Abstract

An axicon and a lens are combined to form an optical system producing a ring-shaped pattern. The purpose of this paper is to show that when a lens–axicon combination is illuminated by a Gaussian beam, the transverse distribution of the focal ring is also a Gaussian distribution. The typical width of this distribution was found to be, in the case of the lens–axicon combination, 1.65 times greater than the typical width of the Gaussian beam obtained by focusing the same beam using the lens alone. This focusing system is well suited for the drilling of good quality large diameter holes using a high power laser beam.

© 1978 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. J. H. McLeod, J. Opt. Soc. Am. 50, 106 (1960).
    [Crossref]
  3. R. B. Barber, “Laser Optical Apparatus for Cutting Holes,” U. S. Patents3,419,321 (December1968).
  4. P. A. Bélanger, M. Rioux, Can. J. Phys. 54, 1774 (1976).
    [Crossref]
  5. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).
  7. J. B. Goodell, Appl. Opt. 8, 2506 (1969).
    [Crossref]
  8. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  9. Y. L. Luke, Integral of Bessel Functions (McGraw-Hill, New York, 1962).
  10. A. Laflamme, Rev. Sci. Instrum. 41, 1578 (1970).
    [Crossref]

1976 (1)

P. A. Bélanger, M. Rioux, Can. J. Phys. 54, 1774 (1976).
[Crossref]

1970 (1)

A. Laflamme, Rev. Sci. Instrum. 41, 1578 (1970).
[Crossref]

1969 (1)

J. B. Goodell, Appl. Opt. 8, 2506 (1969).
[Crossref]

1960 (1)

J. H. McLeod, J. Opt. Soc. Am. 50, 106 (1960).
[Crossref]

1954 (1)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Barber, R. B.

R. B. Barber, “Laser Optical Apparatus for Cutting Holes,” U. S. Patents3,419,321 (December1968).

Bélanger, P. A.

P. A. Bélanger, M. Rioux, Can. J. Phys. 54, 1774 (1976).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Goodell, J. B.

J. B. Goodell, Appl. Opt. 8, 2506 (1969).
[Crossref]

Laflamme, A.

A. Laflamme, Rev. Sci. Instrum. 41, 1578 (1970).
[Crossref]

Luke, Y. L.

Y. L. Luke, Integral of Bessel Functions (McGraw-Hill, New York, 1962).

McLeod, J. H.

J. H. McLeod, J. Opt. Soc. Am. 50, 106 (1960).
[Crossref]

J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954).
[Crossref]

Rioux, M.

P. A. Bélanger, M. Rioux, Can. J. Phys. 54, 1774 (1976).
[Crossref]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Appl. Opt. (1)

J. B. Goodell, Appl. Opt. 8, 2506 (1969).
[Crossref]

Can. J. Phys. (1)

P. A. Bélanger, M. Rioux, Can. J. Phys. 54, 1774 (1976).
[Crossref]

J. Opt. Soc. Am. (2)

J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954).
[Crossref]

J. H. McLeod, J. Opt. Soc. Am. 50, 106 (1960).
[Crossref]

Rev. Sci. Instrum. (1)

A. Laflamme, Rev. Sci. Instrum. 41, 1578 (1970).
[Crossref]

Other (5)

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Y. L. Luke, Integral of Bessel Functions (McGraw-Hill, New York, 1962).

R. B. Barber, “Laser Optical Apparatus for Cutting Holes,” U. S. Patents3,419,321 (December1968).

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

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Figures (9)

Fig. 1
Fig. 1

Holes drilled in an acrylic plate (3 mm thick) by means of various lens–axicon combinations illuminated by a CO2–TEA laser beam.

Fig. 2
Fig. 2

Path of the light beam for a lens–axicon combination: (a) convergent axicon with angle α and refractive index n; (b) divergent axicon with angle α and refractive index n. The radius of the focal ring R0 is given by the equation R0= (n − 1)αF, where F is the focal length of the lens.

Fig. 3
Fig. 3

Normalized intensity distribution in the radial direction in the vicinity of the focal ring of radius R0.

