Claudio Rivolta, "Intensity and resolution at the diffraction focus of optical systems with rotationally symmetric aberration functions," Appl. Opt. 31, 4444-4452 (1992)

The intensities and resolutions at the diffraction focus of optical systems with rotationally symmetric minimum-variance aberration functions as well as functions with slight variations from this shape and a prescribed fixed maximum deviation value at the edge of the exit pupil are calculated for circular and annular exit pupils. The results for the intensity at the diffraction focus obtained either by numerical integration of the diffraction integral or by using the Maréchal formula show that the maxima of these intensities are achieved for optical systems with wave aberration functions of minimum variance, thus verifying the Maréchal criterion. The resolutions (first zeros of the intensities in the radial direction) at this diffraction focus are also presented.

José Antonio Díaz and Virendra N. Mahajan Appl. Opt. 52(6) 1136-1147 (2013)

References

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Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

3.752

−9.337

5.835

0.1894

88.485

88.180

1.7845

100

0.2

3.748

−9.155

5.657

0.1554

82.762

82.549

2.6949

92.16

0.3

3.735

−8.867

5.382

0.1174

76.019

75.898

3.7377

82.81

0.4

3.677

−8.333

4.906

0.0782

66.778

66.734

4.7851

70.56

0.5

3.505

−7.449

4.194

0.0428

54.814

54.806

5.3001

56.25

0.6

3.180

−6.243

3.313

0.0171

40.637

40.637

4.8903

40.96

0.707

2.705

−4.821

2.367

0.0241

24.986

24.986

3.5139

25.02

The W_{2}_{n} coefficients of the spherical aberration polynomials are the values that minimize the variance for W(1) = λ/4. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 2

Intensity (I_{NI}) at the Diffraction Focus (U_{DE}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

7.505

−18.675

11.670

0.2340

60.241

57.179

2.0939

100

0.2

7.496

−18.310

11.314

0.1576

59.175

56.892

2.5282

92.16

0.3

7.470

−17.733

10.763

0.1019

58.374

56.973

3.2571

82.81

0.4

7.354

−16.665

9.811

0.0738

56.449

55.898

4.2341

70.56

0.5

7.009

−14.898

8.388

0.0527

50.693

50.585

4.9597

56.25

0.6

6.360

−12.486

6.626

0.0273

39.683

39.675

4.7847

40.96

0.707

5.409

−9.642

4.733

0.0069

24.898

24.898

3.5020

25.02

The W_{2}_{n} coefficients of the spherical aberration polynomials are the values that minimize the variance for W(1) = λ/2. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{Fs}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 3

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

11.257

−28.012

17.505

18.4146

34.251

20.374

0.4683

100

0.2

11.244

−27.465

16.971

−0.3502

32.268

24.711

2.5786

92.16

0.3

11.205

−26.601

16.146

−0.3245

36.635

31.444

2.7610

82.81

0.4

11.031

−24.999

14.718

−0.1523

42.267

39.965

3.4975

70.56

0.5

10.515

−22.347

12.582

−0.0103

44.421

43.926

4.4433

56.25

0.6

9.540

−18.729

9.939

0.0235

38.137

38.098

4.6136

40.96

0.707

8.115

−14.463

7.101

0.0701

24.751

24.750

3.4822

25.02

The W_{2}_{n} coefficients of the spherical aberration polynomials are the values that minimize the variance for W(1) = 3λ/4. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 4

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

2.752

−8.337

5.835

−1.2733

82.510

78.892

1.3347

100

0.2

2.748

−8.155

5.657

−1.1756

77.247

74.519

2.2854

92.16

0.3

2.735

−7.867

5.382

−0.9901

71.086

69.380

3.3744

82.81

0.4

2.677

−7.333

4.906

−0.6143

62.723

62.063

4.4672

70.56

0.5

2.505

−6.449

4.194

0.0425

51.921

51.759

5.0331

56.25

0.6

2.180

−5.243

3.313

1.0944

39.033

38.565

4.7089

40.96

0.707

1.705

−3.821

2.367

2.8395

24.449

23.410

3.4412

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 1 are, respectively, ΔW_{4} = −1 and ΔW_{6} = +1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 5

