Abstract

This paper examines the nature of the interaction of rapidly scanning, coherent light beams with linear dispersive optical elements. The analytical technique used is to represent a scanning beam by a sum of plane waves of different frequencies, traveling in different directions. The response of the optical element to each component plane wave is computed and then the responses are added. It is found, for example, that a dispersive element can transform a scanning beam into an amplitude-modulated plane wave, or conversely, transform an amplitude-modulated plane wave into a scanning beam.

© 1965 Optical Society of America

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Figures (13)

Fig. 1
Fig. 1

Scanning light field.

Fig. 2
Fig. 2

Spatial envelope of a scanning beam.

Fig. 3
Fig. 3

Diffraction-grating relations: sinθm=sinθi+mλ/d, where d is the grating period.

Fig. 4
Fig. 4

mth-order field of a diffraction grating scanned with velocity v=c/sinθ0m.

Fig. 5
Fig. 5

Spatial envelopes of fields when a diffraction grating is scanned perpendicular to the rulings by a light beam. (a) Case I, v=0; (b) Case II, v=c sinθ0m; (c) Case III, v=c/sinθ0m; (d) Case IV, v→∞.

Fig. 6
Fig. 6

Refraction at a dielectric interface; η(λ) sinθt=sinθi.

Fig. 7
Fig. 7

Spatial envelopes when a dielectric interface is scanned by a light beam; vg(λ)=c/[η(λ)−λη′(λ)].

Fig. 8
Fig. 8

Refraction of a plane wave by a prism; sin ( θ e + α ) = [ η 2 - sin 2 θ i ] 1 2 sin α + sin θ i cos α.

Fig. 9
Fig. 9

Spatial envelopes of fields when a prism is scanned by a light beam; γ = [ ( c / v ) cos α - λ 0 η ( λ 0 ) sin α ] [ 1 - η 2 ( λ 0 ) sin 2 α ] - 1 2; sin(θ0e+α)=η0) sinα.

Fig. 10
Fig. 10

Spatial envelopes of fields when a prism is scanned by a light beam; Case I with normal dispersion:

Fig. 11
Fig. 11

Spatial envelopes of fields when a prism is scanned by a light beam; Case I with anomalous dispersion:

Fig. 12
Fig. 12

Spatial envelopes of fields when a prism is scanned by a light beam; Case II with normal dispersion: v→∞;

Fig. 13
Fig. 13

Spatial envelopes of fields when a prism is scanned by a light beam; Case II with anomalous dispersion: v→∞;

Equations (107)

