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Lah-Laguerre Optics

 Chromatic Dispersion Orders In Closed Form

 Published in arXiv:2011, 30 October 2020
Published in Optics Express 30, 22, 20 October 2022

 

Operating with ultrashort, broadband laser pulses presents a challenge to maintain the pulse phase intact so that the pulse duration or shape does not change and the peak power does not decrease substantially.   

Ultrashort pulse stemming from an anti-resonant fiber

The description of uniform systems allows for the exact implementation of the accumulated phase, i.e., pulse propagation in waveguides or optical fibers. However, balancing the pulse phase becomes an optimization problem when several systems are interconnected. It is no longer practical due to computational speed to match phases ‘shapes’ but rather individual perturbative orders of dispersion. The cited papers have lay the foundation of analytical and more robust description for the chromatic dispersion phenomena that will aid the design of novel optical systems and materials.

The perturbative description of the chromatic dispersion involves several hypergeometrical transforms, the most famous amongst, are the Lah, and Laguerre transforms. The Lah, Laguerre, and several other unnamed forward and inverse transforms form the core of the Lah-Laguerre dispersive optics. In Lah-Laguerre optics, such algorithms allow for incredibly faster computations when solving complicated optimization problems involving phase balancing of different optical systems, at the extereme single cycle waform synthesis.

Using the Lah-Laguerre approach gives the mathematical foundation to evaluate, optimize, and design systems, to output a pulse desired pulse duration balanced to an anticipated chromatic dispersion order. However, due to the experimental uncertainty in measurements of the refractive index or due to the simplified level of description of an optical system, these relations can leave some ambiguity in the estimated chromatic dispersion orders. Fortunately, the inverse transforms relate the Taylor coefficients of the refractive index or optical path to the phase or wavevector. Thus, to put in perspective, a single point phase measurement, provides information for the refractive index or optical path in an extended vicinity of measured frequency. Consequently, this formalism can also facilitate more precise interferometric measurements of the refractive index and aid the design of novel optical materials, nanostructures, and optical systems based on desired dispersion. Furthermore, from a practical point of view, the evaluation speed of the simple hypergeometric series can be competitive even against algorithms such as the fast Fourier transform (FFT). Numerically, the highest order that can be evaluated is solely limited by the computer architecture’s ability to allocate the smallest /largest/ floating-point number.

When pulses have substantial bandwidth it is essential to consider also the higher orders of dispersion. A few example of evaluated chromatic dispersion orders follow below.

The first 10 material dispersion orders of CaF2CaF2 material are shown below. More data plots for the material dispersion orders of various materials can be found in Ref.[2].

Material dispersion orders for CaF2
 

The first 10 chromatic dispersion orders of conventional genuine laser pulse compressors are illustrated below.

 

Chromatic dispersion orders in A) grating compressor B) Prism-pair compressor  

For more information, please visit the original papers Ref [1], Ref [2], Ref [3].

Mathematical description. 

In dispersive systems, the phase velocity of a wave depends on its frequency. Effectively a single wavelength monochromatic light will accumulate phase and will experience a time delay compared to the same wave traveling in a vacuum. Such idealized waves have infinite pulse duration – continuous waves. In contrast, when considering many monochromatic waves that interfere constructively over a small spatial extent, create a wavepacket that will experience more dramatic changes. In such cases, we can define a group delay and group velocity dispersion that is a result of collective behavior of the wavelets. The propagation of pulses in matter leads to changes in the phase and pulse duration. In linear light-matter interactions, the spectrum of the wavepacket does not change. The phenomenon is known as chromatic dispersion. The chromatic dispersion has worked both in favor and against some of the most significant innovations of the past decades.

