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Broadband spatiotemporal Gaussian Schell-model pulse trains Rahul Dutta,1,* Minna Korhonen,1 Ari T. Friberg,1 Göery Genty,2 and Jari Turunen1 1
Institute of Photonics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland 2 Optics Laboratory, Tampere University of Technology, FI-33101 Tampere, Finland *Corresponding author: [email protected] Received October 28, 2013; revised January 2, 2014; accepted January 17, 2014; posted January 22, 2014 (Doc. ID 200078); published February 24, 2014
A new class of partially coherent model sources is introduced on the basis of the second-order coherence theory of nonstationary optical fields. These model sources are spatially fully coherent at each frequency but can have broadband spectra and variable spectral coherence properties, which lead to reduced spatiotemporal coherence in the time domain. The source model is motivated by the spectral coherence properties of supercontinuum pulse trains generated in single-spatial-mode optical fibers. We demonstrate that such broadband light is highly (but not completely) spatially coherent, even though the spectral and temporal coherence properties may vary over a wide range. The model sources introduced here are convenient in assessing the spatiotemporal coherence of broadband pulses in optical systems. © 2014 Optical Society of America OCIS codes: (030.1640) Coherence; (050.1940) Diffraction; (060.2310) Fiber optics; (320.6629) Supercontinuum generation. http://dx.doi.org/10.1364/JOSAA.31.000637
1. INTRODUCTION Supercontinuum (SC) pulse trains generated in nonlinear optical fibers [1–4] are generally thought to be spatially coherent. This is indeed true at each individual frequency, i.e., in the space-frequency domain, provided that the SC light is produced in a fiber with a single spatial mode. Full spatial coherence in the space-time domain is also guaranteed if the SC pulses are fully spectrally (and therefore also temporally) coherent. However, recent research [5–7] clearly shows that SC pulse trains with widely variable spectral coherence properties can be created in nonlinear fibers, depending on initial conditions such as excitation pulse power and duration, and propagation distance in the fiber. In view of these studies, SC light can be described as a superposition of quasi-coherent and quasi-stationary contributions, of which the latter is dominant in a broad range of excitation conditions. Then, from a fundamental point of view [8], full spatial coherence in the space-time domain is no longer guaranteed even if the SC light is spatially fully coherent at each individual frequency. Since the spectral and temporal coherence properties of SC light generated in nonlinear fibers can be determined from simulated ensembles of individual realizations [3,5,6], and the (frequency-dependent) transverse mode distribution of the fiber can be deduced from its refractive-index profile, it is possible to develop a fully realistic spatiotemporal model for the coherence of SC light. However, past experience has shown that simplified source models are useful in optical coherence theory. Models of this type allow analytical solutions in terms of continuously variable parameters that enable the description of fields with different states of coherence. They have been developed for spatially, spectrally, and temporally partially coherent fields, and, in particular, Gaussian distributions of various field properties have been found 1084-7529/14/030637-07$15.00/0
convenient in establishing quantitative and qualitative results (see, e.g., [9–12]). The above considerations motivate us to introduce a simplified analytical model for statistical description of broadband pulse trains generated in single-mode fibers, taking into account the frequency dependence of the transverse scale of the fiber mode and the spectral coherence properties of the pulse trains. We assume that the spatial mode of the fiber is Gaussian with a spectral scale that ensures isodiffracting (the Rayleigh range is independent of frequency) freespace propagation of the output field. The spectrum of the field as well as the spectral coherence function are also taken as Gaussian, the latter depending on the frequency difference only; hence the model may be viewed as an extension of the Gaussian Schell-model plane-wave pulse [13].
2. SPECTRAL AND SPATIAL COHERENCE OF NONSTATIONARY FIELDS Let us denote the time-domain representation of a single realization of the light field arriving at the exit plane, z constant, of a nonlinear optical fiber by V ρ; t, where ρ x; y, the z coordinate is omitted for brevity, and t denotes time. We assume that the temporal center of the initial pump pulse determines the origin of time for each realization. The temporal realizations V ρ; t have Fourier representations of Z V ρ; t
∞ 0
V ρ; ω exp−iωtdω;
(1)
where V ρ; ω depends on the transverse position ρ, and angular frequency ω in a way dictated by the refractive-index profile of the fiber. The cross-spectral density function © 2014 Optical Society of America
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ZZ
(CSD) of the pulse train is defined as the ensemble average over all individual space-frequency-domain realizations, i.e., W ρ1 ; ρ2 ; ω1 ; ω2 hV ρ1 ; ω1 V ρ2 ; ω2 i:
Γ ρ1 ; ρ2 ; t1 ; t2
∞
0
W ρ1 ; ρ2 ; ω1 ; ω2
× expiω1 t1 − ω2 t2 dω1 dω2 :
(2)
(9)
In analogy with Eqs. (3) and (4), we may then introduce the temporal intensity,
The spectral density S ρ; ω W ρ; ρ; ω; ω
(3)
I ρ; t Γ ρ; ρ; t; t;
(10)
and the complex degree of spectral coherence and the complex degree of coherence, W ρ1 ; ρ2 ; ω1 ; ω2 μ ρ1 ; ρ2 ; ω1 ; ω2 p S ρ1 ; ω1 S ρ2 ; ω2
(4)
of the field can then be evaluated. If the pulses are generated in a single-mode fiber, the spectral degree of spatial coherence at any single frequency ω is complete, i.e., jμ ρ1 ; ρ2 ; ω; ωj 1 for any pair of points ρ1 ; ρ2 , but its two-frequency dependence may be quite arbitrary. More specifically, if the (deterministic) spatial profile of the single eigenmode propagating in the fiber is U ρ; ω, the space-frequency field may be written as V ρ; ω aωU ρ; ω, where aω is a random function representing the complex amplitude of the modal field. Since we are dealing with a single-mode field, it is natural that aω is independent of the lateral position. On substituting the mode field into Eq. (2), the CSD takes on the form, W ρ1 ; ρ2 ; ω1 ; ω2 W ω1 ; ω2 U ρ1 ; ω1 U ρ2 ; ω2 ;
(5)
where W ω1 ; ω2 ha ω1 aω2 i is the spectral correlation function associated with the field. This quantity is analogous to the CSD used to analyze the spectral coherence of planewave fields, e.g., in [13]. In SC modeling, the CSD function W ρ1 ; ρ2 ; ω1 ; ω2 can be determined by numerical simulations as described in [5,6]. From the knowledge of this function, one may straightforwardly evaluate, e.g., the overall degree of spectral coherence μ¯ defined according to μ¯ 2
RR ∞ 0
jW ω1 ; ω2 j2 dω1 dω2 R ; 0∞ Sωdω2
(6)
where Sω Wω; ω. Generalizing this definition, we introduce a quantity, RR ∞ jW ρ1 ; ρ2 ; ω1 ; ω2 j2 dω1 dω2 R∞ μ¯ 2 ρ1 ; ρ2 R ∞ 0 ; 0 S ρ1 ; ω1 dω1 0 S ρ2 ; ω2 dω2
