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Rotating scale-invariant electromagnetic fields

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Abstract

The concept of scalar fields with uniformly rotating intensity distributions and propagation-invariant radial scales is extended to the case of electromagnetic fields with rotating but otherwise propagation-invariant states of polarization. It is shown that the conditions for field rotation are different for scalar and electromagnetic fields and that the electromagentic analysis brings in new aspects such as the possibility that different components of a rotating electromagnetic field can rotate in opposite directions.

©2001 Optical Society of America

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Supplementary Material (4)

Media 1: MOV (1568 KB)     
Media 2: MOV (1388 KB)     
Media 3: MOV (1249 KB)     
Media 4: MOV (1404 KB)     

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Figures (4)

Fig. 1.
Fig. 1. (1.53 MB) The squared absolute values of the central parts of Eρ , Ez , and Ex as a function of z-coordinate, calculated with the parameters given in Table 1 (left). The movie should be viewed repeatedly.
Fig. 2.
Fig. 2. (1.35 MB) Same as Fig. 1, except that the parameters used in the calculations given in Table 1 (right). The movie should be viewed repeatedly.
Fig. 3.
Fig. 3. (1.21 MB) Same as Fig. 1, except that the parameters used in the calculations given in Table 2 (left). The movie should be viewed repeatedly.
Fig. 4.
Fig. 4. (1.37 MB) Same as Fig. 1, except that the parameters used in the calculations given in Table 2 (right). The movie should be viewed repeatedly.

Tables (2)

Tables Icon

Table 1. The parameters assumed in Figs. 1 (left) and 2 (right). The constant βc must be less than k but is otherwise arbitrary.

Tables Icon

Table 2. The parameters assumed in Figs. 3 (left) and 4 (right). The constant βc must be less than k but is otherwise arbitrary.

Equations (33)

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U ( ρ , ϕ , z ) = 0 0 2 π α A ( α , ψ ) exp [ i α ρ cos ( ϕ ψ ) + i β z ] d α d ψ ,
A ( α , ψ ) = 1 2 π 0 0 2 π ρ U ( ρ , ϕ , 0 ) exp [ i α ρ cos ( ϕ ψ ) ] d ρ d ϕ ,
β = { ( k 2 α 2 ) 1 2 if α k i ( α 2 k 2 ) 1 2 if k < α .
U ( ρ , ϕ + η Δ z , z + Δ z ) = exp [ i ξ ( ρ , ϕ , Δ z ) ] U ( ρ , ϕ , z ) .
0 = m = 0 a m ( α ) J m ( αρ ) exp ( i m ϕ )
× { exp [ i ξ ( ρ , ϕ , Δ z ) ] exp [ i ( + β ) Δ z ] } d α ,
a m ( α ) = i m α 0 2 π A ( α , ψ ) exp ( i ) d ψ .
exp ( i ϑ cos φ ) = m =- i m J m ( ϑ ) exp ( i ) .
ξ ( ρ , ϕ , Δ z ) = ξ ( Δ z ) = ( + β ) Δ z + 2 π q ,
β m = β 0 ,
U ( ρ , ϕ , z ) = m M a m J m ( α m ρ ) exp [ i ( + β m z ) ] ,
z T = 2 π / η .
z T / Q = 2 π / η Q ,
I ( ρ , ϕ + 2 π q / Q , z ) = I ( ρ , ϕ , z ) ,
E ρ ( ρ , ϕ , z ) = E x ( ρ , ϕ , z ) cos ϕ + E y ( ρ , ϕ , z ) sin ϕ
E ϕ ( ρ , ϕ , z ) = E x ( ρ , ϕ , z ) sin ϕ + E y ( ρ , ϕ , z ) cos ϕ
{ E ρ ( ρ , ϕ + γ Δ z , z + Δ z ) = exp [ i ξ ( ρ , ϕ , Δ z ) ] E ρ ( ρ , ϕ , z ) E ϕ ( ρ , ϕ + γ Δ z , z + Δ z ) = exp [ i ξ ( ρ , ϕ , Δ z ) ] E ϕ ( ρ , ϕ , z ) ,
E ρ ( ρ , ϕ , z ) = m = 0 [ a m ( α ) cos ϕ + b m ( α ) sin ϕ ]
× J m ( αρ ) exp [ i ( + βz ) ] d α ,
E ϕ ( ρ , ϕ , z ) m = 0 [ a m ( α ) sin ϕ + b m ( α ) cos ϕ ]
× J m ( αρ ) exp [ i ( + β z ) ] d α ,
{ g m ( ρ , ϕ , Δ z ) a m ( α ) + h m ( ρ , ϕ , Δ z ) b m ( α ) = 0 h m ( ρ , ϕ , Δ z ) a m ( α ) + g m ( ρ , ϕ , Δ z ) b m ( α ) = 0 ,
{ g m ( ρ , ϕ , Δ z ) = cos ( ϕ + γ Δ z ) exp [ i ( m γ + β ) Δ z ] cos ϕ exp [ i ξ ( ρ , ϕ , Δ z ) ] h m ( ρ , ϕ , Δ z ) = sin ( ϕ + γ Δ z ) exp [ i ( m γ + β ) Δ z ] sin ϕ exp [ i ξ ( ρ , ϕ , Δ z ) ] .
h m ( ρ , ϕ , Δ z ) = ± i g m ( ρ , ϕ , Δ z )
b m ( α ) = ± i a m ( α ) ,
[ ( m ± 1 ) γ + k z ] Δ z = ξ ( ρ , ϕ , Δ z ) .
β mn = β 0 n ( m + n ) γ ,
E ρ ( ρ , ϕ , z ) = m M n = 1,1 a mn J m ( α mn ρ ) exp { i [ ( m + n ) ϕ + β mn z ] }
E ϕ ( ρ , ϕ , z ) = m M n = 1,1 i na mn J m ( α mn ρ ) exp { i [ ( m + n ) ϕ + β mn z ] } ,
· E ( ρ , ϕ , z ) = 1 ρ ρ [ ρ E ρ ( ρ , ϕ , z ) ] + 1 ρ ϕ E ϕ ( ρ , ϕ , z ) + z E z ( ρ , ϕ , z ) = 0 ,
E z ( ρ , ϕ , z ) = m M n = 1,1 na mn α mn i β mn J m + n ( α mn ρ ) exp { i [ ( m + n ) ϕ + β mn z ] } .
{ β j = β 0 = β 0 n γ ( m + n j ) , β m = β 0 = β 0 n γ ( m + n m ) .
η γ = P Q = m j + n m n j m j .
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