• Photonics Research
  • Vol. 10, Issue 11, 2460 (2022)
Atri Halder* and Jari Turunen
Author Affiliations
  • Institute of Photonics, University of Eastern Finland, FI-80101 Joensuu, Finland
  • show less
    DOI: 10.1364/PRJ.461314 Cite this Article Set citation alerts
    Atri Halder, Jari Turunen. Spectral coherence of white LEDs[J]. Photonics Research, 2022, 10(11): 2460 Copy Citation Text show less

    Abstract

    We address space–frequency domain coherence properties of broadband light-emitting diodes (white LEDs) and fields radiated by them. Inverse-source techniques are employed to determine the spectral degree of spatial coherence of an effective planar source representing a real LED, and coherent elementary fields associated with it. By fitting with experimental measurements, we formulate simple analytical coherence models that can be used as a basis for theoretical and experimental studies of the coherence of polychromatic stationary light in free space and in various optical systems. In particular, we find that radiation from white LEDs follows closely Wolf’s scaling law for spectral invariance [Phys. Rev. Lett.56, 1370 (1986)PRLTAO0031-900710.1103/PhysRevLett.56.1370] in the blue and the phosphor-generated parts of the spectrum separately, but not across the entire white-light spectrum.
    W(ρ¯,Δρ,ω)=S(ρ¯,ω)μ(Δρ,ω),

    View in Article

    T(κ1,κ2,ω)=1(2π)4W(ρ1,ρ2,ω)×exp[i(κ1·ρ1κ2·ρ2)]d2ρ1d2ρ2,

    View in Article

    T(κ¯,Δκ,ω)=1(2π)4W(ρ¯,Δρ,ω)×exp[i(Δκ·ρ¯+κ¯·Δρ)]d2ρ¯d2Δρ

    View in Article

    T(κ¯,Δκ,ω)=S˜(κ¯,ω)μ˜(Δκ,ω),

    View in Article

    S˜(Δκ,ω)=1(2π)2S(ρ¯,ω)exp(iΔκ·ρ¯)d2ρ¯

    View in Article

    μ˜(κ¯,ω)=1(2π)2μ(Δρ,ω)exp(iκ¯·Δρ)d2Δρ

    View in Article

    W()(rs^1,rs^2,ω)=(2πωrc)2sz1sz2T(ωcs1,ωcs2,ω),

    View in Article

    W()(rs^1,rs^2,ω)=(2πωrc)2sz1sz2S˜(ωcΔs,ω)μ˜(ωcs¯,ω).

    View in Article

    S()(rs^,ω)=(2πωrc)2sz2S˜(0,ω)μ˜(ωcΔs,ω)

    View in Article

    μ()(rs^1,rs^2,ω)=S˜[(ω/c)Δs,ω]S˜(0,ω),

    View in Article

    S˜(0,ω)=1(2π)2S(ρ,ω)d2ρ

    View in Article

    J(s^,ω)=limrrS()(rs^,ω)=(2π)2sz2(ω/c)2S˜(0,ω)μ˜(ωcs,ω)

    View in Article

    μ(Δρ,ω)=|κ|<ω/cμ˜(κ,ω)exp(iκ·Δρ)d2κ.

    View in Article

    μ(Δρ,ω)=1(2π)2S˜(0,ω)×LFJ(s^,ω)1s2exp(iωcs·Δρ)d2s,

    View in Article

    μ(Δρ,ω)=LF(1s2)1J(s^,ω)exp[i(ω/c)s·Δρ]d2sLF(1s2)1J(s^,ω)d2s.

    View in Article

    μ(Δρ,ω)=01s(1s2)1J(s,ω)J0[(ω/c)Δρs]ds01s(1s2)1J(s,ω)ds,

    View in Article

    μ(Δρ,ω)=1C2f(ρ¯ρΔρ/2,ω)×f(ρ¯ρ+Δρ/2,ω)d2ρ,

    View in Article

    C2=|f(ρ,ω)|2d2ρ.

    View in Article

    W(ρ1,ρ2,ω)=f(ρ1,ω)f(ρ2,ω)

    View in Article

    f˜(κ,ω)=1(2π)2f(ρ,ω)exp(iκ·ρ)d2ρ.

    View in Article

    J(s^,ω)=(2π)2sz2(ω/c)2|f˜(ωcs,ω)|2.