(11)
in the space-time domain. The instantaneous (complex) degree of spatial coherence is defined as the equal-time correlation function, γ ρ1 ; ρ2 ; t; t. Combining Eqs. (5) and (9), we find ZZ Γ ρ1 ; ρ2 ; t; t
0
∞
W ω1 ; ω2
× U ρ1 ; ω1 U ρ2 ; ω2 expiω1 − ω2 tdω1 dω2 : (12) If W ω1 ; ω2 is separable, i.e., the field is spectrally fully coherent, then Γρ1 ; ρ2 ; t; t is spatially separable, and the field therefore is spatially fully coherent in the time domain. In general, partial spectral coherence implies partial spatiotemporal coherence, even if the field is spatially fully coherent at each individual frequency. However, in the special case of a scale-invariant modal profile of the form U ρ; ω U ρ, we obtain from Eq. (12): Γ ρ1 ; ρ2 ; t; t U ρ1 U ρ2 ZZ ∞ × W ω1 ; ω2 expiω1 − ω2 tdω1 dω2 ; (13) 0
which immediately implies that jγ ρ1 ; ρ2 ; t; tj 1 for all ρ1 and ρ2 , i.e., full spatial coherence in the space-time domain. This would, however, require an achromatic fiber in the sense that the modal profile is the same at all frequencies within the wide spectral band considered in this work. Instantaneous field quantities are difficult to measure at optical frequencies; hence time-integrated quantities such as
(7)
which represents the frequency-integrated degree of spatial coherence between the wave fields at any two transverse points, ρ1 and ρ2 . The space-time field realizations are connected to the frequency-domain realizations by the Fourier-transform relationship of Eq. (1), and the space-time correlation properties of the field are characterized by the mutual coherence function (MCF): Γ ρ1 ; ρ2 ; t1 ; t2 hV ρ1 ; t1 V ρ2 ; t2 i:
Γ ρ1 ; ρ2 ; t1 ; t2 ; γ ρ1 ; ρ2 ; t1 ; t2 p I ρ1 ; t1 I ρ2 ; t2
(8)
By inserting from Eq. (1) into Eq. (8) and using Eq. (2), we obtain a relationship between CSD and MCF in the form of the generalized Wiener–Khintchine theorem:
¯ ρ1 ; ρ2 Γ γ¯ ρ1 ; ρ2 p ; ¯ ρ1 I ¯ ρ2 I
(14)
with ¯ ρ1 ; ρ2 Γ
Z
∞
−∞
Γ ρ1 ; ρ2 ; t; tdt
(15)
and ¯ ρ Γ ¯ ρ; ρ I
Z
∞
−∞
I ρ; tdt
(16)
are of interest. In view of Eq. (9) and using the integral representation of the Dirac delta function, we have in general:
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¯ ρ1 ; ρ2 2π Γ
Z
∞
0
W ρ1 ; ρ2 ; ω; ωdω
(17)
¯ ω1 ω2 ∕2 and Δω ω2 − ω1 , respectively. Then ω Eq. (6) takes the form of
and ¯ ρ 2π I
Z
∞ 0
μ¯ 2 S ρ; ωdω:
(18)
This implies that all information on the spectral coherence properties of the field is lost in a time-averaged measurement and, consequently, γ¯ ρ1 ; ρ2 should not in general be considered as a true measure of the degree of spatial coherence: it depends only on the field’s spectral properties and one obtains values of j¯γ ρ1 ; ρ2 j smaller than unity even if the field is spectrally fully coherent and thereby also spatially fully coherent.
R ∞ R 2ω¯ 0
¯ Δωj2 dωdΔω ¯ jW ω; −2ω¯ R ; ∞ 2 0 Sωdω
(24)
and Eq. (7) is modified analogously. In order to obtain analytical results, we will replace the finite integration limits by −∞; ∞, which can be justified if the ratio Ω∕ω0 is sufficiently ¯ and Δω, we small (see Appendix A for further details). Using ω may collect Eqs. (20)–(22) in the form 2 2 T ¯ − ω0 2 exp − Δω2 ; (25) ¯ Δω S 0 exp − 2 ω W ω; 8 Ω with a new constant T defined by writing
3. SPECTRAL-DOMAIN FIELD MODEL Let us proceed by introducing a model source that mimics the properties of real SC sources at least in a qualitative sense. To this end, we assume that the SC is generated in a graded-index fiber with a parabolic refractive-index distribution, so that the modal profile in the frequency domain can be expressed in the Gaussian form: U ρ; ω U 0 exp−aωρ2 ;
639
(19)
where U 0 is a constant and a is a real quantity that depends on the refractive-index profile of the fiber. Comparing Eq. (19) with a Gaussian beam propagating in free space, we may express the constant a in the form a 2czR −1 , where c is the speed of light, and zR is the Rayleigh range of an isodiffracting beam. In the model to be developed, we consider a spectral correlation function of the form: W ω1 ; ω2 Sω1 Sω2 1∕2 μω1 ; ω2 ;
(20)
2 Sω S 0 exp − 2 ω − ω0 2 Ω
(21)
ω − ω 2 μω1 ; ω2 exp − 1 2 2 : 2Σ
(22)
with
and
T2 1 1 1 2 2 2 4; 4 Ω Σ Ω μ¯
(26)
where r 2 −1∕4 2 Ω μ¯ 1 ΩT Σ
(27)
is the integrated degree of spectral coherence given by Eq. (24) with infinite integration limits. The physical meaning of the constant T will become clear in Section 4. In the fully coherent limit Σ → ∞, we have μ¯ → 1, and in the quasip stationary case Σ ≪ Ω, we may approximate μ¯ ≈ Σ∕Ω. It should be stressed that Eq. (27) holds only if the integration limits in Eq. (24) are extended to infinity, and the validity of this approximation is discussed in the Appendix A. When we express the product U ρ1 ; ω1 Uρ2 ; ω2 in Eq. (5) using the average and difference frequency coordinates, the total CSD takes the form of ¯ ρ21 ρ22 ω ¯ Δω W 0 exp − W ρ1 ; ρ2 ; ω; ω0 w2 2 2 T ¯ − ω0 2 exp − Δω2 × exp − 2 ω 8 Ω 2 2 1 Δω ρ2 − ρ1 × exp − ; (28) 2 ω0 w2
Thus Ω and Σ represent the spectral width and the spectral coherence width of the field, respectively, and the ratio Σ∕Ω can be understood as a measure of the degree of spectral coherence. It will prove convenient to write the transverse scale constant a in Eq. (19) in the form a 1∕ω0 w2 , where w represents the 1∕e modal half-width at the peak frequency ω0 of the Gaussian spectrum. We therefore rewrite Eq. (19) as
where W 0 S 0 jU 0 j2 and the frequency-integrated degree of spatial coherence (again with infinite integration limits) is given by
ω ρ2 : U ρ; ω U 0 exp − ω0 w2
The exponential term in this expression arises from the spectral variation of the beam size. In the coherent limit μ¯ 1, we obviously obtain μ ¯ ρ1 ; ρ2 μ. ¯ The same is always true if the points ρ1 and ρ2 are symmetrically located with respect to the optical axis. Comparing an off-axis and an axial point, we find a nontrivial result:
(23)
Furthermore, it will be advantageous to employ the average and difference frequency coordinates defined as
1 Ω2 1 − μ¯ 4 ρ21 − ρ22 2 μ¯ ρ1 ; ρ2 μ¯ exp − : 8 w4 ω20
(29)
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1.0
Thus, at every transverse position, the pulse has a Gaussian temporal shape with characteristic length T that depends on the spectral coherence. In the coherent limit T 2∕Ω and in the quasi-stationary case T ≈ 2∕Σ. The spatial profile diverges for large values of ρ in a nonphysical manner due to the use of infinite integration limits, but we demonstrate in the Appendix A that Eq. (33) is an excellent approximation of the exact result in the region where Sρ; ω has significant values. Next we evaluate the complex degree of temporal coherence, which turns out to be
0.8 0.6 0.4 0.2 0.0 0.0
0.5
1.0
1.5
2.0
Fig. 1. Spatial variation of the normalized integrated degree of coherence μ¯ ρ; 0∕μ¯ given by Eq. (30) with Ω∕ω0 0.5 and μ¯ 4 0 (red), 0.25 (green), 0.5 (blue), and 0.75 (black). The dashed lines illustrate the normalized Gaussian spectral densities Sρ∕S 0 at ω ω0 (red), ω ω0 − Ω (green), and ω ω0 Ω (blue).
