Abstract

The transmitting axicon is examined for the axial field distribution of a regular axicon when the incident wave is plane and uniform, and when it has a gaussian irradiance distribution. In the microwave region, the focal depth of an axicon does not differ very much from that of a perfect lens. In the visible region, the axicon possesses a much greater focal depth, which can be greatly changed by changing the curvature of the slanting side of the axicon. For an axicon with a straight slanting side and small height, when a gaussian incident wave is used, the irradiance at the principal focus varies inversely with the focal length.

© 1973 Optical Society of America

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References

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  1. J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954).
    [Crossref]
  2. O. Bryngdahl, J. Opt. Soc. Am. 60, 915 (1970);J. Opt. Soc. Am. 61, 169 (1971).
    [Crossref]
  3. S. Cornblee, Proc. Inst. Electr. Eng. 117, 869 (1970).
    [Crossref]
  4. J. O. Stoner, Appl. Opt. 9, 53 (1970).
    [Crossref] [PubMed]
  5. J. W. Y. Lit and E. Brannen, J. Opt. Soc. Am. 60, 370 (1970).
    [Crossref]
  6. J. W. Y. Lit, J. Opt. Soc. Am. 60, 1001 (1970).
    [Crossref]
  7. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. Government Printing Office, Washington, D. C.1964;Dover, New York, 1965), p. 304, Eqs. 7.4.38 and 39.
  8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 307, Eqs. 3.322.1 and 2.

1970 (5)

1954 (1)

Brannen, E.

Bryngdahl, O.

Cornblee, S.

S. Cornblee, Proc. Inst. Electr. Eng. 117, 869 (1970).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 307, Eqs. 3.322.1 and 2.

Lit, J. W. Y.

McLeod, J. H.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 307, Eqs. 3.322.1 and 2.

Stoner, J. O.

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Proc. Inst. Electr. Eng. (1)

S. Cornblee, Proc. Inst. Electr. Eng. 117, 869 (1970).
[Crossref]

Other (2)

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. Government Printing Office, Washington, D. C.1964;Dover, New York, 1965), p. 304, Eqs. 7.4.38 and 39.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 307, Eqs. 3.322.1 and 2.

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Figures (6)

F. 1
F. 1

Geometry of a ray transmitted by an axicon.

F. 2
F. 2

An axicon with curved slanting side.

F. 3
F. 3

Axial irradiance distribution. Axicon has a = 21.3λ, h = 1λ. Incident wave is plane, uniform.

F. 4
F. 4

Same as in Fig. 3 but with an incident wave that has a gaussian irradiance distribution with ω0 = a.

F. 5
F. 5

Axial irradiance distribution. Axicon has a = 104λ, h = 5λ. Incident wave has a gaussian irradiance distribution with ω0 = a. Curve D: results for a perfect lens.

F. 6
F. 6

Irradiance at principal focus of different axicons. Numbers in brackets are the heights h (in λ) of axicons. All axicons have a = 104λ. Incident waves have gaussian irradiance distributions. Line A joins results for incident wave with ω0 = a/2. Line B joins those with ω0 = a/3.

Equations (37)

