optical fields that propagate in free space without any change of shape or scale [1, 2], the interest in these propagation-invariant fields rapidly expanded in several directions. The propagation characteristics of various finite-aperture approximations of these fields, including so-called Bessel–Gauss beams [3], were investigated. A wide range of methods were introduced for generating such fields using spatial filters, refractive and diffractive optics, and resonators [4–6]. The concept of propagation-invariance was extended to electromagnetic [7–9] and spatially partially coherent [10] fields, and the propagation of these fields through paraxial lens systems was investigated [11, 12]. Pulsed versions of coherent propagation-invariant fields were also introduced [13–17] and the connection of monochromatic propagation-invariant fields to certain previously known pulsed solutions of the wave equations, known as focus wave modes [18, 19], was noticed [20, 21]. More complete descriptions of the developments and state of the art in propagation-invariant optics can be found in recent review papers [22, 23].
Compared to the well-studied monochromatic (or quasi-monochromatic) case, the interest in wideband polychromatic propagation-invariant fields is rather recent, being motivated by the fascinating progress in development of high-brightness super-luminescent diodes (SLDs) and supercontinuum sources. Fischer and coworkers [24, 25] presented a seminal series of experiments on generation of propagation-invariant fields with stationary and pulsed wideband sources. In these experiments, an axicon was used to form approximations of polychromatic Bessel fields using sources with spectra ranging from supercontinuum through ‘white light’ (halogen lamps) to width characteristic of SLDs (tens of nanometers).
One goal of this paper is to take some steps towards better understanding of the recent experimental observations, in which the coherence of light plays a crucial role. However, the main goal is not to consider any particular light source or generation method; rather, certain general aspects of coherence of stationary polychromatic propagation-invariant fields are investigated. A mathematically convenient power spectrum is assumed, with enough flexibility to roughly model sources of widely different spectral widths (including supercontinuum sources and SLDs). Our second assumption is more fundamental: the field is assumed to be completely coherent at each (angular) frequency, i.e., in the space-frequency domain [26]. Determination of the coherence properties of such field in the space-time domain then leads to a concrete illustration of one of the less well-appreciated aspects of optical coherence theory, namely the relationship between coherence in space-frequency and space-time domains. In particular, it is demonstrated that complete coherence in the space-frequency domain does not guarantee a high degree of spatial coherence in space-time domain: the coherence area in the latter domain can be strikingly small (radius of the order of the wavelength) for perfectly realistic wideband fields.
2. Polychromatic stationary propagation-invariant fields
A stationary optical field that is completely coherent in the space-frequency domain is characterized by a cross-spectral density function that, at any angular frequency ω, is of a factorized form (see [27] and Section 4.5 of [28])
We assume that the field U(x,y,z;ω) in Eq. (1) is propagation-invariant [1, 2], i.e., expressible as
with
and
where s(ω), α(ω), and β(ω) are non-negative real functions, A(ϕ) is an arbitrary (well-behaved) complex function, and c is the speed of light in vacuum.
The space-time-domain coherence properties of a scalar optical field are described by the mutual coherence function Γ(x 1,y 1,z 1,x 2,y 2,z 2;τ), related to the cross-spectral density function through the generalizedWiener-Khintchine theorem
Defining, as usual, a normalized quantity
known as the complex degree of coherence, the degrees of transverse spatial coherence, longitudinal spatial coherence, and temporal coherence are characterized by functions γ(x 1,y 1,z,x 2,y 2,z;0), γ(x,y,z 1,x,y,z 2;0) and γ(x,y,z,x,y,z;τ), respectively.
Inserting Eqs. (1) and (2) into Eq. (5) immediately yields a general space-time-domain expression for stationary polychromatic fields that are completely coherent and propagation-invariant in the space-frequency domain:
where Δz=z 2-z 1 and V(x,y;ω) is given by Eq. (3). The optical intensity I(x,y,z)=Γ(x,y, z,x,y, z;0) of the field takes the form
where
is the spectral density of the field at point (x,y).