1 Ω2 1 − μ¯ 4 ρ 4 : μ¯ ρ; 0∕μ¯ exp − 8 w ω20
(30)
Figure 1 illustrates the spatial deviation of the integrated degree of coherence from its plane-wave value μ¯ . For comparison, the (normalized) spectral densities at ω ω0 and at ω ω0 Ω are also shown. These results, plotted assuming a broad spectrum (Ω∕ω0 0.5), show that the spatial variation becomes significant in the region ρ > w, though it is observable even for somewhat smaller values of ρ. The effect is less significant for smaller values of Ω∕ω0 , and it becomes hardly noticeable in the region ρ < 2w if Ω∕ω0 < 0.1.
4. TIME-DOMAIN FIELD MODEL Introducing average and difference temporal coordinates, ¯t t1 t2 ∕2 and Δt t2 − t1 , the generalized Wiener– Khintchine theorem, Eq. (9), may be rewritten in the form of Γ ρ1 ; ρ2 ; ¯t; Δt
Z
∞ 0
Z
2ω¯
−2ω¯
¯ Δω W ρ1 ; ρ2 ; ω;
¯ Δω¯tdωdΔω: ¯ × exp−iωΔt
(31)
We then obtain, by again extending the limits of Δω integration to −∞; ∞, 2 ρ ρ2 Γ ρ1 ; ρ2 ; ¯t; Δt I 0 exp − 1 2 2 exp−iω0 Δt w 2 2 2 Ω ρ1 ρ22 × exp iΔt 8 ω0 w2 2 2 2 ρ − ρ22 ¯t ; − i × exp 2 1 T 2ω0 w2
where μ¯ 2 Σ Θ T p T 4 Ω 1 − μ¯
(32)
2 −1∕4 T μ¯ 1 ; Θ
(33)
(36)
in addition to Eq. (27). In view of Eq. (34), the complex degree of coherence has a phase that generally depends on all spatiotemporal coordinates. More often, however, one is interested in the absolute value jγρ1 ; ρ2 ; ¯t; Δtj ≡ jγρ1 ; ρ2 ; Δtj 1 Ω2 1 − μ¯ 4 ρ21 − ρ22 2 Δt2 exp − ; exp − 8 w4 ω20 2Θ2 (37) which is independent of average time, equals unity in the coherent case μ¯ 1, and reduces to the plane-wave expression if ρ2 −ρ1 . We also see that the absolute value of the complex degree of temporal coherence, (38)
is space invariant and equal to the plane-wave result. However, the absolute value of the complex degree of spatial coherence between an axial and an off-axis point is of space-variant form, 1 Ω2 1 − μ¯ 4 ρ 4 jγρ; 0; 0j exp − : 8 w ω20
(35)
is a measure of the coherence time in the plane-wave limit w → ∞. The integrated degree of spectral coherence can therefore also be expressed in the form of
Δt2 jγρ; ρ; Δtj exp − 2 ; 2Θ
where I 0 πS 0 jU 0 j2 Ω2 μ¯ 2 . Hence the instantaneous spacetime intensity distribution defined by Eq. (10) is given by 2 ρ I ρ; t I 0 exp −2 w 2 4 1 Ω ρ 2t2 × exp exp − 2 : 2 ω0 w T
1 Ω2 1 − μ¯ 4 ρ21 − ρ22 2 γ ρ1 ; ρ2 ; ¯t; Δt exp − 8 w4 ω20 i Ω¯μ2 ρ21 − ρ22 ¯t i Ω ρ21 ρ22 Δt × exp − exp 2 ω0 w2 T 2 ω0 μ¯ 2 w2 T 2 Δt (34) × exp − 2 exp−iω0 Δt; 2Θ
(39)
Comparing this expression with Eq. (30), we see that the righthand sides are precisely the same. Hence
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jγρ; 0; 0j μρ; ¯ 0∕μ; ¯
q wz w A2 B2 ∕z2R
(40)
and the conclusions drawn from Fig. 1 apply also to the spatial variation of the absolute value of the complex degree of spatial coherence. In general, the spatial coherence profile jγρ; 0; 0j depends on the degree of spectral coherence of the field through the quantity μ, ¯ but in the quasi-stationary regime μ¯ ≪ 1, it depends essentially only on the fractional spectral width Ω∕ω0 .
