Abstract

The propagation of focused beams having an on-axis irradiance null is considered in the presence of thermal blooming. Two cases are treated: (a) beam profiles that have an irradiance zero in the beam center at the focal plane as well as the transmitter aperture; (b) beam profiles that have an on-axis irradiance null only at the transmitter. It is demonstrated that none of the beam profiles considered in case (a) has a meaningful advantage over a Gaussian beam profile. Some of the case (b) profiles do produce a larger bloomed irradiance in the focal plane, particularly when the initial intensity distribution is very uniform in its nonzero regions. Addition of a simple central obscuration to a “filled” irradiance distribution is found to have no advantage, however, for the cases considered.

© 1976 Optical Society of America

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References

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  1. J. P. Gordon, C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
    [Crossref]
  2. A concise review of this topic for gaseous media is presented in J. N. Hayes, P. B. Ulrich, and A. H. Aitken, Appl. Opt. 11, 257 (1972).
    [Crossref] [PubMed]
  3. P. B. Ulrich, J. Opt. Soc. Am. 64, 549 (1974).
  4. L. W. Casperson and M. S. Shekhani, Appl. Opt. 14, 1653 (1975).
  5. W. P. Brown (unpublished).
  6. G. D. Boyd and J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
    [Crossref]
  7. G. Goubau and F. Schwering, IRE Trans. Antennas and Propag. AP-9, 248 (1961).
    [Crossref]
  8. W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
    [Crossref]
  9. W. P. Brown (unpublished).
  10. J. Wallace, I. Itzkam, and J. Camm, J. Opt. Soc. Am. 64, 1123 (1974).
    [Crossref]
  11. C. Yeh, J. E. Pearson, and W. P. Brown, “Enhanced Target Irradiance in the Presence of Thermal Blooming,” Appl. Opt. (to be published).
  12. L. C. Bradley and J. Herrman, Appl. Opt. 13, 331 (1974).
    [Crossref] [PubMed]

1975 (2)

L. W. Casperson and M. S. Shekhani, Appl. Opt. 14, 1653 (1975).

W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
[Crossref]

1974 (3)

1972 (1)

1965 (1)

J. P. Gordon, C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

1961 (2)

G. D. Boyd and J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
[Crossref]

G. Goubau and F. Schwering, IRE Trans. Antennas and Propag. AP-9, 248 (1961).
[Crossref]

Aitken, A. H.

Boyd, G. D.

G. D. Boyd and J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
[Crossref]

Bradley, L. C.

Bridges, W. B.

W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
[Crossref]

Brown, W. P.

W. P. Brown (unpublished).

W. P. Brown (unpublished).

C. Yeh, J. E. Pearson, and W. P. Brown, “Enhanced Target Irradiance in the Presence of Thermal Blooming,” Appl. Opt. (to be published).

Camm, J.

Casperson, L. W.

L. W. Casperson and M. S. Shekhani, Appl. Opt. 14, 1653 (1975).

Gordon, J. P.

J. P. Gordon, C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

G. D. Boyd and J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
[Crossref]

Goubau, G.

G. Goubau and F. Schwering, IRE Trans. Antennas and Propag. AP-9, 248 (1961).
[Crossref]

Hayes, J. N.

Herrman, J.

Itzkam, I.

Leite, C. C.

J. P. Gordon, C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Moore, R. S.

J. P. Gordon, C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Pearson, J. E.

W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
[Crossref]

C. Yeh, J. E. Pearson, and W. P. Brown, “Enhanced Target Irradiance in the Presence of Thermal Blooming,” Appl. Opt. (to be published).

Porto, S. P. S.

J. P. Gordon, C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Schwering, F.

G. Goubau and F. Schwering, IRE Trans. Antennas and Propag. AP-9, 248 (1961).
[Crossref]

Shekhani, M. S.

L. W. Casperson and M. S. Shekhani, Appl. Opt. 14, 1653 (1975).

Ulrich, P. B.

Wallace, J.

Whinnery, J. R.

J. P. Gordon, C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Yeh, C.

C. Yeh, J. E. Pearson, and W. P. Brown, “Enhanced Target Irradiance in the Presence of Thermal Blooming,” Appl. Opt. (to be published).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
[Crossref]

Bell Syst. Tech. J. (1)

G. D. Boyd and J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
[Crossref]

IRE Trans. Antennas and Propag. (1)

G. Goubau and F. Schwering, IRE Trans. Antennas and Propag. AP-9, 248 (1961).
[Crossref]

J. Appl. Phys. (1)

J. P. Gordon, C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

J. Opt. Soc. Am. (2)

Other (3)

W. P. Brown (unpublished).

W. P. Brown (unpublished).

C. Yeh, J. E. Pearson, and W. P. Brown, “Enhanced Target Irradiance in the Presence of Thermal Blooming,” Appl. Opt. (to be published).