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

4.752

−10.337

5.835

1.4409

83.278

78.862

2.4720

100

0.2

4.748

−10.155

5.657

1.2761

77.655

74.536

3.1492

92.16

0.3

4.735

−9.867

5.382

1.0270

71.202

69.411

3.9369

82.81

0.4

4.677

−9.333

4.906

0.6076

62.628

62.045

4.7292

70.56

0.5

4.505

−8.449

4.194

−0.0648

51.821

51.753

5.1078

56.25

0.6

4.180

−7.243

3.313

−1.1116

38.998

38.573

4.7196

40.96

0.707

3.705

−5.821

2.367

−2.8049

24.451

23.448

3.4424

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 1 are, respectively, ΔW_{4} = +1 and ΔW_{6} = −1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 6

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

6.505

−17.675

11.670

−1.6084

57.789

49.777

1.6102

100

0.2

6.496

−17.310

11.314

−1.5102

56.588

50.252

2.1024

92.16

0.3

6.470

−16.733

10.763

−1.2716

55.563

51.340

2.8849

82.81

0.4

6.354

−15.665

9.811

−0.7875

53.548

51.644

3.9196

70.56

0.5

6.009

−13.898

8.388

−0.0470

48.205

47.670

4.6998

56.25

0.6

5.360

−11.486

6.626

1.0726

38.154

37.625

4.6059

40.96

0.707

4.409

−8.642

4.733

2.8149

24.365

23.339

3.4295

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 2 are, respectively, ΔW_{4} = −1 and ΔW_{6} = +1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 7

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

8.505

−19.675

11.670

1.5790

56.561

49.699

2.9635

100

0.2

8.496

−19.310

11.314

1.3310

55.437

50.282

3.1706

92.16

0.3

8.470

−18.733

10.763

1.0195

54.656

51.378

3.6513

82.81

0.4

8.354

−17.665

9.811

0.5752

52.948

51.601

4.3252

70.56

0.5

8.009

−15.898

8.388

−0.0892

47.928

47.643

4.8297

56.25

0.6

7.360

−13.486

6.626

−1.1224

38.077

37.640

4.6257

40.96

0.707

6.409

−10.642

4.733

−2.8283

24.363

23.348

3.4310

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 2 are, respectively, ΔW_{4} = +1 and ΔW_{6} = −1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 8

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

10.257

−27.012

17.505

−2.7092

31.598

16.066

2.9402

100

0.2

10.244

−26.465

16.971

−2.6241

33.659

20.400

2.5550

92.16

0.3

10.205

−25.601

16.146

−2.1793

36.917

27.281

2.6485

82.81

0.4

10.031

−23.999

14.718

−1.2925

41.132

36.382

3.3034

70.56

0.5

9.515

−21.347

12.582

−0.2112

42.561

41.209

4.2260

56.25

0.6

8.540

−17.729

9.939

1.0312

36.721

36.086

4.4429

40.96

0.707

7.115

−13.463

7.101

2.8701

24.219

23.145

3.4093

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 3 are, respectively, ΔW_{4} = −1 and ΔW_{6} = +1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 9

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

12.257

−29.012

17.505

19.4870

37.994

16.008

0.3794

100

0.2

12.244

−28.465

16.971

20.1466

33.810

20.428

0.1442

92.16

0.3

12.205

−27.601

16.146

0.6607

33.787

27.340

3.1294

82.81

0.4

12.031

−25.999

14.718

0.3410

39.499

36.342

3.6751

70.56

0.5

11.515

−23.347

12.582

−0.1796

41.983

41.192

4.3761

56.25

0.6

10.540

−19.729

9.939

−1.1449

36.593

36.108

4.4695

40.96

0.707

9.115

−15.463

7.101

−2.7708

24.226

23.258

3.4130

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 3 are, respectively, ΔW_{4} = +1 and ΔW_{6} = −1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 10