The dispersion relation for the phase is:

φ(ω|λ)=k(ω)z=ωcn(ω)z=2πλn(λ)z=ωcOP(ω)=2πλOP(λ)=ωτ(ω)=2πτ0τ(ω)
where OP(ω|λ) is the optical path, n(ω|λ) is the refractive index of the medium, τ(ω|λ) is the corresponding temporal interval, and τ0  is the single-cycle pulse duration for a wavelet with a wavelength λ. In such a picture, the spectral content of the signal does not change but rather rearranges temporally over a certain band around an average frequency ω0. Historically, the dispersion orders have been defined in the frequency space through the Taylor expansion of the phase φ(ω|λ), or the wavevector k(ω|λ) around the pulse intensity-averaged frequency or the carrier frequency at the pulse peak

The dispersion orders are defined by the Taylor expansion of the phase or the wavevector.

φ(ω)=φ |ω0+ φω|ω0(ωω0)+12 2φω2|ω0(ωω0)2 ++1p! pφωp|ω0(ωω0)p+=
=φ |ω0+τg |ω0(ωω0)+12GDD(ωω0)2++1p!POD(ωω0)p+Rp
Such a representation is convenient as it requires knowledge only of a small number of spectral derivatives at a single point. The first derivative φω|ω0=τg corresponds to the group delay (GD), which results in a temporal lag of the pulse envelope. The presence of higher orders causes distortions in the temporal pulse shape. The second term 2φω2|ω0=ωτg(ω)|ω0corresponds to the group delay dispersion (GDD), and in general,  pφωp|ω0=p1ωp1τg(ω)|ω0 is the pth order dispersion (POD). Finally, Rp=maxωξω0p+1φ(ξ)ξp+1 (ωω0)p+1(p+1)!is the Lagrange error after the first p terms.

 In a similar fashion, the wavevector k(ω|λ) can be expanded in a Taylor series :

k(ω)=k|ω0+kω|ω0(ωω0)+122kω2|ω0(ωω0)2 ++1p!pkωp|ω0(ωω0)p+=
=k0+v1gr(ωω0)+12GDD(ωω0)2++1p!POD(ωω0)p+Rp
Again, the lowest term kω|ω0=1vg=τgz represents the inverse group velocity, whereas the second term 2kω2|ω0=ωv1gr|ω0 represents the GDD. In general, pkωp|ω0=p1ωp1v1gr|ω0 is the pth order chromatic dispersion (POD).

The chromatic dispersion orders can be easily evaluated in the frequency domain by obtaining the successive derivatives of the wavevector k(ω) or the phase φ(ω). In general: 

pωpk(ω)=1c(pp1ωp1n(ω)+ωpωpn(ω))   
pωpφ(ω)=1c(pp1ωp1OP(ω)+ωpωpOP(ω))
The derivatives of any differentiable function f(ω|λ) in the wavelength or the frequency space is specified through a Lah transform as:

pωpf(ω)=(1)p(λ2πc)ppm=0A(p,m)λmmλmf(λ)
pλpf(λ)=(1)p(ω2πc)ppm=0A(p,m)ωmmωmf(ω)  
The matrix elements of the transformation are the Lah coefficients: A(p,m)=p!(pm)!m!(p1)!(m1)!

Written for the GDD the above expression states that a constant with wavelength GDD, will have zero higher orders. From a practical point of view, when the GDD data is experimentally or numerically accessible in the wavelength space, the dispersion orders can be expressed as:

p+2ωp+2φ(ω)=pωpGDD=(1)p(λ2πc)ppm=0A(p,m) λmmλmGDD(λ)
Substituting equation (6) and (7) expressed for the refractive index n or optical path OP into equation (4) and (5) results in closed-form expressions for the dispersion orders. In general the pth  order dispersion (POD) is a Laguerre type transform of negative order two:

POD(n)=pωpk(ω)=(1)p1c(λ2πc)p1pm=0B(p,m) λmmλmn(λ)
POD(OP)=pωpφ(ω)=(1)p1c(λ2πc)p1pm=0B(p,m) λmmλmOP(λ)
 The inverse transforms relate the Taylor coefficients of the refractive index or the optical path to the wavevector or the phase.