Rz
where Uρ; z; ω
ω iωD 2 expiωL∕c exp ρ i2πcB 2cB Z∞ iωA 02 ρ Uρ0 ; ω exp × 2cB −∞ iω × exp − ρ · ρ0 d2 ρ0 cB
iω 2 ρ ; 2cq0
S 0 jU 0 j2 z2R expiΔωL A2 z2R B2 2 i ρ22 ρ2 ¯ − ω0 2 ¯ × exp − 2 ω − 1 ω 2c qz q z Ω 2 2 ρ22 T i ρ1 × exp − Δω2 Δω : 4c q z qz 8 (49)
The transformation into the space-time domain is then accomplished by using the Wiener–Khintchine theorem given in Eq. (31). The MCF takes the form Ω exp−iω0 Δt T iω0 ρ22 ρ2 − 1 × exp 2c qz q z 2 Ω2 1 ρ22 ρ2 × exp − − 1 − Δt 8 2c qz q z 2 2 ρ22 2 1 ρ1 ¯tr × exp − 2 ; − T 4c q z qz (50)
(42)
where
U 0 q0 iωL iω exp ρ2 ; exp c 2cqz Aq0 B
(44)
Aq0 B Cq0 D
(45)
where qz
2πS 0 jU 0 j2 z2R ; A2 z2R B2
(51)
and ¯tr ¯t − L∕c;
(52)
is the retarded mean time. From the explicit CSD and MCF expressions, as given in Eqs. (49) and (50), all the various normalized measures of coherence discussed in previous sections can readily be evaluated in an arbitrary paraxial optical system.
6. CONCLUSIONS
is the q parameter at the output plane z of the system. If we introduce the beam width wz at the center frequency ω0 and the radius of curvature Rz by writing iω ω iω − ; 2cqz ω0 w2 z 2cRz
I0
(43)
where q0 −izR is the well-known q parameter at the plane z 0, and zR ω0 w2 ∕2c is the Rayleigh range. It then follows from Eq. (42) that Uρ; z; ω
(48)
Γρ1 ; ρ2 ; z; ¯t; Δt I 0
is the usual paraxial lens-system diffraction formula, and L is the axial optical path through the system. It is convenient to express the isodiffracting field defined in Eqs. (19) or (23) in the form of Uρ; ω U 0 exp
A2 z2R B2 : ACz2R BD
Then the CSD in Eq. (41) can be expressed as
We next consider the propagation of the fields discussed above in optical systems that can be described by the usual 2 × 2 ABCD matrix. If we assume that the CSD at the plane z 0 is of the form of Eq. (5), the CSD at any plane z > 0 can be written in the form of W ρ1 ; ρ2 ; z; ω1 ; ω2 W ω1 ; ω2 × U ρ1 ; z; ω1 Uρ2 ; z; ω2 ; (41)
(47)
and
¯ Δω W ρ1 ; ρ2 ; z; ω;
5. PROPAGATION IN ABCD SYSTEMS
641
(46)
and compare this with Eq. (45), we get explicit expressions for the propagation parameters wz and Rz at the output plane of the system:
Inspired by the coherence properties of ultra-broadband supercontinuum pulses generated by nonlinear interactions in optical fibers, we have developed a simplified analytical model that accounts for the spectral correlations of the field in the space-frequency representation. A gradient-index fiber supporting a single-frequency-dependent isodiffracting transverse mode is considered. The optical field is spatially completely coherent in the frequency domain, but owing to the spatial variation of the fiber mode as a function of frequency and the partial correlations among different frequency
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components, contrary to conventional thinking, the field is not generally spatially fully coherent in the time domain. Making use of Gaussian spectra and Gaussian spectral correlations in the pulse-train model, we have evaluated analytically the frequency-averaged spatial coherence and the space-time degree of spatial coherence. We showed that with ultrawide spectra (as appearing in supercontinuuum), the field differs from full spatial coherence, but the effect is less noticeable for narrower spectral profiles. We have also extended the Gaussian field model for pulse propagation in arbitrary optical systems represented by 2 × 2 ABCD matrices, thereby allowing easy assessment of the spatial coherence properties in a variety of paraxial photonic devices. Whereas in our treatment, we have explicitly considered pulse trains of the Gaussian Schell-model type, other profiles for the fiber mode, pulse shape, and field correlations could be employed as well. Examples are effective step-index fibers and more accurate SC spectra and spectral correlations as obtained from detailed numerical simulations. The field model that we have proposed, which can be regarded as a generalization of the Gaussian Schell-model plane-wave pulse, may thus be expected to be useful in paving the way for the analysis of spatial coherence properties in realistic broadband optical systems.
In the main text, we replaced the finite integration limits in all frequency integrals with infinite limits to obtain analytical results. In this appendix, we investigate the domain of validity of these replacements. Considering first the overall degree of spectral coherence defined in Eq. (24), we obtained the result given in Eq. (27) by assuming infinite integration limits. Evaluating analytically the Δω integral with finite limits in the numerator of Eq. (24), defining a quantity
0.04 0.02 0.00 0.5
1.0
1.5
2.0
Fig. 2. Overall degree of spectral coherence as a function of the fractional spectral width of the pulse for different states of spectral coherence. Blue solid line: β 0.99. Orange long-dashed line: β 0.75. Green short-dashed line: β 0.5. Blue-dotted line: β 0.1.
1.0 0.8 0.6
0.2 0.0
0
(A1)
(A2)
Figure 2 illustrates the difference between this formula and the result μ¯ 2 β obtained with infinite integration limits for several values of β as a function of the effective spectral width Ω∕ω0 of the field. Clearly, the difference in the results is marginal when Ω∕ω0 ∼ 0.5 or smaller. Hence, no significant error is made by using infinite limits in Fig. 1. The expression in Eq. (33) for the spatial intensity profile obtained with infinite integration limits is in principle unsatisfactory since it diverges when ρ → ∞. Keeping the finite limits while performing the Δω integral analytically, we get the result
1
2
3
4
Fig. 3. Normalized transverse intensity profile at t 0 for the fully coherent case β 1 if Ω∕ω0 0.5, calculated with finite integration limits (solid blue line) and infinite limits (dashed red line).
Iρ; t
¯ and denoting x ω∕Ω, we arrive at the result 8β 1 p μ¯ 2 p π 1 erf 2ω0 ∕Ω2 Z∞ ω 2 2x erf dx: × exp −4 x − 0 β Ω 0
0.06
0.4
APPENDIX A: FINITE INTEGRATION LIMITS
2 −1∕2 Ω 2 β 1 ; Σ ΩT
0.08
p Ω 2t2 2π jU 0 j2 S 0 exp − 2 T T Z∞ Ω ρ 2 ω 2 exp −2 x exp −2 x − 0 × ω0 w Ω 0 p p 2t 2x 2t 2x × erf i − erf i dx; (A3) − β β T T
where T, Ω, and β are related by Eq. (A1). As illustrated in Fig. 3, the transverse intensity profiles with finite and infinite limits are virtually indistinguishable for Ω∕ω0 0.5, although the latter diverges in the region where the true intensity is vanishingly small. The same general conclusion holds for all values of β and for smaller values of the ratio Ω∕ω0 , but the analytical results given in the main text are not accurate if Ω∕ω0 > 0.5.
ACKNOWLEDGMENTS This research was funded by the Academy of Finland (grants 135027, 272692, and 134998).