It is obvious from Eqs. (3), (4) and (7) that although the polychromatic, stationary propagation-invariant field is propagation-invariant in the space-time domain (the intensity distribution is independent on z), it is in general neither spatially nor temporally coherent. The coherence properties of the field depend of the chosen dispersion relation, i.e., on the functional form of β(ω), and on the spectral weight function s(ω) of the different monochromatic components. In this work we use a model weight function of the form
or, in wavelength scale,
where the parameter n is assumed real [n>-1/2 in Eq. (10) and n>1/2 in Eq. (11)], Γ denotes the Gamma function, ω̄ and λ̄=2πc/ω̄ are the peak angular frequency and peak wavelength of the spectrum, respectively, and we have normalized ∫∞ 0 s(ω)dω=∫∞ 0 s(λ)dλ. Figure 1 illustrates the function s(λ) for three values of n; a peak wavelength λ̄=0.55 µm is chosen to illustrate the spectra in (and around) the visible range. With the smallest value n=1 we have an ultra-wideband (supercontinuum-like) field: the distribution s(λ) has a full width at half maximum (FWHM) of approximately 2.35λ̄. With n=10 we have essentially white light with FWHM equal to 0.54λ̄, and with n=100 the FWHM is 0.17λ̄. Larger values of n could be used to simulate, e.g., spectra of SLDs, but in this range it is more convenient (both analytically and numerically) to use a Gaussian spectral weight of the same FWHM since the difference between the two is negligible.
Fig. 1. Spectral weight given by Eq. (11) for n=1 (solid line), n=10 (dashed line), and n=100 (dotted line), plotted for λ̄=0.55 µm.
3. Transversely achromatic fields
In view of Eqs. (9), the (normalized) spectrum of the field is independent of position if the function V(x,y;ω) is independent of ω, i.e., if α(ω)=α=constant. Equation (4) then immediately gives the dispersion relation for transversely achromatic fields in the form
Such field are conical at each frequency, but the cone angle θ(ω) depends on frequency according to
Approximations of fields of this type can be produced, for example, in the first order of a circular diffraction grating with radial period d=2π/α, illuminated by a collimated polychromatic field. Leach et al. [29] recently generated these fields using essentially the latter approach, with an off-axis diffractive element written in a spatial light modulator, accompanied with a prism to compensate for the dispersion caused by the carrier grating.
It should be noted that the dispersion relation, Eq. (12), has been studied in connection with coherent pulsed fields, with each frequency component following a Bessel profile. It has been show, e.g., that these so-called Bessel pulses propagate with subluminal group velocity
and exhibit anomalous dispersion [13, 30, 31]. We next consider stationary fields that satisfy Eq. (12), stressing that even though ν g can still be defined by Eq. (14), it loses its meaning as a group velocity.
Since V(x,y;ω)≡V(x,y) is independent of ω, it follows from Eq. (7) that the stationary achromatic solutions satisfying Eq. (12) are described in the space-time domain by a mutual coherence function of the form
where V(x,y) is given by Eq. (3) with α(ω)=α. This is a product of two terms containing the transverse spatial dependence and the longitudinal-temporal dependence, respectively. Consequently, the complex degree of coherence defined in Eq. (6) factors in the form
where the first factor
is precisely the complex degree of transverse spatial coherence in the space-time domain, while the second factor
gives the complex degree of longitudinal spatial coherence if τ=0 and the complex degree of temporal coherence if z 1=z 2. In writing Eq. (18) we assumed that the frequency-integrated spectral weight is normalized to unity, as in case of Eq. (10).
Some simple general conclusions can be drawn immediately from Eqs. (17) and (18). Since |γ(x 1,y 1,z 1,x 2,y 2,z 2;τ)|=1, the field is completely transversely spatially coherent in the space-time domain (as in the space-frequency domain). The complex degree of temporal coherence is, at every transverse point, exactly the same as that of a polychromatic plane wave with a power spectrum s(ω). The complex degree of longitudinal spatial coherence must, in general, be calculated numerically, but it is independent of position. Such simple conclusions only hold for the particular dispersion relation, Eq. (12). The space-time coherence properties of conical waves become considerably more elaborate if other dispersion relations satisfying the conical-wave constraint, Eq. (4), are considered.
4. Fields with achromatic cone angle
Fields characterized by the same cone angle θ(ω)=θ=constant at each angular frequency ω must satisfy the relations
Waves of this type can be realized, e.g., in Durnin’s original setup [2] or by illuminating an achromatic axicon with a collimated polychromatic wave.