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

A

0

4.758

−11.840

7.399

0.2237

82.032

81.357

0.317

0.2

5.112

−12.488

7.716

0.1727

75.312

74.703

0.341

0.3

5.707

−13.548

8.223

0.1373

67.686

67.144

0.382

0.4

7.001

−15.865

9.340

0.0808

57.660

57.200

0.476

0.5

9.730

−20.679

11.643

0.0125

45.974

45.603

0.694

0.6

15.964

−31.340

16.631

−0.0646

33.485

33.209

1.255

0.707

35.168

−62.690

30.774

−0.1440

20.458

20.285

3.251

The W_{2}_{n} polynomial coefficients were calculated for the edge polynomial values of W(1) = Aλ listed in the last column to obtain the constant variance value of
$\sqrt{V}=0.071$ for each obscuration ratio, i.e., its upper permissible limit for calculating the intensity with the Maréchal formula.

Table 11

Resolution (V_{R}) and Intensity (I_{R}) at the Diffraction Focus Plane (u = u_{DF}) for Optical Systems with the Spherical Aberration Coefficients Listed in Tables 1, 4, and 5a

The second column lists the resolution at the paraxial image plane for a diffraction-limited (DL) system.

Table 12

Resolution (V_{R}) and Intensity (I_{R}) at the Diffraction Focus Plane (U = U_{DF}) for Optical Systems with the Spherical Aberration Coefficients Listed in Tables 2, 6, and 7a

The second column lists the resolution at the paraxial image plane for a diffraction-limited (DL) system.

Table 13

Resolution (V_{R}) and Intensity (I_{R}) at the Diffraction Focus Plane (U = U_{DF}) for Optical Systems with the Spherical Aberration Coefficients Listed in Tables 3, 8, and 9a

The second column lists the resolution at the paraxial image plane for a diffraction-limited (DL) system.

Tables (13)

Table 1

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

3.752

−9.337

5.835

0.1894

88.485

88.180

1.7845

100

0.2

3.748

−9.155

5.657

0.1554

82.762

82.549

2.6949

92.16

0.3

3.735

−8.867

5.382

0.1174

76.019

75.898

3.7377

82.81

0.4

3.677

−8.333

4.906

0.0782

66.778

66.734

4.7851

70.56

0.5

3.505

−7.449

4.194

0.0428

54.814

54.806

5.3001

56.25

0.6

3.180

−6.243

3.313

0.0171

40.637

40.637

4.8903

40.96

0.707

2.705

−4.821

2.367

0.0241

24.986

24.986

3.5139

25.02

The W_{2}_{n} coefficients of the spherical aberration polynomials are the values that minimize the variance for W(1) = λ/4. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 2

Intensity (I_{NI}) at the Diffraction Focus (U_{DE}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

7.505

−18.675

11.670

0.2340

60.241

57.179

2.0939

100

0.2

7.496

−18.310

11.314

0.1576

59.175

56.892

2.5282

92.16

0.3

7.470

−17.733

10.763

0.1019

58.374

56.973

3.2571

82.81

0.4

7.354

−16.665

9.811

0.0738

56.449

55.898

4.2341

70.56

0.5

7.009

−14.898

8.388

0.0527

50.693

50.585

4.9597

56.25

0.6

6.360

−12.486

6.626

0.0273

39.683

39.675

4.7847

40.96

0.707

5.409

−9.642

4.733

0.0069

24.898

24.898

3.5020

25.02

The W_{2}_{n} coefficients of the spherical aberration polynomials are the values that minimize the variance for W(1) = λ/2. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{Fs}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 3