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E ( x , z , t ) z = 0 = E ( x , 0 , t ) = { e i ω 0 t f [ 1 T ( t - x v ) ] , 0 x L 0 , otherwise ,
f ( ξ + n ) = f ( ξ ) ,
L = v T .
f ( ξ ) = n A n e i 2 π n ξ ,
A n = - 1 2 1 2 f ( ξ ) e - i 2 π n ξ d ξ ,
A - n = A n * .
E ( x , 0 , t ) = { n A n e i ( ω n t - β n x ) , 0 x L 0 , otherwise ,
ω n = ω 0 + n ω s ,
ω s = 2 π / T = 2 π v / L ,
β n = 2 π n / L = n ω s / v .
( 2 x 2 + 2 y 2 - 1 c 2 2 t 2 ) E ( x , z , t ) = 0 ,
E ( x , z , t ) = n - E n ( β ) e i ( ω n t - β x - γ z ) d β ,
γ = ( k n 2 - β 2 ) 1 2
k n = ω n / c .
E n ( β ) = A n [ sin [ ( β - β n ) L / 2 ] π ( β - β n ) ] e i ( β - β n ) L / 2 .
lim L [ sin [ ( β - β n ) L / 2 ] π ( β - β n ) ] = δ ( β - β n ) ,
E n ( β ) A n δ ( β - β n ) .
E ( x , z , t ) = A n e i ( ω n t - β n x - γ n z ) ,
γ n = ( k n 2 - β n 2 ) 1 2 .
E ( x , z , t ) = A n exp [ i ω n ( t - x c sin θ n - z c cos θ n ) ] ,
sin θ n = β n / k n = n λ n / L = ( n ω s / ω n ) ( c / v ) = n λ 0 / L 1 + ( v / c ) ( n λ 0 / L ) ,
λ n = 2 π k n = λ 0 1 + n ω s / ω 0 = λ 0 1 + ( v / c ) ( n λ 0 / L ) ,
λ 0 = 2 π c / ω 0 .
- 1 < sin θ n < 1.
- [ ( L / λ 0 ) / ( 1 + v / c ) ] < n < ( L / λ 0 ) / ( 1 - v / c ) .
n < ( L / λ 0 ) / ( 1 + v / c ) .
n = ± L / b .
N = L / b ,
sin θ ± N = ( λ 0 / b ) / [ 1 ± ( v / c ) ( λ 0 / b ) ] .
- 1 < sin θ ± N < 1 ,
b > λ 0 ( 1 + v / c ) .
b λ 0
b ( v / c ) λ 0 ,
sin θ n 1
cos θ n 1
E ( x , z , t ) = A n exp [ i ω n ( t - n λ n x L c - z c ) ] = e i ω 0 ( t - z / c ) A n exp [ i n ω s ( t - x v - z c ) ] = e i ω 0 ( t - z / c ) S ( x , z , t ) ,
S ( x , z , t ) = f [ 1 T ( t - x v - z c ) ] .
E m ( x , z , t ) = α m ( θ i , ω ) exp [ i ω ( t - x c sin θ m - z c cos θ m ) ] ,
sin θ m = sin θ i + m λ / d ,
E m ( x , z , t ) = n α m ( θ n , ω n ) A n × exp [ i ω n ( t - x c sin θ n m - z c cos θ n m ) ] ,
ω n = ω 0 + n ω s
sin θ n m = sin θ n + m λ n d = λ n ( m d + n L ) = ω 0 ω n sin θ 0 m + n ω s ω n c v ,
sin θ 0 m = m λ 0 / d .
sin θ n m = [ m λ 0 d ( m d + n L ) ] / ( m d + n L v c m λ 0 d ) .
v / c = d / m λ 0 = 1 / sin θ 0 m ,
sin θ n m = m λ 0 / d = sin θ 0 m ;
E m ( x , z , t ) = exp [ i ω 0 ( t - x c sin θ 0 m - z c cos θ 0 m ) ] × n α m ( θ n , ω n ) A n exp [ i n ω s ( t - x c sin θ 0 m - z c cos θ 0 m ) ]
E m ( x , z , t ) = exp [ i ω 0 ( t - x c sin θ 0 m - z c cos θ 0 m ) ] × g [ 1 T ( t - x c sin θ 0 m - z c cos θ 0 m ) ] ,
g ( ξ ) = α m ( θ n , ω n ) A n e i 2 π n ξ .
ω n sin θ n m = ω 0 sin θ 0 m + n ω s ( c / v ) .
ω n cos θ n m = ( ω n 2 - ω n 2 sin 2 θ n m ) 1 2 = ω 0 cos θ 0 m + n ω s { [ 1 - ( c / v ) sin θ 0 m ] / cos θ 0 m } .
α m ( θ n , ω n ) α m ( 0 , ω 0 ) .
E m ( x , z , t ) = α m ( 0 , ω 0 ) exp [ i ω 0 ( t - x c sin θ 0 m - z c cos θ 0 m ) ] × n A n exp ( i n ω s { t - x v - z c [ 1 - ( c / v ) sin θ 0 m cos θ 0 m ] } ) ;
E m ( x , z , t ) = α m ( 0 , ω 0 ) × exp [ i ω 0 ( t - x c sin θ 0 m - z c cos θ 0 m ) ] S m ( x , z , t ) ,
S m ( x , z , t ) = f ( 1 T { t - x v - z c [ 1 - ( c / v ) sin θ 0 m cos θ 0 m ] } ) .
S ( x , z , t ) = f ( - x / L ) .
S m ( x , z , t ) = f [ ( - x + z tan θ 0 m ) / L ] .
S ( x , z , t ) = f [ 1 T ( t - x c sin θ 0 m - z c ) ] ;
S m ( x , z , t ) = f { T - 1 [ t - ( x / v ) ] } .