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8. A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space– frequency domains,” Opt. Lett. 20, 623–625 (1995). 9. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980). 10. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982). 11. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–927 (1982). 12. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988). 13. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
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Broadband spatiotemporal Gaussian Schell-model pulse trains Rahul Dutta,1,* Minna Korhonen,1 Ari T. Friberg,1 Göery Genty,2 and Jari Turunen1 1
Institute of Photonics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland 2 Optics Laboratory, Tampere University of Technology, FI-33101 Tampere, Finland *Corresponding author: [email protected] Received October 28, 2013; revised January 2, 2014; accepted January 17, 2014; posted January 22, 2014 (Doc. ID 200078); published February 24, 2014
A new class of partially coherent model sources is introduced on the basis of the second-order coherence theory of nonstationary optical fields. These model sources are spatially fully coherent at each frequency but can have broadband spectra and variable spectral coherence properties, which lead to reduced spatiotemporal coherence in the time domain. The source model is motivated by the spectral coherence properties of supercontinuum pulse trains generated in single-spatial-mode optical fibers. We demonstrate that such broadband light is highly (but not completely) spatially coherent, even though the spectral and temporal coherence properties may vary over a wide range. The model sources introduced here are convenient in assessing the spatiotemporal coherence of broadband pulses in optical systems. © 2014 Optical Society of America OCIS codes: (030.1640) Coherence; (050.1940) Diffraction; (060.2310) Fiber optics; (320.6629) Supercontinuum generation. http://dx.doi.org/10.1364/JOSAA.31.000637
1. INTRODUCTION Supercontinuum (SC) pulse trains generated in nonlinear optical fibers [1–4] are generally thought to be spatially coherent. This is indeed true at each individual frequency, i.e., in the space-frequency domain, provided that the SC light is produced in a fiber with a single spatial mode. Full spatial coherence in the space-time domain is also guaranteed if the SC pulses are fully spectrally (and therefore also temporally) coherent. However, recent research [5–7] clearly shows that SC pulse trains with widely variable spectral coherence properties can be created in nonlinear fibers, depending on initial conditions such as excitation pulse power and duration, and propagation distance in the fiber. In view of these studies, SC light can be described as a superposition of quasi-coherent and quasi-stationary contributions, of which the latter is dominant in a broad range of excitation conditions. Then, from a fundamental point of view [8], full spatial coherence in the space-time domain is no longer guaranteed even if the SC light is spatially fully coherent at each individual frequency. Since the spectral and temporal coherence properties of SC light generated in nonlinear fibers can be determined from simulated ensembles of individual realizations [3,5,6], and the (frequency-dependent) transverse mode distribution of the fiber can be deduced from its refractive-index profile, it is possible to develop a fully realistic spatiotemporal model for the coherence of SC light. However, past experience has shown that simplified source models are useful in optical coherence theory. Models of this type allow analytical solutions in terms of continuously variable parameters that enable the description of fields with different states of coherence. They have been developed for spatially, spectrally, and temporally partially coherent fields, and, in particular, Gaussian distributions of various field properties have been found 1084-7529/14/030637-07$15.00/0
convenient in establishing quantitative and qualitative results (see, e.g., [9–12]). The above considerations motivate us to introduce a simplified analytical model for statistical description of broadband pulse trains generated in single-mode fibers, taking into account the frequency dependence of the transverse scale of the fiber mode and the spectral coherence properties of the pulse trains. We assume that the spatial mode of the fiber is Gaussian with a spectral scale that ensures isodiffracting (the Rayleigh range is independent of frequency) freespace propagation of the output field. The spectrum of the field as well as the spectral coherence function are also taken as Gaussian, the latter depending on the frequency difference only; hence the model may be viewed as an extension of the Gaussian Schell-model plane-wave pulse [13].
2. SPECTRAL AND SPATIAL COHERENCE OF NONSTATIONARY FIELDS Let us denote the time-domain representation of a single realization of the light field arriving at the exit plane, z constant, of a nonlinear optical fiber by V ρ; t, where ρ x; y, the z coordinate is omitted for brevity, and t denotes time. We assume that the temporal center of the initial pump pulse determines the origin of time for each realization. The temporal realizations V ρ; t have Fourier representations of Z V ρ; t
∞ 0
V ρ; ω exp−iωtdω;
(1)
where V ρ; ω depends on the transverse position ρ, and angular frequency ω in a way dictated by the refractive-index profile of the fiber. The cross-spectral density function © 2014 Optical Society of America
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ZZ
(CSD) of the pulse train is defined as the ensemble average over all individual space-frequency-domain realizations, i.e., W ρ1 ; ρ2 ; ω1 ; ω2 hV ρ1 ; ω1 V ρ2 ; ω2 i:
Γ ρ1 ; ρ2 ; t1 ; t2
∞
0
W ρ1 ; ρ2 ; ω1 ; ω2
× expiω1 t1 − ω2 t2 dω1 dω2 :
(2)
(9)
In analogy with Eqs. (3) and (4), we may then introduce the temporal intensity,
The spectral density S ρ; ω W ρ; ρ; ω; ω
(3)
I ρ; t Γ ρ; ρ; t; t;
(10)
and the complex degree of spectral coherence and the complex degree of coherence, W ρ1 ; ρ2 ; ω1 ; ω2 μ ρ1 ; ρ2 ; ω1 ; ω2 p S ρ1 ; ω1 S ρ2 ; ω2
(4)
of the field can then be evaluated. If the pulses are generated in a single-mode fiber, the spectral degree of spatial coherence at any single frequency ω is complete, i.e., jμ ρ1 ; ρ2 ; ω; ωj 1 for any pair of points ρ1 ; ρ2 , but its two-frequency dependence may be quite arbitrary. More specifically, if the (deterministic) spatial profile of the single eigenmode propagating in the fiber is U ρ; ω, the space-frequency field may be written as V ρ; ω aωU ρ; ω, where aω is a random function representing the complex amplitude of the modal field. Since we are dealing with a single-mode field, it is natural that aω is independent of the lateral position. On substituting the mode field into Eq. (2), the CSD takes on the form, W ρ1 ; ρ2 ; ω1 ; ω2 W ω1 ; ω2 U ρ1 ; ω1 U ρ2 ; ω2 ;
(5)
where W ω1 ; ω2 ha ω1 aω2 i is the spectral correlation function associated with the field. This quantity is analogous to the CSD used to analyze the spectral coherence of planewave fields, e.g., in [13]. In SC modeling, the CSD function W ρ1 ; ρ2 ; ω1 ; ω2 can be determined by numerical simulations as described in [5,6]. From the knowledge of this function, one may straightforwardly evaluate, e.g., the overall degree of spectral coherence μ¯ defined according to μ¯ 2