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

11.257

−28.012

17.505

18.4146

34.251

20.374

0.4683

100

0.2

11.244

−27.465

16.971

−0.3502

32.268

24.711

2.5786

92.16

0.3

11.205

−26.601

16.146

−0.3245

36.635

31.444

2.7610

82.81

0.4

11.031

−24.999

14.718

−0.1523

42.267

39.965

3.4975

70.56

0.5

10.515

−22.347

12.582

−0.0103

44.421

43.926

4.4433

56.25

0.6

9.540

−18.729

9.939

0.0235

38.137

38.098

4.6136

40.96

0.707

8.115

−14.463

7.101

0.0701

24.751

24.750

3.4822

25.02

The W_{2}_{n} coefficients of the spherical aberration polynomials are the values that minimize the variance for W(1) = 3λ/4. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 4

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

2.752

−8.337

5.835

−1.2733

82.510

78.892

1.3347

100

0.2

2.748

−8.155

5.657

−1.1756

77.247

74.519

2.2854

92.16

0.3

2.735

−7.867

5.382

−0.9901

71.086

69.380

3.3744

82.81

0.4

2.677

−7.333

4.906

−0.6143

62.723

62.063

4.4672

70.56

0.5

2.505

−6.449

4.194

0.0425

51.921

51.759

5.0331

56.25

0.6

2.180

−5.243

3.313

1.0944

39.033

38.565

4.7089

40.96

0.707

1.705

−3.821

2.367

2.8395

24.449

23.410

3.4412

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 1 are, respectively, ΔW_{4} = −1 and ΔW_{6} = +1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 5

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

4.752

−10.337

5.835

1.4409

83.278

78.862

2.4720

100

0.2

4.748

−10.155

5.657

1.2761

77.655

74.536

3.1492

92.16

0.3

4.735

−9.867

5.382

1.0270

71.202

69.411

3.9369

82.81

0.4

4.677

−9.333

4.906

0.6076

62.628

62.045

4.7292

70.56

0.5

4.505

−8.449

4.194

−0.0648

51.821

51.753

5.1078

56.25

0.6

4.180

−7.243

3.313

−1.1116

38.998

38.573

4.7196

40.96

0.707

3.705

−5.821

2.367

−2.8049

24.451

23.448

3.4424

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 1 are, respectively, ΔW_{4} = +1 and ΔW_{6} = −1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 6

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

6.505

−17.675

11.670

−1.6084

57.789

49.777

1.6102

100

0.2

6.496

−17.310

11.314

−1.5102

56.588

50.252

2.1024

92.16

0.3

6.470

−16.733

10.763

−1.2716

55.563

51.340

2.8849

82.81

0.4

6.354

−15.665

9.811

−0.7875

53.548

51.644

3.9196

70.56

0.5

6.009

−13.898

8.388

−0.0470

48.205

47.670

4.6998

56.25

0.6

5.360

−11.486

6.626

1.0726

38.154

37.625

4.6059

40.96

0.707

4.409

−8.642

4.733

2.8149

24.365

23.339

3.4295

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 2 are, respectively, ΔW_{4} = −1 and ΔW_{6} = +1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 7

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

8.505

−19.675

11.670

1.5790

56.561

49.699

2.9635

100

0.2

8.496

−19.310

11.314

1.3310

55.437

50.282

3.1706

92.16

0.3

8.470

−18.733

10.763

1.0195

54.656

51.378

3.6513

82.81

0.4

8.354

−17.665

9.811

0.5752

52.948

51.601

4.3252

70.56

0.5

8.009

−15.898

8.388

−0.0892

47.928

47.643

4.8297

56.25

0.6

7.360

−13.486

6.626

−1.1224

38.077

37.640

4.6257

40.96

0.707

6.409

−10.642

4.733

−2.8283

24.363

23.348

3.4310

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 2 are, respectively, ΔW_{4} = +1 and ΔW_{6} = −1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 8

