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The cross correlation function of partially coherent vortex beam

Pan Feng Ding and Jixiong Pu

Open Access Open Access

Abstract

This work presents theoretical analysis on the cross correlation function (CCF) of partially coherent vortex beam (PCVB), where the relation of the number of the rings of CCF dislocations and orbital angular momentum (OAM) of PCVB is analyzed in detail. It is shown that rings of CCF dislocations do not always exist, and depend on the coherence length, the order of PCVB and location of observation plane, although the CCF indicates topological charge to some degree. Comprehensive analysis of the CCF of PCVB and numerical simulations all validate such phenomenon.

© 2014 Optical Society of America

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Figures (5)
Fig. 1
Fig. 1 Distributions of normalized CCF of the 1st order PCVB in near and far field for different coherence length. (a) z = 0.01ZR; (b) z = 0.5 ZR; (c) z = ZR; (d) z = 20 ZR.

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Fig. 2
Fig. 2 Distributions of normalized CCF of the 2nd order PCVB in near and far field for different coherence length. (a) z = 0.01 ZR; (b) z = 0.5 ZR; (c) z = ZR; (d) z = 20 ZR.

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Fig. 3
Fig. 3 Distributions of normalized CCF of higher order PCVB (m = 3, 4, 5) with low coherence length (Lc = 0.2w). (a) z = 0.01 ZR; (b) z = 0.02 ZR; (c) z = 0.04 ZR; (d) z = 0.5 ZR.

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Fig. 4
Fig. 4 Distributions of normalized CCF of higher order PCVB (m = 3, 4, 5) with moderate coherence length (Lc = w). (a) z = 0.05 ZR; (b) z = 0.2 ZR; (c) z = 0.5 ZR; (d) z = ZR.

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Fig. 5
Fig. 5 Distributions of normalized CCF of higher order PCVB (m = 3, 4, 5) with high coherence length (Lc = 3w). (a) z = 0.05 ZR; (b) z = 0.5 ZR; (c) z = 5 ZR; (d) z = 20 ZR.

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Equations (9)

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Γ( r ⇀ 1 , r ⇀ 2 ,0)= E 0 2 ( r 1 r 2 w 2 ) m exp[ − | r ⇀ 1 − r ⇀ 2 | 2 L c 2 − r 1 2 + r 2 2 w 2 −im( ϕ 2 − ϕ 1 ) ],
Γ( ρ ⇀ 1 , ρ ⇀ 2 ,z)= (k/ 2πz ) 2 ∬ Γ( r ⇀ 1 , r ⇀ 2 ,0)exp( − ik 2z [ ( ρ ⇀ 1 − r ⇀ 1 ) 2 − ( ρ ⇀ 2 − r ⇀ 2 ) 2 ] ) d r ⇀ 1 d r ⇀ 2 ,
χ ′ ( ρ ⇀ )=Γ( ρ ⇀ ,− ρ ⇀ ,z)= E 0 2 π exp( −Δζ ) Δ 1 ∑ n=0 m n! ( C m n ) 2 Δ 2 n ( − Δ 3 ζ ) m−n ,
{ Δ= 2 k 2 w 4 (2 w 2 + L c 2 ) k 2 w 4 L c 2 +4 z 2 (2 w 2 + L c 2 ) , Δ 1 = ( 4z k w 2 ) 2m [ k 2 w 4 L c 2 k 2 w 4 L c 2 +4 z 2 (2 w 2 + L c 2 ) ] m+1 , Δ 2 = 4 z 2 k 2 w 2 L c 2 ,ζ= ρ 2 w 2 , C m n = m! n!(m−n)! , Δ 3 = k 2 w 4 L c 4 +4 z 2 (2 w 2 + L c 2 ) 2 k 2 w 4 L c 4 +4 z 2 L c 2 (2 w 2 + L c 2 ) .
χ ′ (ζ)∝exp( −Δζ )( Δ 2 − Δ 3 ζ).
ρ 0 = Δ 2 / Δ 3 w.
χ ′ (ζ)∝exp( −Δζ )(2 Δ 2 2 −4 Δ 2 Δ 3 ζ+ Δ 3 2 ζ 2 ).
ρ 0 = 2± 2 Δ 2 / Δ 3 w.
χ ′ (ζ)∝exp( −Δζ )(6 Δ 2 3 −18 Δ 2 2 Δ 3 ζ+9 Δ 2 Δ 3 2 ζ 2 − Δ 3 3 ζ 3 ).
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