Coherent propagation-invariant pulses satisfying Eqs. (19) and (20), known as X waves, were introduced by Lu and Greenleaf [15] and studied further in, e.g., [16,17,32,33]. They propagate with constant (ω-independent) superluminal group velocity
and are therefore completely dispersion-free in addition to having shape- and scale-invariant (but chromatic) transverse profiles. Hence these superluminal pulses, also known as ‘localized waves’, are propagation-invariant in a stricter sense than transversely achromatic Bessel pulses.
With Eqs. (3), (19), and (20), and rearranging the order of integrations, Eq. (7) for stationary propagation-invariant fields can be written in the form
where
is the Fourier transform of the spectral weight function. Now there is coupling between transverse, longitudinal, and temporal coherence properties of the field. In general, the field is transversely partially spatially coherent, while the longitudinal and temporal coherence properties depend on transverse position, unlike in the case of achromatic fields considered in Section 3.
It follows from Eq. (22) that, on the optical axis, a simple Fourier-transform relationship holds between the spectral weight function and the longitudinal/temporal coherence function:
Thus the on-axis field has exactly the same temporal coherence properties as a plane wave with the same power spectrum, and the same longitudinal coherence properties as that plane would have if it propagated in an effective medium of refractive index N=cosθ. Since N<1, the on-axis coherence length of the conical wave is greater than that of the plane wave (but of course the difference is significant only for non-paraxial cone angles θ). This is a reflection of the superluminal group velocity of the corresponding X wave, but like the latter [17], is only an ‘apparent’ property due to the conical-wave character of the field.
5. Polychromatic stationary Bessel fields
In order to illustrate more explicitly the space-time coherence properties of fields that satisfy Eqs. (19) and (20), we turn our attention to the special case A(ϕ)=1/2π, which leads to the fundamental, rotationally symmetric J 0 Bessel-mode solution
where α(ω̄)=(ω̄/c) sin θ and . With s(ω) given by Eq. (10), the spectral density in Eq. (9) takes the form
Defining β(ω̄)=(ω̄/c)cos θ and changing variables as Ω=ω/ω̄, we obtain from Eq. (7) an expression for the mutual coherence function in the form
The integral does not have a general analytical solution. However, the optical intensity distribution I(ρ)=Γ(ρ,z,ρ,z;0) is given by
where P F Q is a generalized hypergeometric function. The complex degree of transverse spatial coherence between an arbitrary point ρ 1=ρ and an axial point ρ 2=0 is of the form
Fig. 2. Fig. 2. Square root of the radial intensity distribution for conical waves given by Eq. (28) if n=1 (solid line), n=10 (dashed line), and n=100 (dotted line).
Furthermore, the complex degrees of longitudinal and temporal coherence of the field at an arbitrary transverse point ρ 1=ρ 2=ρ can be determined by setting either τ or Δz equal to zero in the expression
where g(Δz;τ)=β(ω̄)Δz-ω̄τ. On the optical axis Eq. (30) reduces to the form
which follows also from Eq. (24).
The radial intensity distributions given by Eq. (28) are illustrated in Fig. 2 for the spectral weight distributions shown in Fig. 1. In order to better see the differences in sidelobe structures for different values of n, the square root of I(ρ) is plotted. The characteristic ring structure of the monochromatic Bessel field is completely washed out in case of an ultra-wideband field (n=1); no sidelobes are seen and the radial profile decreases monotonously, resembling the 1/ρ envelope of the J 0 profile. With increasing n, sidelobes begin to take shape, starting from those closest to the optical axis in case of white light (n=10), and growing in number when the FWHM of s(ω) is decreased further (n=100).
Figure 3 illustrates the absolute value of the complex degree of transverse spatial coherence between an arbitrary and an axial point, given by Eq. (29). The function γ(ρ,z,0,z;0) is real and changes sign (phase change of π radians) at the zeroes of the plot in Fig. 3. However, we chose to plot the absolute value because it is the quantity measured in Young’s double-slit experiment and because the main features of the spatial coherence properties of the field are easier to appreciate from this plot.