Fig. 4
Fig. 4

Comparison between the function |F(ρ)|2 and a Gaussian function of the same typical width. —|F(ρ)|2;- - - - exp[−2ρ2/(1.05)2].

Fig. 5
Fig. 5

The function | F [ ρ ( 1 + i Δ ) 1 / 2 ] | 2 ( 1 + Δ 2 ) ¾vs the dimensionless variable ρ = (πWR0)/(2λF){1 − [(r2F2)/(R02z2)]}.

Fig. 6
Fig. 6

Variation of the normalized width of the intensity at 1/e2 vs the defocusing parameter Δ.

Fig. 7
Fig. 7

Intensity distribution of the laser output on the lens–axicon combination. The full line corresponds to a Gaussian distribution having a width of 5 mm at 1/e2.

Fig. 8
Fig. 8

Scintillation of a ring pattern obtained at the focus of a germanium lens–axicon combination illuminated by an ir laser beam.

Fig. 9
Fig. 9

Intensity distribution measured in the plane of the focal ring. The full line is the Gaussian distribution predicted by the theory.

Equations (62)

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U ( r o ) = ( I o ) 1 / 2 exp ( ik r 0 2 2 q ) ,
1 q = 1 R 2 i k W 2
W 2 = W 0 2 [ 1 + ( λ d π W 0 2 ) 2 ] ,
R = d [ 1 + ( π W 0 2 λ d ) 2 ] .
I 0 = ( 2 E ) / ( π W 2 ) ,
k = ( 2 π ) / λ .
U ( r , z ) = k z 0 a U A ( r 0 ) exp ( ik r 0 2 2 z ) J 0 ( kr r 0 z ) r 0 d r 0 .
U A ( r 0 ) = exp { ik [ r 0 2 2 q r 0 2 2 F + ( n 1 ) α r 0 ] } ,
R 0 = ( n 1 ) α F .
U A ( r 0 ) = ( I 0 ) 1 / 2 exp { ik [ r 0 2 2 q ( 1 q 1 F ) + r 0 R 0 F ] } ,
U ( r , z ) = k ( I 0 ) 1 / 2 z 0 a exp { ik [ r 0 2 2 q ( 1 q + 1 z 1 F ) + R 0 r 0 F ] } × J 0 ( kr r 0 z ) r 0 d r 0 .
I AL ( r , z ) = 0.24 EWF R 0 λ z 2 | F ( ρ ( 1 + i Δ ) 1 / 2 ) | 2 ( 1 + Δ 2 ) 3 / 4 ,
Δ = π W 2 λ ( 1 R + 1 z 1 F ) .
ρ = R 0 π W 2 F λ ( 1 r 2 F 2 R 0 2 z 2 ) .
1 z F = 1 F 1 R .
I AL ( r , F ) = 0.24 EW R 0 λ F | F ( ρ ) | 2 ,
ρ = π W λ F R 0 2 ( 1 r 2 R 0 2 ) .
r 2 R 0 2 = 1 2 λ F ρ π W R 0 .
r R 0 = 1 λ F ρ π W R 0 .
ρ = π W λ F ( r R 0 ) .
Δ AL = 3.3 [ ( λ F ) / ( π W ) ] .
Δ L = 2 [ ( λ F ) / ( π W ) ] .
I AL ( R 0 , F ) = 0.24 [ ( EW ) / ( R 0 λ F ) ] .
I L = 2 π E ( W F λ ) 2 .
I AL I L = 0.04 λ E R 0 W .
| F ( ρ ) | 2 = exp { 2 ρ 2 ( 1.65 ) 2 } .
E ( r 1 , r 2 ) = 2 π × 0.24 E ρ 1 ρ 2 | F ( ρ ) | 2 d ρ .
