Virendra N. Mahajan, "Strehl ratio for primary aberrations: some analytical results for circular and annular pupils," J. Opt. Soc. Am. 72, 1258-1266 (1982)
Imaging systems with circular and annular pupils aberrated by primary aberrations are considered. Both classical and balanced (Zernike) aberrations are discussed. Closed-form solutions are derived for the Strehl ratio, except in the case of coma, for which the integral form is used. Numerical results are obtained and compared with Maréchal’s formula for small aberrations. It is shown that, as long as the Strehl ratio is greater than 0.6, the Maréchal formula gives its value with an error of less than 10%. A discussion of the Rayleigh quarter-wave rule is given, and it is shown that it provides only a qualitative measure of aberration tolerance. Nonoptimally balanced aberrations are also considered, and it is shown that, unless the Strehl ratio is quite high, an optimally balanced aberration does not necessarily give a maximum Strehl ratio.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Ai is the coefficient of the ith abberation (measured in radians) and ∊ is the obscuration ratio of an annular pupil. ∊ ≤ ρ ≤ 1, 0 < θ ≤ 2π,
, H(a) = [J02(a) + J12(a)]1/2, α(a) = tan−1([J1(a)/J0(a)].
Table 2
Aberration Coefficient, Absolute Peak Value, and Peak-to-Peak Value for Primary Aberrations (∊ = 0)
Aberration
Aberration Coefficient
Absolute Peak Value |Wp|
Peak-to-Peak Value Wp−p
Spherical
As
|As|
As
Balanced spherical
As
|As|/4
As/4
Coma
Ac
|Ac|
2Ac
Balanced coma
Ac
|Aa|/3
2Ac/3
Astigmatism
Aa
|Aa|
Aa
Balanced astigmatism
Aa
|Aa|/2
Aa
Table 3
Strehl Ratio for a Quarter-Wave Absolute Peak Value of a Primary Aberration (|Wp| = λ/4)a
Aberration
Ai(λ)
S
Spherical
0.25
0.8003
Balanced spherical
1
0.8003
Coma
0.25
0.7374
Balanced coma
0.75
0.7317
Astigmatism
0.25
0.8572
Balanced astigmatism
0.5
0.6602
The corresponding aberration coefficient Ai is given in units of wavelength (∊ = 0).
Table 4
Strehl Ratio for a Quarter-Wave Peak-to-Peak Value of a Primary Aberration (Wp−p = λ/4)a
Aberration
Ai (λ)
S
Spherical
0.25
0.8003
Balanced spherical
1
0.8003
Coma
0.125
0.92
Balanced coma
0.375
0.92
Astigmatism
0.25
0.8572
Balanced astigmatism
0.25
0.9021
The corresponding aberration coefficient Ai is given in units of wavelength (∊ = 0).
Table 5
Aberration Coefficient Ai, Absolute Peak Value |Wp|, and Peak-to-Peak Value Wp−p, All in Units of Wavelength, for a Strehl Ratio of 0.80 (∊ = 0)
Aberration
Ai(λ)
|Wp| (λ)
Wp−p (λ)
Spherical
0.25
0.25
0.25
Balanced spherical
1
0.25
0.25
Coma
0.21
0.21
0.42
Balanced coma
0.63
0.21
0.42
Astigmatism
0.30
0.30
0.30
Balanced astigmatism
0.37
0.18
0.37
Table 6
Strehl Ratio for Annular Pupils Aberrated with One Wave of Spherical Aberration Optimally Balanced with Defocus for Annular and Circular Pupils
∊
SA
SB
0
0.8003
0.8003
0.1
0.8074
0.8069
0.2
0.8279
0.8239
0.3
0.8589
0.8407
0.4
0.8957
0.8452
0.5
0.9326
0.8315
0.6
0.9637
0.8082
0.7
0.9852
0.7993
0.8
0.9962
0.8340
0.9
0.9995
0.9240
Tables (6)
Table 1
Standard Deviation and Strehl Ratio for Primary Aberrationsa
Ai is the coefficient of the ith abberation (measured in radians) and ∊ is the obscuration ratio of an annular pupil. ∊ ≤ ρ ≤ 1, 0 < θ ≤ 2π,
, H(a) = [J02(a) + J12(a)]1/2, α(a) = tan−1([J1(a)/J0(a)].
Table 2
Aberration Coefficient, Absolute Peak Value, and Peak-to-Peak Value for Primary Aberrations (∊ = 0)
Aberration
Aberration Coefficient
Absolute Peak Value |Wp|
Peak-to-Peak Value Wp−p
Spherical
As
|As|
As
Balanced spherical
As
|As|/4
As/4
Coma
Ac
|Ac|
2Ac
Balanced coma
Ac
|Aa|/3
2Ac/3
Astigmatism
Aa
|Aa|
Aa
Balanced astigmatism
Aa
|Aa|/2
Aa
Table 3
Strehl Ratio for a Quarter-Wave Absolute Peak Value of a Primary Aberration (|Wp| = λ/4)a
Aberration
Ai(λ)
S
Spherical
0.25
0.8003
Balanced spherical
1
0.8003
Coma
0.25
0.7374
Balanced coma
0.75
0.7317
Astigmatism
0.25
0.8572
Balanced astigmatism
0.5
0.6602
The corresponding aberration coefficient Ai is given in units of wavelength (∊ = 0).
Table 4
Strehl Ratio for a Quarter-Wave Peak-to-Peak Value of a Primary Aberration (Wp−p = λ/4)a
Aberration
Ai (λ)
S
Spherical
0.25
0.8003
Balanced spherical
1
0.8003
Coma
0.125
0.92
Balanced coma
0.375
0.92
Astigmatism
0.25
0.8572
Balanced astigmatism
0.25
0.9021
The corresponding aberration coefficient Ai is given in units of wavelength (∊ = 0).
Table 5
Aberration Coefficient Ai, Absolute Peak Value |Wp|, and Peak-to-Peak Value Wp−p, All in Units of Wavelength, for a Strehl Ratio of 0.80 (∊ = 0)
Aberration
Ai(λ)
|Wp| (λ)
Wp−p (λ)
Spherical
0.25
0.25
0.25
Balanced spherical
1
0.25
0.25
Coma
0.21
0.21
0.42
Balanced coma
0.63
0.21
0.42
Astigmatism
0.30
0.30
0.30
Balanced astigmatism
0.37
0.18
0.37
Table 6
Strehl Ratio for Annular Pupils Aberrated with One Wave of Spherical Aberration Optimally Balanced with Defocus for Annular and Circular Pupils