S ( x , z , t ) = f [ 1 T ( t - x c sin θ 0 m - z c ) ] ;
S m ( x , z , t ) = f [ 1 T ( t - x c sin θ 0 m - z c cos θ 0 m ) ] .
S ( x , z , t ) = f { T - 1 [ t - ( z / c ) ] } ;
S m ( x , z , t ) = f { T - 1 [ t - ( z / c cos θ 0 m ) ] } .
E t ( x , z , t ) = T ( θ i , ω ) × exp { i ω [ t - η ( λ ) x c sin θ t - η ( λ ) z c cos θ t ] } ,
sin θ t = sin θ i / η ( λ ) ,
E t ( x , z , t ) = n T ( θ n , ω n ) A n × exp { i ω n [ t - η ( λ n ) x c sin θ n t - η ( λ n ) z c cos θ n t ] } ,
ω n = ω 0 + n ω s ,
η ( λ n ) sin θ n t = sin θ n ,
η ( λ n ) cos θ n t = [ η 2 ( λ n ) - sin 2 θ n ] 1 2 .
θ n n λ 0 / L = ( n ω s / ω 0 ) ( c / v )
λ n λ 0 ( 1 - θ n v / c ) = λ 0 ( 1 - n ω s / ω 0 ) .
η ( λ n ) cos θ n t η ( λ 0 ) - η ( λ 0 ) λ 0 θ n ( v / c ) = η ( λ 0 ) - η ( λ 0 ) λ 0 n ω s / ω 0 ,
η ( λ 0 ) = [ d η ( λ ) / d λ ] λ = λ 0 .
T ( θ n , ω n ) T ( 0 , ω 0 ) .
E t ( x , z , t ) = T ( 0 , ω 0 ) exp { i ω 0 [ t - z c η ( λ 0 ) ] } × n A n exp ( i n ω s { t - x v - z c [ η ( λ 0 ) - λ 0 η ( λ 0 ) ] } ) ,
E t ( x , z , t ) = T ( 0 , ω 0 ) × exp { i ω 0 [ t - z c η ( λ 0 ) ] } S t ( x , z , t ) ,
S t ( x , z , t ) = n A n exp ( i n ω s { t - x v - z c [ η ( λ 0 ) - λ 0 η ( λ 0 ) ] } ) = f { 1 T [ t - x v - z v g ( λ 0 ) ] } ,
v g ( λ 0 ) = c / [ η ( λ 0 ) - λ 0 η ( λ 0 ) ] .
v g = ( d β d ω ) - 1 = [ d d ω ( ω v p ) ] - 1 = [ d d ω ( ω η c ) ] - 1 = ( n c + ω c d η d λ d λ d ω ) - 1 = ( η - λ η c ) - 1 .
E e ( x , z , t ) = T e ( θ i , ω ) exp [ i ω ( t - x c sin θ e - z c cos θ e ) ] ,
sin ( θ e + α ) = [ η 2 ( λ ) - sin 2 θ i ] 1 2 sin α + sin θ i cos α ,
E e ( x , z , t ) = n T e ( θ n , ω n ) × exp [ i ω n ( t - x c sin θ n e - z c cos θ n e ) ] ,
ω n = ω 0 + n ω s
sin ( θ n e + α ) = [ η 2 ( λ n ) - sin 2 θ n ] 1 2 sin α + sin θ n cos α .
sin θ n e sin θ 0 e + [ cos α - η ( λ 0 ) ( v / c ) λ 0 sin α [ 1 - η 2 ( λ 0 ) sin 2 α ] 1 2 ] θ n cos θ 0 e
cos θ n e cos θ 0 e - [ cos α - η ( λ 0 ) ( v / c ) sin α [ 1 - η 2 ( λ 0 ) sin 2 α ] 1 2 ] θ n sin θ 0 e .
T e ( θ n , ω n ) T e ( 0 , ω 0 ) .
E e ( x , z , t ) = T e ( 0 , ω 0 ) × exp [ i ω 0 ( t - x c sin θ 0 e - z c cos θ 0 e ) ] S e ( x , z , t ) ,
S e ( x , z , t ) = f { 1 T [ t - x c ( sin θ 0 e + γ cos θ 0 e - z c ( cos θ 0 e - γ sin θ 0 e ) ] } ,
γ = ( c / v ) cos α - η ( λ 0 ) λ 0 sin α [ 1 - η 2 ( λ 0 ) sin 2 α ] 1 2 .
x = x cos θ 0 e + z sin θ 0 e ,
z = z cos θ 0 e - x sin θ 0 e ,
E e ( x , z , t ) = T e ( 0 , ω 0 ) e i ω 0 ( t - z / c ) S e ( x , z , t ) ,
S e ( x , z , l ) = f [ 1 T ( t - x γ c - z c ) ] ,
v = c cot α / λ 0 η ( λ 0 ) .
S o ( x , z , t ) = f { T - 1 [ t - ( z / c ) ] } .
γ = - λ 0 η ( λ 0 ) sin α [ 1 - η 2 ( λ 0 ) sin 2 α ] 1 2 ,
v x = c γ = - c [ 1 - η 2 ( λ 0 ) sin 2 α ] 1 2 λ 0 η ( λ 0 ) sin α
E ( x , z , t ) = n F n ( x , z ) e i ω n t ,
F n ( x , z ) = A n exp [ - i ω n c ( x sin θ n + z cos θ n ) ]
F n ( x , z ) = A n .
F n ( x , z ) e - i ω n z / c
E ( x , z , t ) e i ω 0 ( t - z / c ) f { T - 1 [ t - ( z / c ) ] } .
v = c [ λ 0 η ( λ 0 ) ] - 1 cot α ; η ( λ 0 ) < 0.
v = c [ λ 0 η ( λ 0 ) ] - 1 cot α ; η ( λ 0 ) > 0.
γ = - [ 1 - η 2 ( λ 0 ) sin 2 α ] - 1 2 λ 0 η ( λ 0 ) sin α ; η ( λ 0 ) < 0.
λ = - [ 1 - η 2 ( λ 0 ) sin 2 α ] 1 2 λ 0 η ( λ 0 ) sin α ; η ( λ 0 ) > 0.