RR ∞ 0
jW ω1 ; ω2 j2 dω1 dω2 R ; 0∞ Sωdω2
(6)
where Sω Wω; ω. Generalizing this definition, we introduce a quantity, RR ∞ jW ρ1 ; ρ2 ; ω1 ; ω2 j2 dω1 dω2 R∞ μ¯ 2 ρ1 ; ρ2 R ∞ 0 ; 0 S ρ1 ; ω1 dω1 0 S ρ2 ; ω2 dω2
(11)
in the space-time domain. The instantaneous (complex) degree of spatial coherence is defined as the equal-time correlation function, γ ρ1 ; ρ2 ; t; t. Combining Eqs. (5) and (9), we find ZZ Γ ρ1 ; ρ2 ; t; t
0
∞
W ω1 ; ω2
× U ρ1 ; ω1 U ρ2 ; ω2 expiω1 − ω2 tdω1 dω2 : (12) If W ω1 ; ω2 is separable, i.e., the field is spectrally fully coherent, then Γρ1 ; ρ2 ; t; t is spatially separable, and the field therefore is spatially fully coherent in the time domain. In general, partial spectral coherence implies partial spatiotemporal coherence, even if the field is spatially fully coherent at each individual frequency. However, in the special case of a scale-invariant modal profile of the form U ρ; ω U ρ, we obtain from Eq. (12): Γ ρ1 ; ρ2 ; t; t U ρ1 U ρ2 ZZ ∞ × W ω1 ; ω2 expiω1 − ω2 tdω1 dω2 ; (13) 0
which immediately implies that jγ ρ1 ; ρ2 ; t; tj 1 for all ρ1 and ρ2 , i.e., full spatial coherence in the space-time domain. This would, however, require an achromatic fiber in the sense that the modal profile is the same at all frequencies within the wide spectral band considered in this work. Instantaneous field quantities are difficult to measure at optical frequencies; hence time-integrated quantities such as
(7)
which represents the frequency-integrated degree of spatial coherence between the wave fields at any two transverse points, ρ1 and ρ2 . The space-time field realizations are connected to the frequency-domain realizations by the Fourier-transform relationship of Eq. (1), and the space-time correlation properties of the field are characterized by the mutual coherence function (MCF): Γ ρ1 ; ρ2 ; t1 ; t2 hV ρ1 ; t1 V ρ2 ; t2 i:
Γ ρ1 ; ρ2 ; t1 ; t2 ; γ ρ1 ; ρ2 ; t1 ; t2 p I ρ1 ; t1 I ρ2 ; t2
(8)
By inserting from Eq. (1) into Eq. (8) and using Eq. (2), we obtain a relationship between CSD and MCF in the form of the generalized Wiener–Khintchine theorem:
¯ ρ1 ; ρ2 Γ γ¯ ρ1 ; ρ2 p ; ¯ ρ1 I ¯ ρ2 I
(14)
with ¯ ρ1 ; ρ2 Γ
Z
∞
−∞
Γ ρ1 ; ρ2 ; t; tdt
(15)
and ¯ ρ Γ ¯ ρ; ρ I
Z
∞
−∞
I ρ; tdt
(16)
are of interest. In view of Eq. (9) and using the integral representation of the Dirac delta function, we have in general:
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¯ ρ1 ; ρ2 2π Γ
Z
∞
0
W ρ1 ; ρ2 ; ω; ωdω
(17)
¯ ω1 ω2 ∕2 and Δω ω2 − ω1 , respectively. Then ω Eq. (6) takes the form of
and ¯ ρ 2π I
Z
∞ 0
μ¯ 2 S ρ; ωdω:
(18)
This implies that all information on the spectral coherence properties of the field is lost in a time-averaged measurement and, consequently, γ¯ ρ1 ; ρ2 should not in general be considered as a true measure of the degree of spatial coherence: it depends only on the field’s spectral properties and one obtains values of j¯γ ρ1 ; ρ2 j smaller than unity even if the field is spectrally fully coherent and thereby also spatially fully coherent.
R ∞ R 2ω¯ 0
¯ Δωj2 dωdΔω ¯ jW ω; −2ω¯ R ; ∞ 2 0 Sωdω
(24)
and Eq. (7) is modified analogously. In order to obtain analytical results, we will replace the finite integration limits by −∞; ∞, which can be justified if the ratio Ω∕ω0 is sufficiently ¯ and Δω, we small (see Appendix A for further details). Using ω may collect Eqs. (20)–(22) in the form 2 2 T ¯ − ω0 2 exp − Δω2 ; (25) ¯ Δω S 0 exp − 2 ω W ω; 8 Ω with a new constant T defined by writing
3. SPECTRAL-DOMAIN FIELD MODEL Let us proceed by introducing a model source that mimics the properties of real SC sources at least in a qualitative sense. To this end, we assume that the SC is generated in a graded-index fiber with a parabolic refractive-index distribution, so that the modal profile in the frequency domain can be expressed in the Gaussian form: U ρ; ω U 0 exp−aωρ2 ;
639
(19)
where U 0 is a constant and a is a real quantity that depends on the refractive-index profile of the fiber. Comparing Eq. (19) with a Gaussian beam propagating in free space, we may express the constant a in the form a 2czR −1 , where c is the speed of light, and zR is the Rayleigh range of an isodiffracting beam. In the model to be developed, we consider a spectral correlation function of the form: W ω1 ; ω2 Sω1 Sω2 1∕2 μω1 ; ω2 ;
(20)
2 Sω S 0 exp − 2 ω − ω0 2 Ω
(21)
ω − ω 2 μω1 ; ω2 exp − 1 2 2 : 2Σ
(22)
with
and
T2 1 1 1 2 2 2 4; 4 Ω Σ Ω μ¯
(26)
where r 2 −1∕4 2 Ω μ¯ 1 ΩT Σ
(27)
is the integrated degree of spectral coherence given by Eq. (24) with infinite integration limits. The physical meaning of the constant T will become clear in Section 4. In the fully coherent limit Σ → ∞, we have μ¯ → 1, and in the quasip stationary case Σ ≪ Ω, we may approximate μ¯ ≈ Σ∕Ω. It should be stressed that Eq. (27) holds only if the integration limits in Eq. (24) are extended to infinity, and the validity of this approximation is discussed in the Appendix A. When we express the product U ρ1 ; ω1 Uρ2 ; ω2 in Eq. (5) using the average and difference frequency coordinates, the total CSD takes the form of ¯ ρ21 ρ22 ω ¯ Δω W 0 exp − W ρ1 ; ρ2 ; ω; ω0 w2 2 2 T ¯ − ω0 2 exp − Δω2 × exp − 2 ω 8 Ω 2 2 1 Δω ρ2 − ρ1 × exp − ; (28) 2 ω0 w2
Thus Ω and Σ represent the spectral width and the spectral coherence width of the field, respectively, and the ratio Σ∕Ω can be understood as a measure of the degree of spectral coherence. It will prove convenient to write the transverse scale constant a in Eq. (19) in the form a 1∕ω0 w2 , where w represents the 1∕e modal half-width at the peak frequency ω0 of the Gaussian spectrum. We therefore rewrite Eq. (19) as
where W 0 S 0 jU 0 j2 and the frequency-integrated degree of spatial coherence (again with infinite integration limits) is given by
ω ρ2 : U ρ; ω U 0 exp − ω0 w2
The exponential term in this expression arises from the spectral variation of the beam size. In the coherent limit μ¯ 1, we obviously obtain μ ¯ ρ1 ; ρ2 μ. ¯ The same is always true if the points ρ1 and ρ2 are symmetrically located with respect to the optical axis. Comparing an off-axis and an axial point, we find a nontrivial result:
(23)
Furthermore, it will be advantageous to employ the average and difference frequency coordinates defined as
1 Ω2 1 − μ¯ 4 ρ21 − ρ22 2 μ¯ ρ1 ; ρ2 μ¯ exp − : 8 w4 ω20
(29)
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1.0
Thus, at every transverse position, the pulse has a Gaussian temporal shape with characteristic length T that depends on the spectral coherence. In the coherent limit T 2∕Ω and in the quasi-stationary case T ≈ 2∕Σ. The spatial profile diverges for large values of ρ in a nonphysical manner due to the use of infinite integration limits, but we demonstrate in the Appendix A that Eq. (33) is an excellent approximation of the exact result in the region where Sρ; ω has significant values. Next we evaluate the complex degree of temporal coherence, which turns out to be
0.8 0.6 0.4 0.2 0.0 0.0
0.5
1.0
1.5
2.0
Fig. 1. Spatial variation of the normalized integrated degree of coherence μ¯ ρ; 0∕μ¯ given by Eq. (30) with Ω∕ω0 0.5 and μ¯ 4 0 (red), 0.25 (green), 0.5 (blue), and 0.75 (black). The dashed lines illustrate the normalized Gaussian spectral densities Sρ∕S 0 at ω ω0 (red), ω ω0 − Ω (green), and ω ω0 Ω (blue).
