Optical Design Using Excel E-Book

Hiroshi Nakajima

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 402

Veröffentlichungsjahr: 2015

Beschreibung

A practical introductory guide to optical design covering geometrical optics, simple wave-optics and diffraction, using Excel software * explains practical calculation methods for designing optical systems with fully worked-out examples and avoiding complex mathematical methods * includes practical calculations for ray tracing, laser beam (Gaussian beam) focusing, and diffraction calculations; the ray tracing and the diffraction calculations are done by using the VBA program which Excel provides as a supporting tool * describes basic optical theory and application methods, and provides readers with calculation methods for designing laser optical systems with numerous practical calculation examples. After finishing the book, even inexperienced readers should have the ability to design laser optical systems * covers large areas of geometrical optics and diffraction theory, providing a good overview and reference for beginners or non-specialist engineers * accompanied by a website including password protected electronic files

Leseprobe

Table of Contents

Cover

Title Page

Copyright

About the Author

Preface

OUTLINE OF CONTENTS

Chapter 1: Geometrical Optics

1.1 Characteristics of Lasers

1.2 The Three Fundamental Characteristics of Light Which Form the Basis of Geometrical Optics

1.3 Fermat's Principle

1.4 Principle of Reversibility

1.5 Paraxial Theory Using Thin Lenses

1.6 The Five Seidel Aberrations

1.7 The Sine Condition

1.8 Aplanatic Lenses

1.9 Reflection and Transmission

References

Chapter 2: Examples of Simple Optical Design Using Paraxial Theory

2.1 Types of Lenses

2.2 Applied Calculations for Simple Optical Systems

2.3 Considerations Relating to the Design of Laser Optical Systems

Chapter 3: Ray Tracing Applications of Paraxial Theory

3.1 Deriving the Equations for Ray Tracing Using Paraxial Theory

3.2 Problems of Ray Tracing Calculations Using Paraxial Theory

Chapter 4: Two-Dimensional Ray Tracing

4.1 Ray Tracing for a Spherical Surface

4.2 Ray Tracing for a Plane Surface

4.3 Ray Tracing for an Aspheric Surface (Using VBA Programming)

4.4 Ray Tracing for an Aberration-Free Lens

4.5 Optical Path Length Calculation for an Aberration-Free Lens

4.6 Ray Tracing for an Optical System Which Is Set at a Tilt

4.7 How to Use the Ray Trace Calculation Table

4.8 A Method for Generating a Ray Trace Calculation Table Using a VBA Program

4.9 Sample Ray Tracing Problems

References

Chapter 5: Three-Dimensional Ray Tracing

5.1 Three-Dimensional Ray Tracing for a Spherical Surface

5.2 Three-Dimensional Ray Tracing for a Cylindrical Surface

5.3 Simulation for Two Cylindrical Lenses Which Are Fixed Longitudinally (or Laterally) but Allowed to Rotate Slightly around the Optical Axis

