Abstract

A theory is developed for multidither adaptive optical systems that employ N discrete actuators driving a deformable control and dither mirror with the same shape of influence function. The major emphasis is on mirrors with a Gaussian influence function. It is shown, as a special case, that single-actuator displacements or errors of such systems yield error signals that can be described in closed form, via sine integral functions of the actuator displacement. Closed-form expressions for both the first- and second-harmonic content in the dither signal outputs are developed. Expressions for servo cross coupling (from adjacent actuator deformations) are developed and it is shown how the coupling is typically larger than the mechanical (actuator center) coupling. Effective modulation index comparisons are established between Gaussian and piston mirrors, for piston and Gaussian phasing error components. It is shown how a potential secondary maxima lock-up condition (the 2 problem) can occur when operating hill-climbing adaptive optical systems with deformable mirrors. In some cases a major loss in system Strehl ratio may result.

© 1977 Optical Society of America

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References

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  1. T. R. O’Meara, “The multidither principle in adaptive optics,” J. Opt. Soc. Am. 67, 306–315 (1977) (this issue).
    [Crossref]
  2. W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
    [Crossref] [PubMed]
  3. J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 13, 611 (1976).
    [Crossref]
  4. J. E. Pearson and S. Hansen, “Experimental studies of a deformable-mirror adaptive optical system,” J. Opt. Soc. Am. 67, 325–333 (1977) (following paper).
    [Crossref]
  5. J. E. Pearson and et al., “High Power Closed Loop, Adaptive System (HICLAS) Study,” Contract N60921-76-C-0008, June1976, available from DDC or HSWC.
  6. Exact in the sense that a near central actuator on a very large mirror is assumed.
  7. Dave Kocher, Lincoln Laboratory (private communication).
  8. Servo ambiguities analogous to the 2Nπ states of interest here can also occur for “modal” systems like Zernike polynomial control.
  9. This conclusion was independently reached by Dave Kocher of Lincoln Laboratory, who also showed that the relative width and center-center alignment of dither mirror and corrector mirror influence functions is important for the 2Nπ problem.

1977 (2)

1976 (1)

J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 13, 611 (1976).
[Crossref]

1974 (1)

Bridges, W. B.

Brown, W. P.

Brunner, P. T.

Hansen, S.

Kocher, Dave

Dave Kocher, Lincoln Laboratory (private communication).

Lazzara, S. P.

Nussmeier, T. A.

O’Meara, T. R.

Pearson, J. E.

J. E. Pearson and S. Hansen, “Experimental studies of a deformable-mirror adaptive optical system,” J. Opt. Soc. Am. 67, 325–333 (1977) (following paper).
[Crossref]

J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 13, 611 (1976).
[Crossref]

J. E. Pearson and et al., “High Power Closed Loop, Adaptive System (HICLAS) Study,” Contract N60921-76-C-0008, June1976, available from DDC or HSWC.

Pedinoff, M. E.

J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 13, 611 (1976).
[Crossref]

Sanguinet, J. A.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Other (5)

J. E. Pearson and et al., “High Power Closed Loop, Adaptive System (HICLAS) Study,” Contract N60921-76-C-0008, June1976, available from DDC or HSWC.

Exact in the sense that a near central actuator on a very large mirror is assumed.

Dave Kocher, Lincoln Laboratory (private communication).

Servo ambiguities analogous to the 2Nπ states of interest here can also occur for “modal” systems like Zernike polynomial control.

This conclusion was independently reached by Dave Kocher of Lincoln Laboratory, who also showed that the relative width and center-center alignment of dither mirror and corrector mirror influence functions is important for the 2Nπ problem.

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Figures (5)

FIG. 1
FIG. 1

Outgoing-wave multidither system operating with a deformable mirror. Such a system operates as a hillclimbing servo which drives the received power to a (local) maximum with respect to each of the actuator settings.

FIG. 2
FIG. 2

A comparison of three types of influence functions. All three functions have been employed in our analysis and computer explorations of the 2 problem.

FIG. 3
FIG. 3

Normalized on-axis irradiance as a function of a central single-actuator displacement. (a) Two-dimensional Gaussian influence function (curve c of Fig. 1). (b) Two-dimensional cosine-squared influence function (curve b of Fig. 1). (c) Two-dimensional boundary-value matched influence function (curve a of Fig. 1).

FIG. 4
FIG. 4

An initial deformation state of a deformable mirror. (a) With deformation to a one-dimensional secondary maximum; (b) with an additional phase perturbation via a single actuator.

FIG. 5
FIG. 5

Mirror geometry for 2 investigation. (a) Five actuators displaced as a row to the 2π state; (b) five actuators displaced as a central group to the 2π state.

Tables (3)

Tables Icon

TABLE I Relative modulation index for piston Gaussian influence functions.

Tables Icon

TABLE II Single-actuator 2 convergence (computer simulation) studies.

Tables Icon

TABLE III Blocks of actuators, 2Nπ studies.