I AL ( r , F ) = 0.25 E R 0 Δ AL exp { 2 ( r R 0 ) 2 [ ( Δ AL ) / 2 ] 2 } ,
| F ( ρ ( 1 + i Δ ) 1 / 2 ) | 2 ( 1 + Δ 2 ) 3 / 2
δ F F = 0.75 Δ AL W .
δ F F = 0.8 λ W 2 .
I AL ( r , F ) = I AL ( R 0 , F ) exp { 2 ( r R 0 ) 2 [ ( Δ AL ) / ( 2 ) ] 2 } ,
I AL ( R 0 , F ) = 0.25 E R 0 Δ AL .
δ F F = 0.75 Δ AL W .
| F [ ρ ( 1 + i Δ ) 1 / 2 ] | 2 ( 1 + Δ 2 ) ¾
U ( r , z ) = k I 0 z 0 a exp [ ik ( r 0 2 2 q 0 + R 0 r 0 F ) ] × J 0 ( kr r 0 z ) r 0 d r 0 ,
1 q 0 = 1 q + 1 z 1 F ,
1 q 0 = 1 R + 1 z 1 F i 2 k W 2 .
t = ( x r 0 ) / a ,
x = k R 0 a F .
U ( r , z ) = I 0 F 2 kz R 0 2 0 x exp ( i F 2 t 2 2 k q 0 R 0 2 ) exp ( it ) J 0 ( r F R 0 z t ) tdt .
J 0 ( λ t ) = n = 0 ( 1 λ 2 ) n 2 n n ! t n J n ( t ) .
U ( r , z ) = I 0 F 2 kz R 0 2 n = 0 [ 1 ( rF R 0 z ) 2 ] n n ! 2 n S n ,
S n = 0 x exp ( i δ t 2 ) exp ( it ) J n ( t ) t n + 1 dt
δ = F 2 2 k R 0 2 q 0 .
S n = i = 0 ( i δ ) l l ! G n , l ,
G n , l = 0 x t 2 l + n + 1 exp ( it ) J n ( t ) dt .
G n , l = x 2 l + 2 n + 2 ( 2 l + 2 n + 2 ) 2 n n ! F 2 2 ( n + 1 2 , 2 l + 2 n + 2 2 n + 1 , 2 l + 2 n + 3 2 ix ) .
G n , l = x 2 l + n + 2 ( i ) n ( 2 i π x ) 1 / 2 ( 2 l + n + 3 / 2 )
S n = x 2 ( ix ) n ( 2 i π x ) 1 / 2 l = 0 ( i δ x 2 ) l l ! ( 2 l + n + 3 / 2 ) .
S n = x 2 ( ix ) n ( 2 i π x ) 1 / 2 ( n + 3 / 2 ) F 1 1 ( n 2 + 3 4 n 2 + 7 4 i δ x 2 ) .
( R W ) ,
( i δ x 2 ) = a 2 W 2 + ik a 2 2 ( 1 R + 1 z 1 F ) .
S n = Γ ( n 2 + 3 4 ) ( i ) n 2 ( 2 i π ) 1 / 2 ( i δ ) 3 / 4 + n / 2 .
U ( r , z ) = Γ ( 3 4 ) ( I 0 F W 3 ) 1 / 2 2 z ( i R 0 λ ) 1 / 2 ( 1 + i Δ ) 3 / 4 × n = 0 [ i 2 ( i δ ) 1 / 2 ( 1 r 2 F 2 R 0 2 z 2 ) ] n n ! Γ ( n 2 + 3 4 ) Γ ( 3 4 ) ,
i δ = F 2 R 0 2 k 2 W 2 ( 1 + i Δ )
Δ = k W 2 2 ( 1 R + 1 z 1 F ) .
ρ = R 0 π W 2 F λ ( 1 r 2 F 2 R 0 2 z 2 ) .
U ( r , z ) = B n = 0 ( i 2 ρ ( 1 + i Δ ) 1 / 2 ) n Γ ( n 2 + 3 4 ) Γ ( 3 4 ) n ! ,
B = Γ ( 3 / 4 ) 2 z ( 1 + i Δ ) 3 / 4 ( I 0 F W 3 i R 0 λ ) 1 / 2 .
U ( r , z ) = BF [ ρ ( 1 + i Δ ) 1 / 2 ] .
F ( y ) = F 1 1 ( 3 4 1 2 | y 2 ) i Γ ( 5 / 4 ) Γ ( 3 / 4 ) y 1 F 1 ( 5 4 3 2 | y 2 ) ,

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