5.4 Three-Dimensional Ray Tracing for a Plane Surface Which Is Perpendicular to the Optical Axis

5.5 Three-Dimensional Ray Tracing for an Aberration-Free Lens

5.6 Three-Dimensional Ray Tracing for a Lens Which Is Set at a Tilt

5.7 How to Use the Three-Dimensional Ray Trace Calculation Table

5.8 Operating Instructions Using the Ray Trace Calculation Table, while Running the VBA Program

5.9 Three-Dimensional Ray Tracing Problems

Reference

Chapter 6: Mathematical Formulae for Describing Wave Motion

6.1 Mathematical Formulae for Describing Wave Motion

6.2 Describing Waves with Complex Exponential Functions

6.3 Problems Relating to Wave Motion

Reference

Chapter 7: Calculations for Focusing Gaussian Beams

7.1 What is a Gaussian Beam?

7.2 Equations for Focusing a Gaussian Beam

7.3 The

M

2

(M Squared) Factor

7.4 Sample Gaussian Beam Focusing Problems

References

Chapter 8: Diffraction: Theory and Calculations

8.1 The Concept of Diffraction

8.2 Diffraction at a Slit Aperture

8.3 Diffraction Calculations Using Numerical Integration

8.4 Diffraction at a Rectangular Aperture

8.5 Diffraction at a Circular Aperture

8.6 Diffraction Wave Generated after the Incident Wave Exits a Focusing Lens

8.7 Diffraction Calculation Problems

References

Chapter 9: Calculations for Gaussian Beam Diffraction

9.1 The Power and the Central Irradiance of a Gaussian Beam

9.2 General Equations for Waves Diffracted by an Aperture

9.3 Diffraction Wave Equations for a Focused Beam

9.4 Diffraction Wave Equations for a Collimated Beam

9.5 Diffraction Calculation Program

9.6 Operating Instructions for the Diffraction Calculation Programs

9.7 Gaussian Beam Diffraction Calculation Problems

References

Appendix A: Paraxial Theory: A Detailed Account

A.1 Derivation of Equation (1.39), Equation (1.40), Equation (1.41), and Equation (1.42) for Obtaining Focal Lengths and Principal Points for a Combination of Two Lenses

A.2 Derivation of Equation (1.34), Equation (1.35), and Equation (1.36) for Obtaining Focal Lengths and Principal Points of a Simple Lens

Reference

Appendix B: Table of Refractive Indices for BK7

Reference

Appendix C: Equations for Plane Waves, Spherical Waves, and Gaussian Beams

C.1 Equations for Plane Waves

C.2 Three-dimensional Differential Equation of Wave Motion

C.3 Equations for Spherical Waves

C.4 Helmholtz's Wave Equation

C.5 Equations for Gaussian beams

C.6 Detailed Derivations of Equation (C.21), Equation (C.34), and Equation (C.38) and Gaussian beam equations [Equation (7.1), Equation (7.2), Equation (7.3), and Equation (7.4)]

References

Appendix D: Numerical Integration Methods

D.1 Trapezoidal Rule

D.2 Simpson's Rule (a Better Approximation)

Reference

Appendix E: Fresnel Diffraction and Fraunhofer Diffraction

E.1 How Fresnel and Fraunhofer Diffraction Patterns Are Generated

E.2 Fundamentals of Diffraction Theory

E.3 Fresnel Diffraction

E.4 Fraunhofer Diffraction

References

Appendix F: Wave-Front Conversion by a Lens

F.1 Wave-Front Conversion by a Lens

F.2 Diffraction Field Equation for a Plane Wave after it Exits a Convex Lens

References

Appendix G: List of Excel Calculation Files on the Companion Web Site

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

Chapter 1: Geometrical Optics

Figure 1.1 Pinhole camera

Figure 1.2 Independent action of light rays

Figure 1.3 (a) Reflection of a light ray and (b) refraction of a light ray

Figure 1.4 (a) Rectilinear propagation and (b) law of reflection

Figure 1.5 (a) Angle of refraction and (b) emergency path of a life saver

Figure 1.6 Optical path length in an ideal imaging system

Figure 1.7 Spherical lens surface

Figure 1.8 Refraction of a light ray by a convex lens

Figure 1.9 Refraction by a lens

Figure 1.10 A method of drawing rays in order to obtain their image point

Figure 1.11 Angular magnification

γ

Figure 1.12 Longitudinal magnification

α

Figure 1.13 Illustration of rays, showing the lens, the object, and its image

Figure 1.14 Focal length and principal point of a simple lens

Figure 1.15 Combination of two simple lenses

Figure 1.16 Spherical aberration

Figure 1.17 Coma aberration

Figure 1.18 Astigmatism

Figure 1.19 Aberration caused by field curvature

Figure 1.20 Distortion. (a) Original, (b) pin-cushion distortion, and (c) barrel distortion

Figure 1.21 Ray aberration

Figure 1.22 Abbe sine condition

Figure 1.23 Sine condition for off-axis object and its image

Figure 1.24 Principal surfaces for an off-axis object and its image

Figure 1.25 Principal surfaces for an off-axis object and its image (collimated beam incidence)