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Figures (6)

FIG. 1
FIG. 1

Three-dimensional plots of transmitter irradiance profiles that retain their shape for free-space propagation. The truncated Gaussian beam is used as a comparison reference for all other beams considered. The two “on-axis-null” profiles shown here also have an on-axis null in the focal plane.

FIG. 2
FIG. 2

Pseudogray scale plots of transmitter and focal plane irradiance distributions for the three beams in Fig. 1. A moderate amount of thermal blooming is present in the propagation path.

FIG. 3
FIG. 3

Peak focal plane irradiance versus total transmitter power for the same three cases as in Figs. 1 and 2. The axis units are normalized values as discussed in the Appendix.

FIG. 4
FIG. 4

Three-dimensional plots of transmitter irradiance profiles that have an on-axis null initially that fills in as the beams propagate. Case (III-a) is a mixture of cylindrical modes TEM00 and TEM01 oscillating in phase composition. Case (III-b) is similar to Case (II-b), except |u|2 has no θ dependence. Case (III-c) is a uniform annular beam. Case (III-d) is an annularly truncated Gaussian beam.

FIG. 5
FIG. 5

Pseudogray scale plots of transmitter and focal plane irradiance distributions for the three beams in Fig. 4. A moderate amount of thermal blooming is present in the propagation path.

FIG. 6
FIG. 6

Peak focal plane irradiance versus total transmitter power for the four beams in Figs. 5 and 6. The axis units are normalized as discussed in the Appendix.

Tables (1)

Tables Icon

TABLE AI Values for peak focal plane irradiance calculations.

Equations (19)

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E ( r 1 , θ 1 ) = A m f m ( r 1 ) cos ( m θ 1 ) ,     ( m θ 1 ) ,     ( m = 0 , 1 , 2 ) ,
u ( x 0 , y 0 , z ) = e i k z j λ z exp ( j k 2 z ( x 0 2 + y 0 2 ) ) × - A ( r 1 , θ 1 ) exp ( - j 2 π λ z ( x 0 x 1 + y 0 y 1 ) ) d x 1 d y 1 ,
u ( r 0 , θ 0 , z ) = e j k z j λ z exp ( j k 2 z r 0 2 ) 0 f m ( r 1 ) r 1 d r 1 × 0 2 π cos ( m θ ) exp ( - j 2 π λ z r 1 r 0 cos ( θ 1 - θ 0 ) ) d θ 1 = e j k z j λ z exp ( j k 2 z r 0 2 ) 2 π cos ( m θ 0 ) e - j m ( π / 2 ) I m ( r 0 ) ,
I m ( r 0 ) = 0 A m f m ( r 1 ) J m ( 2 π λ z r 1 r 0 ) r 1 d r 1 .
u ( 0 , θ 0 , z ) 2 = ( 2 π cos ( m θ 0 ) λ z ) 2 I m 2 ( 0 ) .
u 2 = A 1 2 exp ( - r 1 2 / ρ 0 2 ) cos 2 θ 1 for r 1 a 0 , = 0 for r 1 > a 0 ;
u 2 = A 2 2 exp [ - ( r 1 - r m ρ 0 ) 2 1 ρ 2 ] cos 2 θ 1 for b 0 r 1 a 0 and b 0 < r m < a 0 , = 0 otherwise ;
u 2 = A 0 2 exp ( - r 1 2 / ρ 0 2 ) for r 1 a 0 , = 0 otherwise .
u 2 = A 3 2 ( r 1 / ρ 0 ) exp 4 [ - ( r 1 2 / ρ 0 2 ) ] for r 1 < a 0 , = 0 for r 1 > a 0 ;
u 2 = A 4 2 exp [ - ( r 1 - r m ρ 0 ) 2 1 ρ 2 ] for b 0 < r 1 < a 0 , = 0 otherwise ;
u 2 = A 5 2 for b 0 < r 1 < a 0 , = 0 otherwise ;
u 2 = A 6 2 exp [ - ( r 1 2 / ρ 0 2 ) ] for b 0 < r 1 < a 0 , = 0 otherwise .
P T = A T E 2 d A T = 4 π m + 1 ρ 0 2 A m 2 F 1 a 0 ρ 0 ,
F 1 ( a 0 / ρ 0 ) = 0 a 0 / 2 ρ 0 f m 2 ( 2 ρ 0 x ) x d x .
u ( r 0 , θ 0 , z ) 2 = ( 2 π cos ( m θ 0 ) λ z ) 2 I m 2 ( r 0 , z ) ,
u 2 = 4 π ( m + 1 ) ( ρ 0 λ z ) 2 e - α z F 2 2 ( a 0 / ρ 0 , α ) P T F 1 ( a 0 / ρ 0 2 ) ,
F 2 ( a 0 / ρ 0 , β ) = 0 a 0 / 2 ρ 0 f m ( 2 ρ 0 x ) J m ( β x ) x d x
β = 2 π 2 ρ 0 r 0 / λ z .
u 2 = 0.592 ( m + 1 ) ( F 2 2 / F 1 ) ( S . R . ) P T