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.

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A_{i} is the coefficient of the ith abberation (measured in radians) and ∊ is the obscuration ratio of an annular pupil. ∊ ≤ ρ ≤ 1, 0 < θ ≤ 2π,
$C(\text{a})={\int}_{0}^{a}\text{cos}(\pi {x}^{2}/2)\text{d}x,S(a)={\int}_{0}^{a}\text{sin}(\pi {x}^{2}/2)\text{d}x$, H(a) = [J_{0}^{2}(a) + J_{1}^{2}(a)]^{1/2}, α(a) = tan^{−1}([J_{1}(a)/J_{0}(a)].

Table 2

Aberration Coefficient, Absolute Peak Value, and Peak-to-Peak Value for Primary Aberrations (∊ = 0)

Aberration

Aberration Coefficient

Absolute Peak Value |W_{p}|

Peak-to-Peak Value W_{p}_{−}_{p}

Spherical

A_{s}

|A_{s}|

A_{s}

Balanced spherical

A_{s}

|A_{s}|/4

A_{s}/4

Coma

A_{c}

|A_{c}|

2A_{c}

Balanced coma

A_{c}

|A_{a}|/3

2A_{c}/3

Astigmatism

A_{a}

|A_{a}|

A_{a}

Balanced astigmatism

A_{a}

|A_{a}|/2

A_{a}

Table 3

Strehl Ratio for a Quarter-Wave Absolute Peak Value of a Primary Aberration (|W_{p}| = λ/4)^{a}

Aberration

A_{i}(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 A_{i} is given in units of wavelength (∊ = 0).

Table 4

Strehl Ratio for a Quarter-Wave Peak-to-Peak Value of a Primary Aberration (W_{p}_{−}_{p} = λ/4)^{a}

Aberration

A_{i} (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 A_{i} is given in units of wavelength (∊ = 0).

Table 5

Aberration Coefficient A_{i}, Absolute Peak Value |W_{p}|, and Peak-to-Peak Value W_{p}_{−}_{p}, All in Units of Wavelength, for a Strehl Ratio of 0.80 (∊ = 0)

Aberration

A_{i}(λ)

|W_{p}| (λ)

W_{p}_{−}_{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

∊

S_{A}

S_{B}

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 Aberrations^{a}

A_{i} is the coefficient of the ith abberation (measured in radians) and ∊ is the obscuration ratio of an annular pupil. ∊ ≤ ρ ≤ 1, 0 < θ ≤ 2π,
$C(\text{a})={\int}_{0}^{a}\text{cos}(\pi {x}^{2}/2)\text{d}x,S(a)={\int}_{0}^{a}\text{sin}(\pi {x}^{2}/2)\text{d}x$, H(a) = [J_{0}^{2}(a) + J_{1}^{2}(a)]^{1/2}, α(a) = tan^{−1}([J_{1}(a)/J_{0}(a)].

Table 2

Aberration Coefficient, Absolute Peak Value, and Peak-to-Peak Value for Primary Aberrations (∊ = 0)

Aberration

Aberration Coefficient

Absolute Peak Value |W_{p}|

Peak-to-Peak Value W_{p}_{−}_{p}

Spherical

A_{s}

|A_{s}|

A_{s}

Balanced spherical

A_{s}

|A_{s}|/4

A_{s}/4

Coma

A_{c}

|A_{c}|

2A_{c}

Balanced coma

A_{c}

|A_{a}|/3

2A_{c}/3

Astigmatism

A_{a}

|A_{a}|

A_{a}

Balanced astigmatism

A_{a}

|A_{a}|/2

A_{a}

Table 3

Strehl Ratio for a Quarter-Wave Absolute Peak Value of a Primary Aberration (|W_{p}| = λ/4)^{a}

Aberration

A_{i}(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 A_{i} is given in units of wavelength (∊ = 0).

Table 4

Strehl Ratio for a Quarter-Wave Peak-to-Peak Value of a Primary Aberration (W_{p}_{−}_{p} = λ/4)^{a}

Aberration

A_{i} (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 A_{i} is given in units of wavelength (∊ = 0).

Table 5

Aberration Coefficient A_{i}, Absolute Peak Value |W_{p}|, and Peak-to-Peak Value W_{p}_{−}_{p}, All in Units of Wavelength, for a Strehl Ratio of 0.80 (∊ = 0)

Aberration

A_{i}(λ)

|W_{p}| (λ)

W_{p}_{−}_{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