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

10.257

−27.012

17.505

−2.7092

31.598

16.066

2.9402

100

0.2

10.244

−26.465

16.971

−2.6241

33.659

20.400

2.5550

92.16

0.3

10.205

−25.601

16.146

−2.1793

36.917

27.281

2.6485

82.81

0.4

10.031

−23.999

14.718

−1.2925

41.132

36.382

3.3034

70.56

0.5

9.515

−21.347

12.582

−0.2112

42.561

41.209

4.2260

56.25

0.6

8.540

−17.729

9.939

1.0312

36.721

36.086

4.4429

40.96

0.707

7.115

−13.463

7.101

2.8701

24.219

23.145

3.4093

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 3 are, respectively, ΔW_{4} = −1 and ΔW_{6} = +1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 9

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

I_{FS}

I_{DL}

0

12.257

−29.012

17.505

19.4870

37.994

16.008

0.3794

100

0.2

12.244

−28.465

16.971

20.1466

33.810

20.428

0.1442

92.16

0.3

12.205

−27.601

16.146

0.6607

33.787

27.340

3.1294

82.81

0.4

12.031

−25.999

14.718

0.3410

39.499

36.342

3.6751

70.56

0.5

11.515

−23.347

12.582

−0.1796

41.983

41.192

4.3761

56.25

0.6

10.540

−19.729

9.939

−1.1449

36.593

36.108

4.4695

40.96

0.707

9.115

−15.463

7.101

−2.7708

24.226

23.258

3.4130

25.02

The W_{4} and W_{6} coefficients with respect to the coefficients in Table 3 are, respectively, ΔW_{4} = +1 and ΔW_{6} = −1. The last two columns list the intensities at the first secondary maximum in the diffraction focus plane (I_{FS}) and at the paraxial image focus for a diffraction-limited system (I_{DL}).

Table 10

Intensity (I_{NI}) at the Diffraction Focus (U_{DF}) Calculated with a Numerical Integration of the Diffraction Integral and Using the Approximate Maréchal Formula (I_{MF})a

∊

W_{4}

W_{6}

W_{8}

u_{DF}

I_{NI}

I_{MF}

A

0

4.758

−11.840

7.399

0.2237

82.032

81.357

0.317

0.2

5.112

−12.488

7.716

0.1727

75.312

74.703

0.341

0.3

5.707

−13.548

8.223

0.1373

67.686

67.144

0.382

0.4

7.001

−15.865

9.340

0.0808

57.660

57.200

0.476

0.5

9.730

−20.679

11.643

0.0125

45.974

45.603

0.694

0.6

15.964

−31.340

16.631

−0.0646

33.485

33.209

1.255

0.707

35.168

−62.690

30.774

−0.1440

20.458

20.285

3.251

The W_{2}_{n} polynomial coefficients were calculated for the edge polynomial values of W(1) = Aλ listed in the last column to obtain the constant variance value of
$\sqrt{V}=0.071$ for each obscuration ratio, i.e., its upper permissible limit for calculating the intensity with the Maréchal formula.

Table 11

Resolution (V_{R}) and Intensity (I_{R}) at the Diffraction Focus Plane (u = u_{DF}) for Optical Systems with the Spherical Aberration Coefficients Listed in Tables 1, 4, and 5a

The second column lists the resolution at the paraxial image plane for a diffraction-limited (DL) system.

Table 12

Resolution (V_{R}) and Intensity (I_{R}) at the Diffraction Focus Plane (U = U_{DF}) for Optical Systems with the Spherical Aberration Coefficients Listed in Tables 2, 6, and 7a

The second column lists the resolution at the paraxial image plane for a diffraction-limited (DL) system.

Table 13

Resolution (V_{R}) and Intensity (I_{R}) at the Diffraction Focus Plane (U = U_{DF}) for Optical Systems with the Spherical Aberration Coefficients Listed in Tables 3, 8, and 9a