With n=1 in Fig. 3, the absolute value of the complex degree of coherence reaches its first zero at α(ω̄)ρ≈2.83 and then recovers slightly before dying out. Thus the effective width of the transverse coherence function (radius of the transverse coherence area) is of the same order of magnitude as the radius of the main lobe of the monochromatic Bessel field J 0[α(ω̄)ρ] at the center frequency ω̄ (located at α(ω̄)ρ≈2.405). Thus, for large cone angles θ, it is rather strikingly of the order of the center wavelength λ̄ even though the same field is completely coherent Fig. 3. The absolute value of the complex degree of transverse spatial coherence given by Eq. (29) for n=1 (solid line), n=10 (dashed line), and n=100 (dotted line). in the space-frequency domain. Of course, measuring such small coherence areas using two-pinhole setups is a complicated matter not only because Kirchhoff’s boundary conditions fail but also because surface-wave coupling between the pinholes can produce spurious coherence-enhancement effects [34].
Fig. 3. The absolute value of the complex degree of transverse spatial coherence given by Eq. (29) for n = 1 (solid line), n = 10 (dashed line), and n = 100 (dotted line).
Increasing the value of n leads to the appearance of pronounced sidelobes in |γ(ρ,z,0,z;0)|, with zeros close to those of the monochromatic Bessel field at ω̄. The decay rate of the envelope rather than the first zero location should then be considered as a proper measure of the effective coherence area; the zeros do not disappear however small the spectral FWHM becomes (though the minima ultimately get too narrow to be seen in practical two-pinhole experiments). With white-light fields (such as n=10) the effective coherence area covers a few central sidelobes of J 0 [α(ω̄)ρ]. Note (by comparing Figs. 2 and 3) that the effective coherence area is in fact substantially larger than the area where sidelobes of the intensity distribution can be seen; the intensity actually drops close to zero at the minima only if the envelope of |γ(ρ,z,0, z;0)| is practically unity on both sides of its zero.
The fact that the zeros of |γ(ρ,z,0,z;0)| persist however narrow the spectral weight function is can be appreciated by choosing a delta-function spectrum s(ω)=δ(ω-ω̄) instead of the distribution in Eq. (10). Now Eq. (7) gives
Hence
is a binary-phase function, which changes abruptly between zero and π at each zero of the monochromatic Bessel field at ω=ω̄. Any increase of the spectral bandwidth would then create minima of finite width around the phase discontinuities.
Figure 4 illustrates the longitudinal (τ=0) and temporal (Δz=0) coherence properties of the polychromatic Bessel field with n=10 calculated from Eq. (30) as a function of g(Δz;τ). Figure 4(a) illustrates coherence at different radial distances from the axis: shown are results for an axial point (solid line), as well as for distances that correspond to the first zero (dashed line) and first side-lobe maximum (dotted line) of the monochromatic Bessel mode at ω=ω̄. At positions of relatively high intensity the curves are smooth, although the effective coherence length/time depends on position. On the other hand, close to the minima of the transverse distribution (this applies to other minima as well) the longitudinal/temporal coherence shows more complicated behavior. Figure 4(b) displays the transverse variation of longitudinal/temporal coherence more clearly; here a several different values of longitudunal/temporal delay are selected from Fig. 4(a). The transverse variation is seen to be rather sharp and significant especially in the central region of the field, where the envelope of the transverse spatial coherence varies rapidly.
Fig. 4. Longitudinal and temporal coherence given by Eq. (30) for n=10. Absolute value of the complex degree of coherence (a) as a function of longitudinal/temporal delay g(Δz;τ) at radial positions α(ω̄)ρ=0 (solid line), α(ω̄)ρ=2.405 (dashed line), and α(ω̄)ρ=3.83 (dotted line) and (b) as a function of radial position for g(Δz;τ)=2.5 (solid line), g(Δz;τ)=5 (dashed line), and g(Δz;τ)=10 (dotted line).