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[ Q Q P ] z + ( η 1 ) ( 1 ρ ) h + ( ρ a ) 2 / 2 z ,
U = ( i λ ) 1 s ( U i / r ) exp ( i k r ) d S ,
U u ( P ) = ( a 2 / i λ z ) exp ( ikz ) 0 1 0 2 π exp { i k [ ( η 1 ) ( 1 ρ ) h + ( ρ a ) 2 / 2 z ] } ρ d ρ d θ = ( 2 π a 2 / i λ z ) exp { i k [ z + h ( η 1 ) ] } I u ,
I u = 0 1 exp [ i υ ( ρ 2 2 q ρ ) ] ρ d ρ , υ = k a 2 / 2 z , q = z h ( η 1 ) / a 2 .
I u = { exp [ i υ ( 1 2 q ) ] 1 } / ( i 2 υ ) + q 0 1 exp [ i υ ρ ( ρ 2 q ) ] d ρ .
I u = { exp [ i υ ( 1 2 q ) ] 1 } / ( i 2 υ ) + q ( π / 2 υ ) 1 2 exp ( i υ q 2 ) { C [ ( 2 υ / π ) 1 2 ( 1 q ) ] + i S [ ( 2 υ / π ) 1 2 ( 1 q ) ] + C [ ( 2 υ / π ) 1 2 q ] + i S [ ( 2 υ / π ) 1 2 q ] } ,
C ( z ) = 0 z cos ( π t 2 / 2 ) d t , S ( z ) = 0 z sin ( π t 2 / 2 ) d t .
U u ( P ) = G u + C u [ C ( ω 1 ) + i S ( ω 1 ) + C ( ω 2 ) + i S ( ω 2 ) ] ,
G u = { exp [ ikh ( η 1 ) ] exp ( i k a 2 / 2 z ) } exp ( ikz ) ,
C u = i π h ( η 1 ) ( 2 z / λ ) 1 2 a 1 exp { i k [ h ( η 1 ) z h 2 ( η 1 ) 2 / 2 a 2 + z ] } ,
ω 1 = ( 2 / λ z ) 1 2 a [ 1 z h ( η 1 ) / a 2 ] ,
ω 2 = ( 2 z / λ ) 1 2 h ( η 1 ) / a .
U i = ( ω 0 / ω i ) exp [ i ( k z i β i ) ] × exp [ r 2 ( ω i 2 i k / 2 R i ) ] ,
ω i 2 = ω 0 2 [ 1 + ( 2 z i / b ) 2 ] , R i = z i [ 1 + ( b / 2 z i ) 2 ] , β i = tan 1 ( 2 z i / b ) , b = 2 π ω 0 2 / λ .
U g = B I g ,
B = ( 2 π a 2 / i λ z ) ( ω 0 / ω i ) exp { i k [ z + h ( η 1 ) + z i β i / k ] } , I g = 0 1 exp ( u ρ 2 i p ρ ) ρ d ρ , u = ( a 2 / ω i 2 ) ( i k a 2 / 2 ) ( z 1 + R i 1 ) , p = k h ( η 1 ) .
u exp ( x 2 / 4 β γ x ) d x = ( π β ) 1 2 exp ( β γ 2 ) [ 1 erf ( γ β 1 2 + u β 1 2 / 2 ) ]
0 exp ( x 2 / 4 β γ x ) d x = ( π β ) 1 2 exp ( β γ 2 ) [ 1 erf ( γ β 1 2 ) ] ,
Re ( β ) > 0 , u > 0 ,
erf ( z ) = 2 π 1 2 0 z exp ( t 2 ) d t .
I g = [ 1 exp ( u i p ) ] / 2 u ( i p / 4 u ) ( π / u ) 1 2 exp ( w 2 2 ) [ erf ( w 1 ) erf ( w 2 ) ] ,
w 1 = w 2 + u , w 2 = ( i p ) / 2 u .
U g ( P ) = G g C g { erf ( w 1 ) erf ( w 2 ) } ,
G g = ( B / 2 u ) [ 1 exp ( u i p ) ] ,
C g = ( ipB p B / 4 u ) ( π / u ) 1 2 exp ( p 2 / 4 u ) .
U c ( P ) = ( k a 2 / i z ) exp { i k [ z + h ( η 1 ) ] } × 0 1 U i exp { i k [ ( a ρ ) 2 / 2 z h ( η 1 ) ρ s ] } ρ d ρ ,
G g = ( B / 2 u ) [ 1 exp ( u i p ) ] .
B = ( k a 2 / i z ) exp { i k [ z + h ( η 1 ) ] } ,
u = i k a 2 / 2 z ,
p = k h ( η 1 ) .
G g = { exp [ ikh ( η 1 ) ] exp ( i k a 2 / 2 z ) } exp ( ikz ) ,
w 1 = ( π / λ z ) 1 2 ( 1 i ) [ a h z ( η 1 ) / a ] = ( π / 2 ) ( 1 i ) ω 1 , w 2 = ( π / 2 ) ( 1 i ) ω 2 .
C g [ erf ( w 1 ) erf ( w 2 ) ] = C g { erf [ ( π / 2 ) ( 1 i ) ω 1 ] erf [ ( π / 2 ) ( 1 i ) ω 2 ] } .
erf [ ( π / 2 ) ( 1 i ) z ] = 2 [ C ( z ) + i S ( z ) ] / ( 1 + i )
C ( z ) = C ( z ) , S ( z ) = S ( z ) ,
C g [ erf ( ω 1 ) erf ( ω 2 ) ] = C g [ 2 / ( 1 + i ) ] [ C ( ω 1 ) + i S ( ω 1 ) + C ( ω 2 ) + i S ( ω 2 ) ] = C g [ C ( ω 1 ) + i S ( ω 1 ) + C ( ω 2 ) + i S ( ω 2 ) ] ,
C g = [ 2 / ( 1 + i ) ] C g .