    View in Article

    f(ρ,ω)=12πLFJ1/2(s^,ω)(1s2)1/2exp(iωcs·ρ)d2s,

    View in Article

    f(ρ,ω)=f(0,ω)01s(1s2)1/2J1/2(s,ω)J0[(ω/c)ρs]ds01s(1s2)1/2J1/2(s,ω)ds,

    View in Article

    J(s^,ω)=J(ω)szp=J(ω)cospθ,

    View in Article

    01x(1x2)μJ0(ax)dx=2μΓ(μ+1)Jμ+1(a)aμ+1

    View in Article

    01x(1x2)μdx=12(1+μ),

    View in Article

    μ(Δρ,ω)=2p/2Γ(1+p2)Jp/2[(ω/c)Δρ][(ω/c)Δρ]p/2

    View in Article

    f(ρ,ω)=f(0,ω)2q/4+1/2Γ(32+q4)Jq/4+1/2[(ω/c)ρ][(ω/c)ρ]q/4+1/2

    View in Article

    μ(Δρ,ω)=0NAs(1s2)1J(s,ω)J0[(ω/c)Δρs]ds0NAs(1s2)1J(s,ω)ds

    View in Article

    f(ρ,ω)f(0,ω)=0NAs(1s2)1/2J1/2(s,ω)J0[(ω/c)ρs]ds0NAs(1s2)1/2J1/2(s,ω)ds

    View in Article

    M=FF=sinθsinθ=NANA=sinΘsinΘ=ss=ΔρΔρ=ρρ.

    View in Article

    J(θ,ω)cosθ=J(θ,ω)cosθ

    View in Article

    J(s,ω)=J(s,ω)[1s21(s/M)2]1/2=J(s,ω)P(s),

    View in Article

    μ(MΔρ,ω)=0NA/Ms[1(s/M)2]1P(s)J(s,ω)J0[(ω/c)Δρs]ds0NA/Ms[1(s/M)2]1P(s)J(s,ω)ds

    View in Article

    f(Mρ,ω)f(0,ω)=0NA/Ms[1(s/M)2]1/2P(s)J1/2(s,ω)J0[(ω/c)ρs]ds0NA/Ms[1(s/M)2]1/2P(s)J1/2(s,ω)ds,

    View in Article

    s()(rs^,ω)=S()(rs^,ω)0S()(rs^,ω)dω=J(θ,ω)0J(θ,ω)dω.

    View in Article

    s()(rs^,ω)=J(ω)0J(ω)dω,

    View in Article

    μ(Δρ,ω)=H(ωcΔρ)=01s(1s2)1J(s)J0[(ω/c)Δρs]ds01s(1s2)1J(s)ds.

    View in Article

    s(ω)=S˜(0,ω)0S˜(0,ω)dω,

    View in Article

    f(ρ,ω)f(0,ω)=F(ωcρ)=01s(1s2)1/2J1/2(s)J0[(ω/c)ρs]ds01s(1s2)1/2J1/2(s)ds,

    View in Article

    μ()(rs^1,rs^2,ω)=sin[(ω/c)LΔsx/2](ω/c)LΔsx/2sin[(ω/c)LΔsy/2](ω/c)LΔsy/2,

    View in Article

    J(s,ω)=SB(ω)H(s,ωB)+SY(ω)H(s,ωY),

    View in Article

    H(s,ωB)=H(0,ωB)cosmθ=CB(1s2)m/2,

    View in Article

    H(s,ωY)=H(0,ωY)cosnθ=CY(1s2)n/2,

    View in Article

    μ(Δρ,ω)=m1CBSB(ω)LB(Δρ,ω)+n1CYSY(ω)LY(Δρ,ω)m1CBSB(ω)+n1CYSY(ω),

    View in Article

    LB(Δρ,ω)=2m/2Γ(1+m2)Jm/2[(ω/c)Δρ][(ω/c)Δρ]m/2,

    View in Article

    LY(Δρ,ω)=2n/2Γ(1+n2)Jn/2[(ω/c)Δρ][(ω/c)Δρ]n/2.

    View in Article

    f(ρ,ω)f(0,ω)=(1+m/2)1CBSB(ω)KB(ρ,ω)+(1+n/2)1CYSY(ω)KY(ρ,ω)(1+m/2)1CBSB(ω)+(1+n/2)1CYSY(ω),

    View in Article

    LB(ρ,ω)=2m/4+1/2Γ(32+m4)Jm/4+1/2[(ω/c)ρ][(ω/c)ρ]m/4+1/2,

    View in Article

    LY(ρ,ω)=2n/4+1/2Γ(32+n4)Jn/4+1/2[(ω/c)ρ][(ω/c)ρ]n/4+1/2.

    View in Article

    Sj(ω)=exp[2Ωj2(ωωj)2]

    View in Article

    Sj(ω)=exp[2Ωj2(ωω0ωsωj)2],

    View in Article

    ω0=0ωJ(0,ω)dω0J(0,ω)dω

    View in Article

    Atri Halder, Jari Turunen. Spectral coherence of white LEDs[J]. Photonics Research, 2022, 10(11): 2460
    Download Citation