Let us finally illustrate the visual appearance of the polychromatic (colored) fields considered above. For reference, Fig. 5 shows the intensity distributions of the fundamental Bessel fields as they would be seen by an ideal logarithmic detector with a flat spectral response, i.e., two-dimensional plots of log [I(x,y)] are shown for n=1, n=10, and n=100. With n=1 the profile is smooth, but with a distinct axial maximum, and the ring pattern begins to emerge when n is increased. Figures 6 and 7 illustrate the human visual perception of the same propagation-invariant patterns: the intensity seen by the eye is shown in Fig. 6 and the color perception is illustrated in Fig. 7. Here the CIELAB color space is used [35], with reference to a ‘white’ object of the same brightness as the brightest region in the pattern. Comparing Figs. 5 and 6, we see that the eye detects more contrast in broadband patterns than an ideal detector with a flat spectral response. This is clear since the spectral efficiency curve of the eye cuts off the ultraviolet and infrared contributions to, especially, the spectrum assumed in Fig. 5(a). In fact, the intensity distributions with n=1 and n=10 in Figs. 6(a) and 6(b), respectively, appear almost identical, as do the corresponding colored patterns in Figs. 7(a) and 7(b). However, clear differences are seen as soon as the spectrum becomes distinctly narrower than the spectral response of the eye, as is Fig. 7(c); with n=1000, the visual appearance of the spectrum is virtually monochromatic (green, not shown).
6. Final remarks
Perhaps the most striking feature of the fields discussed in this paper is the low degree of transverse spatial coherence in the space-time domain, even though the field is completely coherent in the space-frequency domain. The fact that complete coherence in the space-frequency domain does not imply complete coherence in the space-time domain is well known, in principle; see, e.g., Sections 4.5 and 7.4 in [28]. However, it does not appear to be generally appreciated how different the two degrees of coherence can be, especially for wideband fields. The examples provided here offer a simple, yet illustrating interpretation of this difference. For wideband conical fields considered in Sections 4 and 5, the monochromatic field components at different frequencies ω in the spectrum have widely different transverse scales. Thus, considering Eq. (7) with Δz=τ=0 with fixed (x 2,y 2), the frequency integrations are performed over different spectral distributions at different values of (x 1,y 1). Hence the normalization in Eq. (6) leads in general to |γ | < 1, in contrast to the case of achromatic fields considered in Section 3.
Fig. 5. Intensity distributions of polychromatic fields with (a) n=1, (b) n=10, (c) n=100 as seen by a logarithmic intensity detector with an ideal, flat spectral response.
Fig. 6. Intensity distributions of polychromatic fields with (a) n=1, (b) n=10, (c) n=100 as perceived by the human eye.
Fig. 7. Spectral distributions of polychromatic fields with (a) n=1, (b) n=10, (c) n=100 with colors as perceived by the human eye in normal circumstances.
It is worth stressing that in a practical Young’s experiment the detector is always spectrally selective. Thus the measured degree of transverse spatial coherence that would differ from |γ(ρ,z,0,z;0)|. Indeed, if a bandpass filter simulating the spectral response were assumed, a reduction of the bandwidth would increase the visibility towards unity at the limit of zero bandwidth because of the full coherence in the space-frequency domain. If the field did not possess perfect transverse spatial coherence in the space-frequency domain, a finite value given by the complex degree of spectral coherence between the two points would be approached in the limit of zero bandwidth, as discussed in [36].
The theory presented here used a model spectrum to yield analytic results. It is not directly applicable to practical sources, but in many cases the model spectrum could simply be replaced by a measured spectrum, followed by numerical evaluation of the integrals. In other cases the extension is not as simple, especially if the source is only quasi-stationary or emits trains of ultrashort pulses. For example, supercontinuum sources that are spatially coherent at each frequency feature more complex spectral coherence properties than the stationary sources considered here. Furthermore, pulsed fields produced by chopped stationary fields feature partial spectral coherence [37]. Obviously, therefore, more work needs to be done to fully characterize practical experiments such as those carried out in [24, 25, 29]. One approach is to obtain a connection between fully coherent pulses and stationary polychromatic fields along the lines followed recently in case of polychromatic plane waves [38–40] and Gaussian Schell-model beams [41]. On the other hand, modeling partial spatial coherence in the space-frequency domain is another direction in which the work should be extended to enable the treatment of wideband sources such as halogen bulbs and SLDs. Finally, details of the actual experimental apparatus used, including the finite aperture of the system and the exact dispersion characteristics of the refractive, diffractive or hybrid elements used [42], should be considered.
Acknowledgments
I wish to thank Martti Mäkinen for producing Figs. 6 and 7. The financial support from the Academy of Finland (projects 129155 and 209806) is greatly appreciated. The work was also supported by the Nanophotonics Research and Development Program funded by the Ministry of Education of Finland, and by the Network of Excellence in Micro-Optics (NEMO).
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