Equations (43)

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Φ a - ϕ a I N ¯ a ( x - x a , y - y a ) ,
Φ = a = 1 N a ϕ a I N ¯ a ,
I N ¯ a = I N ¯ ( x - x a , y - y a ) .
U 0 ( ϕ 1 , , ϕ n ) = exp ( a = 1 N a ϕ a D ϕ a ) U 0 | ϕ a = 0 ,
ϕ 1 = ϕ a = ϕ n = 0.
U 0 ( ϕ 1 , , ϕ n ) = mirror d x d y exp ( i ϕ a I N ¯ a ) .
U 0 ( ϕ 1 , , ϕ n ) = [ 1 + a = 1 N a ϕ a D ϕ a + 1 2 ( ϕ a 2 D ϕ a 2 + 2 ϕ a ϕ b D ϕ a D ϕ a ) ] U 0 .
U 0 ( ϕ 1 , , ϕ n ) = U 0 ( 0 ) + i K ϕ a d x d y I N ¯ a - 1 2 K ϕ a 2 d x d y ( I N ¯ a ) 2 - K a a b b ϕ a ϕ b d x d y I N ¯ a I N ¯ b + ,
U 0 ( 0 ) K π R m 2 ,
I N ¯ a = exp - ( ( x - x a ) 2 + ( y - y a ) 2 ( r 0 ) 2 )
mirror d x d y [ I N ¯ a ] n = π r 0 2 / n .
d x d y I N ¯ a I N ¯ b = exp - [ 1 2 ( S γ 0 ) 2 ] d x d y ( I N ¯ a ) 2 = π r 0 2 2 exp - [ 1 2 ( S r 0 ) ] 2 ,
U 0 ( ϕ 1 , , ϕ n ) = K π { R m 2 + r 0 2 a = 1 N a S a = r 0 2 2 exp - [ 1 2 ( S r 0 ) 2 ] ϕ a ϕ b } ,
S a = n = 1 ( i ϕ a ) n n ! n = - C i n ( ϕ a ) + i S i ( ϕ a ) ,
C i n ( ϕ a ) = ln ϕ a + γ - C i ( ϕ a ) ,
I = U 0 2 ( K π ) 2 ( { R m 2 - r 0 2 a = 1 N a C i n ( ϕ a ) - r 0 2 2 exp - [ 1 2 ( S r 0 ) 2 ] a b ϕ a ϕ b } 2 + ( r 0 2 a = 1 N a S i ( ϕ a ) ) 2 ) .
I = ( K π ) 2 { [ R m 2 - r 0 2 C i n ( ϕ a ) ] 2 + r 0 4 S i 2 ( ϕ a ) } = ( K π ) 2 [ R m 4 - 2 r 0 2 R m 2 C i n ( ϕ a ) + r 0 4 C i n 2 ( ϕ a ) + r 0 4 S i 2 ( ϕ a ) ] .
I = ( K π ) 2 R m 4 [ 1 - 2 ( r 0 R m ) 2 C i n ( ϕ a ) ] .
C i n = C i n ϕ a = 1 - cos ϕ a ϕ a = sin ( ϕ a 2 ) sinc ( ϕ a 2 ) ,
C i n = 2 C i n ϕ a 2 = ϕ a sin ϕ a + cos ϕ a - 1 ϕ a 2 .
ϕ a 1 rad ,
I ( ϕ a ) = ( K π ) 2 { R m 2 - r 0 2 a = 1 N a C i n ( ϕ a ) - r 0 2 2 exp - [ 1 2 ( S r 0 ) 2 ] a = 1 N a b = 1 N a ϕ a ϕ b } 2 .
ϕ a = ϕ a e + ψ 0 cos ω m t
I ( ϕ a e + ψ 0 cos ω m t ) = I ( ϕ a e ) + I ϕ a | ϕ a = ϕ a e ψ 0 cos ω m t + 1 2 2 I ϕ a 2 | ϕ a = ϕ a e ψ 0 2 cos 2 ( ω m t )
I ϕ a | ϕ a = ϕ a e = - 2 ( K π ) I r 0 2 × ( C i n ( ϕ a e ) - e - [ 1 / 2 ( s / r 0 ) 2 ] 2 surrounding actuators ϕ b ) ,
2 I ϕ a 2 | ϕ a = ϕ a e = - 2 ( K π ) I r 0 2 C i n ( ϕ a e ) .
S 1 a = - 2 ψ 0 ( I e I ) 1 / 2 [ sin ( ϕ a e 2 ) sinc ( ϕ a e 2 ) + e - [ 1 / 2 ( s / r 0 ) 2 ] 2 a = 1 N s ϕ b e ] ,
S 2 a = - 2 ψ 0 ( I e I ) 1 / 2 ( ϕ a e sin ϕ a e + cos ϕ a e - 1 ϕ a e 2 ) ,
I e = ( K π r 0 2 ) 2
S 1 a = - 2 ψ 0 ( I e I ) 1 / 2 { ϕ a e 2 + 3 ϕ b exp - [ 1 2 ( S r 0 ) 2 ] } ,
C = exp - [ 1 2 ( S r 0 ) 2 ] .
Φ a = Φ m a ( x , y ) + Φ d a ( x , y ) ψ 0 sin ω a t ,
Φ m a ( x , y ) = ϕ a I N ¯ m a ( x , y )
Φ d a ( x , y ) = I N ¯ d a ( x , y ) ,
I = U 0 2 = K 2 [ ( 1 - Φ a 2 2 ) d x d y ] 2 + K 2 ( Φ a d x d y ) 2 K 2 [ A m 2 - A m mirror Φ a 2 d x d y + ( mirror Φ a d x d y ) 2 ] ,
Φ a d x d y A e Φ a ,
A m A e .
U 0 2 K 2 ( A m 2 - A m mirror Φ a 2 d x d y ) .
U 0 2 = K 2 A m 2 [ 1 - 1 A m ( Φ m a 2 + ψ 0 2 Φ d a 2 2 ) d x d y - 2 ψ 0 A m sin ω a t Φ m a Φ d a d x d y ] .
M a = 2 ψ 0 Φ a A m mirror I N ¯ m a ( x , y ) I N ¯ d a ( x , y ) d x d y .
M = ( π S 2 / 2 A m ) ϕ a ψ 0 ,
M = ( π S 2 / A m ) ϕ a ψ 0 .
M = 2 ϕ a A m 0 0 r 0 d θ r d r exp - ( r S ) 2 = π S 2 A m ( 0.442 ) ϕ a ψ 0 .