1 Ω2 1 − μ¯ 4 ρ 4 : μ¯ ρ; 0∕μ¯ exp − 8 w ω20
(30)
Figure 1 illustrates the spatial deviation of the integrated degree of coherence from its plane-wave value μ¯ . For comparison, the (normalized) spectral densities at ω ω0 and at ω ω0 Ω are also shown. These results, plotted assuming a broad spectrum (Ω∕ω0 0.5), show that the spatial variation becomes significant in the region ρ > w, though it is observable even for somewhat smaller values of ρ. The effect is less significant for smaller values of Ω∕ω0 , and it becomes hardly noticeable in the region ρ < 2w if Ω∕ω0 < 0.1.
4. TIME-DOMAIN FIELD MODEL Introducing average and difference temporal coordinates, ¯t t1 t2 ∕2 and Δt t2 − t1 , the generalized Wiener– Khintchine theorem, Eq. (9), may be rewritten in the form of Γ ρ1 ; ρ2 ; ¯t; Δt
Z
∞ 0
Z
2ω¯
−2ω¯
¯ Δω W ρ1 ; ρ2 ; ω;
¯ Δω¯tdωdΔω: ¯ × exp−iωΔt
(31)
We then obtain, by again extending the limits of Δω integration to −∞; ∞, 2 ρ ρ2 Γ ρ1 ; ρ2 ; ¯t; Δt I 0 exp − 1 2 2 exp−iω0 Δt w 2 2 2 Ω ρ1 ρ22 × exp iΔt 8 ω0 w2 2 2 2 ρ − ρ22 ¯t ; − i × exp 2 1 T 2ω0 w2
where μ¯ 2 Σ Θ T p T 4 Ω 1 − μ¯
(32)
2 −1∕4 T μ¯ 1 ; Θ
(33)
(36)
in addition to Eq. (27). In view of Eq. (34), the complex degree of coherence has a phase that generally depends on all spatiotemporal coordinates. More often, however, one is interested in the absolute value jγρ1 ; ρ2 ; ¯t; Δtj ≡ jγρ1 ; ρ2 ; Δtj 1 Ω2 1 − μ¯ 4 ρ21 − ρ22 2 Δt2 exp − ; exp − 8 w4 ω20 2Θ2 (37) which is independent of average time, equals unity in the coherent case μ¯ 1, and reduces to the plane-wave expression if ρ2 −ρ1 . We also see that the absolute value of the complex degree of temporal coherence, (38)
is space invariant and equal to the plane-wave result. However, the absolute value of the complex degree of spatial coherence between an axial and an off-axis point is of space-variant form, 1 Ω2 1 − μ¯ 4 ρ 4 jγρ; 0; 0j exp − : 8 w ω20
(35)
is a measure of the coherence time in the plane-wave limit w → ∞. The integrated degree of spectral coherence can therefore also be expressed in the form of
Δt2 jγρ; ρ; Δtj exp − 2 ; 2Θ
where I 0 πS 0 jU 0 j2 Ω2 μ¯ 2 . Hence the instantaneous spacetime intensity distribution defined by Eq. (10) is given by 2 ρ I ρ; t I 0 exp −2 w 2 4 1 Ω ρ 2t2 × exp exp − 2 : 2 ω0 w T
1 Ω2 1 − μ¯ 4 ρ21 − ρ22 2 γ ρ1 ; ρ2 ; ¯t; Δt exp − 8 w4 ω20 i Ω¯μ2 ρ21 − ρ22 ¯t i Ω ρ21 ρ22 Δt × exp − exp 2 ω0 w2 T 2 ω0 μ¯ 2 w2 T 2 Δt (34) × exp − 2 exp−iω0 Δt; 2Θ
(39)
Comparing this expression with Eq. (30), we see that the righthand sides are precisely the same. Hence
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jγρ; 0; 0j μρ; ¯ 0∕μ; ¯
q wz w A2 B2 ∕z2R
(40)
and the conclusions drawn from Fig. 1 apply also to the spatial variation of the absolute value of the complex degree of spatial coherence. In general, the spatial coherence profile jγρ; 0; 0j depends on the degree of spectral coherence of the field through the quantity μ, ¯ but in the quasi-stationary regime μ¯ ≪ 1, it depends essentially only on the fractional spectral width Ω∕ω0 .
