The calculation of image assessment critiera, e.g., the Strehl ratio, the point spread function, or the optical transfer function, involves the evaluation of an integral where the integrand is highly oscillatory over a large range of integration. Prefaced with a brief description of the well-known numerical quadrature methods adopted for the purpose, this paper presents a new quadrature technique that obviates the need for knowledge of derivatives of the argument of the exponential integrand. Some illustrative numerical results are presented.

You do not have subscription access to this journal. Citation lists with outbound citation links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an OSA 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 OSA 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 OSA 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 OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Article level metrics are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Strehl Ratios for β_{60} in 0(0.5) 4.0 with N = 1a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.0443

0.8260

0.9646

1.0

0.9648

1.0

0.0088

0.4422

0.8622

1.0

0.8659

1.5

0.0002

0.1196

0.7044

1.0

0.7208

2.0

0.0032

0.0022

0.5102

1.0

0.5537

2.5

0.0010

0.0232

0.3064

1.0

0.3886

3.0

0.0002

0.0472

0.1276

1.0

0.2451

3.5

0.0011

0.0268

0.0156

1.0

0.1343

4.0

0.0002

0.0022

0.0204

1.0

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.

Table II

Strehl Ratios for β_{60} in 0(0.5) 4.0 with N = 2a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.6068

0.9501

0.9647

0.9961

0.9648

1.0

0.2192

0.8109

0.8632

0.9845

0.8659

1.5

0.2025

0.6113

0.7087

0.9653

0.7208

2.0

0.1891

0.3920

0.5218

0.9388

0.5537

2.5

0.1421

0.1958

0.3286

0.9055

0.3886

3.0

0.1521

0.0586

0.1582

0.8658

0.2451

3.5

0.1309

0.0014

0.0400

0.8205

0.1343

4.0

0.0671

0.0262

0.0002

0.7702

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.

Table III

Strehl Ratios for β_{60} In 0(0.5) 4.0 with N = 4a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.9355

0.9638

0.9647

0.9888

0.9648

1.0

0.7701

0.8613

0.8630

0.9557

0.8659

1.5

0.5686

0.7098

0.7080

0.9025

0.7208

2.0

0.3904

0.5333

0.5198

0.8322

0.5537

2.5

0.2631

0.3582

0.3246

0.7483

0.3886

3.0

0.1837

0.2070

0.1522

0.6553

0.2451

3.5

0.1354

0.0950

0.0343

0.5577

0.1343

4.0

0.1016

0.0276

0.0018

0.4600

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.

Table IV

Strehl Ratios for β_{60} in 0(0.5) 4.0 with N = 8a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.9629

0.9647

0.9648

0.9719

0.9648

1.0

0.8588

0.8651

0.8654

0.8911

0.8659

1.5

0.7070

0.7180

0.7186

0.7670

0.7208

2.0

0.5337

0.5470

0.5484

0.6142

0.5537

2.5

0.3650

0.3768

0.3798

0.4502

0.3886

3.0

0.2213

0.2284

0.2339

0.2931

0.2451

3.5

0.1138

0.1153

0.1240

0.1597

0.1343

4.0

0.0449

0.0424

0.0533

0.0627

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.

Table V

Strehl Ratios for β_{60} in 0(0.5) 4.0 with N = 16a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.9647

0.9648

0.9648

0.9667

0.9648

1.0

0.8654

0.8658

0.8659

0.8722

0.8659

1.5

0.7200

0.7203

0.7207

0.7315

0.7208

2.0

0.5516

0.5524

0.5534

0.5661

0.5537

2.5

0.3857

0.3864

0.3883

0.3990

0.3886

3.0

0.2414

0.2420

0.2448

0.2504

0.2451

3.5

0.1304

0.1307

0.1344

0.1340

0.1343

4.0

0.0561

0.0561

0.0600

0.0556

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.

Tables (5)

Table I

Strehl Ratios for β_{60} in 0(0.5) 4.0 with N = 1a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.0443

0.8260

0.9646

1.0

0.9648

1.0

0.0088

0.4422

0.8622

1.0

0.8659

1.5

0.0002

0.1196

0.7044

1.0

0.7208

2.0

0.0032

0.0022

0.5102

1.0

0.5537

2.5

0.0010

0.0232

0.3064

1.0

0.3886

3.0

0.0002

0.0472

0.1276

1.0

0.2451

3.5

0.0011

0.0268

0.0156

1.0

0.1343

4.0

0.0002

0.0022

0.0204

1.0

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.

Table II

Strehl Ratios for β_{60} in 0(0.5) 4.0 with N = 2a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.6068

0.9501

0.9647

0.9961

0.9648

1.0

0.2192

0.8109

0.8632

0.9845

0.8659

1.5

0.2025

0.6113

0.7087

0.9653

0.7208

2.0

0.1891

0.3920

0.5218

0.9388

0.5537

2.5

0.1421

0.1958

0.3286

0.9055

0.3886

3.0

0.1521

0.0586

0.1582

0.8658

0.2451

3.5

0.1309

0.0014

0.0400

0.8205

0.1343

4.0

0.0671

0.0262

0.0002

0.7702

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.

Table III

Strehl Ratios for β_{60} In 0(0.5) 4.0 with N = 4a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.9355

0.9638

0.9647

0.9888

0.9648

1.0

0.7701

0.8613

0.8630

0.9557

0.8659

1.5

0.5686

0.7098

0.7080

0.9025

0.7208

2.0

0.3904

0.5333

0.5198

0.8322

0.5537

2.5

0.2631

0.3582

0.3246

0.7483

0.3886

3.0

0.1837

0.2070

0.1522

0.6553

0.2451

3.5

0.1354

0.0950

0.0343

0.5577

0.1343

4.0

0.1016

0.0276

0.0018

0.4600

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.

Table IV

Strehl Ratios for β_{60} in 0(0.5) 4.0 with N = 8a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.9629

0.9647

0.9648

0.9719

0.9648

1.0

0.8588

0.8651

0.8654

0.8911

0.8659

1.5

0.7070

0.7180

0.7186

0.7670

0.7208

2.0

0.5337

0.5470

0.5484

0.6142

0.5537

2.5

0.3650

0.3768

0.3798

0.4502

0.3886

3.0

0.2213

0.2284

0.2339

0.2931

0.2451

3.5

0.1138

0.1153

0.1240

0.1597

0.1343

4.0

0.0449

0.0424

0.0533

0.0627

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.

Table V

Strehl Ratios for β_{60} in 0(0.5) 4.0 with N = 16a

β_{60}

A

B

C

D

E

0

1.0

1.0

1.0

1.0

1.0

0.5

0.9647

0.9648

0.9648

0.9667

0.9648

1.0

0.8654

0.8658

0.8659

0.8722

0.8659

1.5

0.7200

0.7203

0.7207

0.7315

0.7208

2.0

0.5516

0.5524

0.5534

0.5661

0.5537

2.5

0.3857

0.3864

0.3883

0.3990

0.3886

3.0

0.2414

0.2420

0.2448

0.2504

0.2451

3.5

0.1304

0.1307

0.1344

0.1340

0.1343

4.0

0.0561

0.0561

0.0600

0.0556

0.0594

Column A, values obtained by quadrature formula (22) using numerical differentiation; column B, values obtained by quadrature formula (22) using analytical differentiation; column C, values obtained by quadrature formula (26); column D, values obtained by quadrature formula (27); column E, exact values.