λppλpn(λ)=(1)pcωpm=0B(p,m)ωmmωmk(ω)
λppλpOP(λ)=(1)pcωpm=0B(p,m)ωmmωmφ(ω) 
The matrix elements of the transforms are the unsigned Laguerre coefficients of order negative two:  B(p,m)=p!(pm)!m!(p2)!(m2)!
The polynomial sums form sequential polynomials G(α)p(x) . The corresponding generating function can be expressed as:

G(α)p(x)=xαdpdxp(xp+αf(x))=pm=0C(p+α,pm)p!m!xmf(m)(x)
The first four chromatic dispersion orders are well-known in the literature. Using the above-mentioned Lah-Laguerre optical formalism, the first ten dispersion orders, written for the wavevector, can be explicitly written in closed-form expressions as:

C I.GD=ωk(ω)=1c(n(ω)+ωn(ω)ω)=1cG(2)1(λ)=1c(n(λ)λn(λ)λ)=v1gr
The group refractive index ngis defined in terms of the group velocity vgr: ng=cv1gr.

C II.GDD=2ω2k(ω)=1c(2n(ω)ω+ω2n(ω)ω2)=1c(λ2πc)G(2)2(λ)==1c(λ2πc)(λ22n(λ)λ2)
C III.TOD=3ω3k(ω)=1c(32n(ω)ω2+ω3n(ω)ω3)=1c(λ2πc)2G(2)3(λ)==1c(λ2πc)2(3λ22n(λ)λ2+λ33n(λ)λ3)
C IV.FOD=4ω4k(ω)=1c(43n(ω)ω3+ω4n(ω)ω4)=1c(λ2πc)3G(2)4(λ)==1c(λ2πc)3(12λ22n(λ)λ2+8λ33n(λ)λ3+λ44n(λ)λ4)
C V.FiOD=5ω5k(ω)=1c(54n(ω)ω4+ω5n(ω)ω5)=1c(λ2πc)4G(2)5(λ)==1c(λ2πc)4(60λ22n(λ)λ2+60λ33n(λ)λ3+15λ44n(λ)λ4+λ55n(λ)λ5)
C VI.SiOD=6ω6k(ω)=1c(65n(ω)ω5+ω6n(ω)ω6)=1c(λ2πc)5G(2)6(λ)==1c(λ2πc)5(360λ22n(λ)λ2+480λ33n(λ)λ3+180λ44n(λ)λ4+24λ55n(λ)λ5+λ66n(λ)λ6)
C VII.SeOD=7ω7k(ω)=1c(76n(ω)ω6+ω7n(ω)ω7)=1c(λ2πc)6G(2)7(λ)==1c(λ2πc)6(2520λ22n(λ)λ2+4200λ33n(λ)λ3+2100λ44n(λ)λ4+420λ55n(λ)λ5++35λ66n(λ)λ6+λ77n(λ)λ7)
C VIII.EOD=8ω8k(ω)=1c(87n(ω)ω7+ω8n(ω)ω8)=1c(λ2πc)7G(2)8(λ)==1c(λ2πc)7(20160λ22n(λ)λ2+40320λ33n(λ)λ3+25200λ44n(λ)λ4+6720λ55n(λ)λ5++840λ66n(λ)λ6+48λ77n(λ)λ7+λ88n(λ)λ8)
C IX.NOD=9ω9k(ω)=1c(98n(ω)ω8+ω9n(ω)ω9)=1c(λ2πc)8G(2)9(λ)==1c(λ2πc)8(181440λ22n(λ)λ2+423360λ33n(λ)λ3+317520λ44n(λ)λ4+105840λ55n(λ)λ5++17640λ66n(λ)λ6+1512λ77n(λ)λ7+63λ88n(λ)λ8+λ99n(λ)λ9)
 
C X.TeOD=10ω10k(ω)=1c(109n(ω)ω9+ω10n(ω)ω10)=1c(λ2πc)9G(2)10(λ)==1c(λ2πc)9(1814400λ22n(λ)λ2+4838400λ33n(λ)λ3+4233600λ44n(λ)λ4++1693440λ55n(λ)λ5+352800λ66n(λ)λ6+40320λ77n(λ)λ7+2520λ88n(λ)λ8+80λ99n(λ)λ9++λ1010n(λ)λ10)  
Written for the phase φ, the first ten dispersion orders can be expressed as a function of wavelength using the Lah transforms as:

B I.φ(ω)ω=(λ2πc)G(1)1(λ)=(2πcω2)φ(ω)λ=(λ22πc)φ(λ)λ
B II.2φ(ω)ω2=ω(φ(ω)ω)=(λ2πc)2G(1)2(λ)=(λ2πc)2(2λφ(λ)λ+λ22φ(λ)λ2)
B III.3φ(ω)ω3=(λ2πc)3G(1)3(λ)=(λ2πc)3(6λφ(λ)λ+6λ22φ(λ)λ2+λ33φ(λ)λ3)
B IV.4φ(ω)ω4=(λ2πc)4G(1)4(λ)=(λ2πc)4(24λφ(λ)λ+36λ22φ(λ)λ2+12λ33φ(λ)λ3++λ44φ(λ)λ4)
B V.5φ(ω)ω5=(λ2πc)5G(1)5(λ)=(λ2πc)5(120λφ(λ)λ+240λ22φ(λ)λ2+120λ33φ(λ)λ3++20λ44φ(λ)λ4+λ55φ(λ)λ5)
B VI.6φ(ω)ω6=(λ2πc)6G(1)6(λ)=(λ2πc)6(720λφ(λ)λ+1800λ22φ(λ)λ2+1200λ33φ(λ)λ3++300λ44φ(λ)λ4+30λ55φ(λ)λ5 +λ66φ(λ)λ6)
B VII.7φ(ω)ω7=(λ2πc)7G(1)7(λ)=(λ2πc)7(5040λφ(λ)λ+15120λ22φ(λ)λ2++12600λ33φ(λ)λ3+4200λ44φ(λ)λ4+630λ55φ(λ)λ5+42λ66φ(λ)λ6+λ77φ(λ)λ7)
B VIII.8φ(ω)ω8=(λ2πc)8G(1)8(λ)=(λ2πc)8(40320λφ(λ)λ+141120λ22φ(λ)λ2++141120λ33φ(λ)λ3+58800λ44φ(λ)λ4+11760λ55φ(λ)λ5+1176λ66φ(λ)λ6+56λ77φ(λ)λ7++λ88φ(λ)λ8)
B IX.9φ(ω)ω9=(λ2πc)9G(1)9(λ)=(λ2πc)9(362880λφ(λ)λ+1451520λ22φ(λ)λ2++1693440λ33φ(λ)λ3+846720λ44φ(λ)λ4+211680λ55φ(λ)λ5+28224λ66φ(λ)λ6++2016λ77φ(λ)λ7+72λ88φ(λ)λ8+λ99φ(λ)λ9)
B X.10φ(ω)ω10=(λ2πc)10G(1)10(λ)=(λ2πc)10(3628800λφ(λ)λ+16329600λ22φ(λ)λ2++21772800λ33φ(λ)λ3+12700800λ44φ(λ)λ4+3810240λ55φ(λ)λ5+635040λ66φ(λ)λ6++60480λ77φ(λ)λ7+3240λ88φ(λ)λ8+90λ99φ(λ)λ9+λ1010φ(λ)λ10)

For more information, please visit the original papers Ref [1], Ref [2], Ref [3].

 Cite as:

[1]. D. Popmintchev, et al., "Analytical Lah-Laguerre optical formalism for perturbative chromatic dispersion ", Optics Express 30, 22,  pp. 40779-40808, 20 October 2022 DOI: https://doi.org/10.1364/OE.457139, DOI: 10.1364/OE.457139.

[2]. D. Popmintchev, et al., “Dispersion orders of various materials,” figshare (2022), https://doi.org/10.6084/m9.figshare.19236792

[3]. D. Popmintchev, et al., "Theory of the Chromatic Dispersion, Revisited", arXiv:2011, 30  October 2020, https://doi.org/10.48550/arXiv.2011.00066, DOI: 10.48550/arXiv.2011.00066.

 



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