Rz
where Uρ; z; ω
ω iωD 2 expiωL∕c exp ρ i2πcB 2cB Z∞ iωA 02 ρ Uρ0 ; ω exp × 2cB −∞ iω × exp − ρ · ρ0 d2 ρ0 cB
iω 2 ρ ; 2cq0
S 0 jU 0 j2 z2R expiΔωL A2 z2R B2 2 i ρ22 ρ2 ¯ − ω0 2 ¯ × exp − 2 ω − 1 ω 2c qz q z Ω 2 2 ρ22 T i ρ1 × exp − Δω2 Δω : 4c q z qz 8 (49)
The transformation into the space-time domain is then accomplished by using the Wiener–Khintchine theorem given in Eq. (31). The MCF takes the form Ω exp−iω0 Δt T iω0 ρ22 ρ2 − 1 × exp 2c qz q z 2 Ω2 1 ρ22 ρ2 × exp − − 1 − Δt 8 2c qz q z 2 2 ρ22 2 1 ρ1 ¯tr × exp − 2 ; − T 4c q z qz (50)
(42)
where
U 0 q0 iωL iω exp ρ2 ; exp c 2cqz Aq0 B
(44)
Aq0 B Cq0 D
(45)
where qz
2πS 0 jU 0 j2 z2R ; A2 z2R B2
(51)
and ¯tr ¯t − L∕c;
(52)
is the retarded mean time. From the explicit CSD and MCF expressions, as given in Eqs. (49) and (50), all the various normalized measures of coherence discussed in previous sections can readily be evaluated in an arbitrary paraxial optical system.
6. CONCLUSIONS
is the q parameter at the output plane z of the system. If we introduce the beam width wz at the center frequency ω0 and the radius of curvature Rz by writing iω ω iω − ; 2cqz ω0 w2 z 2cRz
I0
(43)
where q0 −izR is the well-known q parameter at the plane z 0, and zR ω0 w2 ∕2c is the Rayleigh range. It then follows from Eq. (42) that Uρ; z; ω
(48)
Γρ1 ; ρ2 ; z; ¯t; Δt I 0
is the usual paraxial lens-system diffraction formula, and L is the axial optical path through the system. It is convenient to express the isodiffracting field defined in Eqs. (19) or (23) in the form of Uρ; ω U 0 exp
A2 z2R B2 : ACz2R BD
Then the CSD in Eq. (41) can be expressed as
We next consider the propagation of the fields discussed above in optical systems that can be described by the usual 2 × 2 ABCD matrix. If we assume that the CSD at the plane z 0 is of the form of Eq. (5), the CSD at any plane z > 0 can be written in the form of W ρ1 ; ρ2 ; z; ω1 ; ω2 W ω1 ; ω2 × U ρ1 ; z; ω1 Uρ2 ; z; ω2 ; (41)
(47)
and
¯ Δω W ρ1 ; ρ2 ; z; ω;
5. PROPAGATION IN ABCD SYSTEMS
641
(46)
and compare this with Eq. (45), we get explicit expressions for the propagation parameters wz and Rz at the output plane of the system:
Inspired by the coherence properties of ultra-broadband supercontinuum pulses generated by nonlinear interactions in optical fibers, we have developed a simplified analytical model that accounts for the spectral correlations of the field in the space-frequency representation. A gradient-index fiber supporting a single-frequency-dependent isodiffracting transverse mode is considered. The optical field is spatially completely coherent in the frequency domain, but owing to the spatial variation of the fiber mode as a function of frequency and the partial correlations among different frequency
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components, contrary to conventional thinking, the field is not generally spatially fully coherent in the time domain. Making use of Gaussian spectra and Gaussian spectral correlations in the pulse-train model, we have evaluated analytically the frequency-averaged spatial coherence and the space-time degree of spatial coherence. We showed that with ultrawide spectra (as appearing in supercontinuuum), the field differs from full spatial coherence, but the effect is less noticeable for narrower spectral profiles. We have also extended the Gaussian field model for pulse propagation in arbitrary optical systems represented by 2 × 2 ABCD matrices, thereby allowing easy assessment of the spatial coherence properties in a variety of paraxial photonic devices. Whereas in our treatment, we have explicitly considered pulse trains of the Gaussian Schell-model type, other profiles for the fiber mode, pulse shape, and field correlations could be employed as well. Examples are effective step-index fibers and more accurate SC spectra and spectral correlations as obtained from detailed numerical simulations. The field model that we have proposed, which can be regarded as a generalization of the Gaussian Schell-model plane-wave pulse, may thus be expected to be useful in paving the way for the analysis of spatial coherence properties in realistic broadband optical systems.
In the main text, we replaced the finite integration limits in all frequency integrals with infinite limits to obtain analytical results. In this appendix, we investigate the domain of validity of these replacements. Considering first the overall degree of spectral coherence defined in Eq. (24), we obtained the result given in Eq. (27) by assuming infinite integration limits. Evaluating analytically the Δω integral with finite limits in the numerator of Eq. (24), defining a quantity
0.04 0.02 0.00 0.5
1.0
1.5
2.0
Fig. 2. Overall degree of spectral coherence as a function of the fractional spectral width of the pulse for different states of spectral coherence. Blue solid line: β 0.99. Orange long-dashed line: β 0.75. Green short-dashed line: β 0.5. Blue-dotted line: β 0.1.
1.0 0.8 0.6
0.2 0.0
0
(A1)
(A2)
Figure 2 illustrates the difference between this formula and the result μ¯ 2 β obtained with infinite integration limits for several values of β as a function of the effective spectral width Ω∕ω0 of the field. Clearly, the difference in the results is marginal when Ω∕ω0 ∼ 0.5 or smaller. Hence, no significant error is made by using infinite limits in Fig. 1. The expression in Eq. (33) for the spatial intensity profile obtained with infinite integration limits is in principle unsatisfactory since it diverges when ρ → ∞. Keeping the finite limits while performing the Δω integral analytically, we get the result
1
2
3
4
Fig. 3. Normalized transverse intensity profile at t 0 for the fully coherent case β 1 if Ω∕ω0 0.5, calculated with finite integration limits (solid blue line) and infinite limits (dashed red line).
Iρ; t
¯ and denoting x ω∕Ω, we arrive at the result 8β 1 p μ¯ 2 p π 1 erf 2ω0 ∕Ω2 Z∞ ω 2 2x erf dx: × exp −4 x − 0 β Ω 0
0.06
0.4
APPENDIX A: FINITE INTEGRATION LIMITS
2 −1∕2 Ω 2 β 1 ; Σ ΩT
0.08
p Ω 2t2 2π jU 0 j2 S 0 exp − 2 T T Z∞ Ω ρ 2 ω 2 exp −2 x exp −2 x − 0 × ω0 w Ω 0 p p 2t 2x 2t 2x × erf i − erf i dx; (A3) − β β T T
where T, Ω, and β are related by Eq. (A1). As illustrated in Fig. 3, the transverse intensity profiles with finite and infinite limits are virtually indistinguishable for Ω∕ω0 0.5, although the latter diverges in the region where the true intensity is vanishingly small. The same general conclusion holds for all values of β and for smaller values of the ratio Ω∕ω0 , but the analytical results given in the main text are not accurate if Ω∕ω0 > 0.5.
ACKNOWLEDGMENTS This research was funded by the Academy of Finland (grants 135027, 272692, and 134998).
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