Figure 1.26 Aplanatic sphere and aplanatic lenses. (a) Aplanatic sphere, (b) convex lens, and (c) concave lens

Figure 1.27 Reflection and refraction of light

Figure 1.28 (a) Calculation results for

r

and

t

and (b) calculation results for

R

and

T

Chapter 2: Examples of Simple Optical Design Using Paraxial Theory

Figure 2.1 (a) Plano-convex lens and (b) plano-concave lens

Figure 2.2 Bi-convex lens

Figure 2.3 (a) Positive meniscus lens and (b) negative meniscus lens

Figure 2.4 (a) Cylindrical convex lens and (b) cylindrical concave lens

Figure 2.5 Achromatic lens

Figure 2.6 Aspheric lens. (a) Collimating and (b) focusing

Figure 2.7 An imaging system using a microscope objective lens

Figure 2.8 Numerical aperture

Figure 2.9 A beam expander consisting of a microscope objective lens and a camera lens

Figure 2.10 Entrance pupil of a camera

Figure 2.11 Beam scanning using an

f-θ

lens

Figure 2.12 An example of a Fresnel lens application

Figure 2.13 Rod lens

Figure 2.14 An imaging system using a camera lens

Figure 2.15 An imaging system using a microscope objective lens

Figure 2.16 An imaging system using a microscope objective lens and an imaging lens

Figure 2.17 Insertion of a glass plate into an imaging system

Figure 2.18 Laser diode beam collimation

Figure 2.19 Beam expander using cylindrical lenses

Figure 2.20 Beam converter (oval beam to circular beam)

Figure 2.21 Beam expander using a microscope objective lens and a camera lens

Figure 2.22 Beam expander using a plano-concave lens and a plano-convex lens

Figure 2.23 Interference between a concave surface and a plane surface

Figure 2.24 Interference caused by a plate beam splitter. (a) Parallel plate beam splitter and (b) wedge shape beam splitter

Figure 2.25 Interference between beams reflected by the front and rear surfaces.

Figure 2.26 Interference caused by rays reflected by the first surface

Figure 2.27 Change in the path of a beam reflected by a mirror

Figure 2.28 Beam shift using a prism

Figure 2.29 A beam trap using a horn-shaped hollow piece of glass

Figure 2.30 Side wall of a lens designed to reduce stray light

Figure 2.31 A serrated wall

Chapter 3: Ray Tracing Applications of Paraxial Theory

Figure 3.1 Ray tracing using paraxial theory

Figure 3.2 (a) Optical system A and (b) optical system B

Figure 3.3 (a) Ray tracing for the imaging system A and (b) ray tracing for the imaging system B

Figure 3.4 Laser beam scanning system

Figure 3.5 Ray tracing for a laser scanning system (a) when and (b) when

Chapter 4: Two-Dimensional Ray Tracing

Figure 4.1 Ray tracing for a spherical surface

Figure 4.2 Ray tracing for a plane surface

Figure 4.3 Aspheric lens used for collimating an LD beam

Figure 4.4 Examples of aspheric surfaces. (a) Second surface of an aspherical lens and (b) first surface of an aspherical lens

Figure 4.5 Ray tracing for an aspheric lens. (a) First surface (convex on the left) and (b) second surface (convex on the right)

Figure 4.6 Newton's method

Figure 4.7 Ray tracing for an aberration-free lens

Figure 4.8 Virtual object point P(

z

,

y

) and virtual image point P′(

z′

,

y′

) for an aberration-free lens

Figure 4.9 Optical path length correction for an aberration-free lens

Figure 4.10 Optical path length correction for parallel incident rays and parallel exiting rays. (a) Incident rays are parallel and (b) exiting rays are parallel

Figure 4.11 Ray tracing for a tilted lens. (a) A tilted lens in the incident ray coordinates (expressed in terms of the incident ray coordinate system). (b) A tilted lens in the lens axis coordinates (expressed in terms of the lens axis coordinate system)

Figure 4.12 Calculation table for the two-dimensional ray tracing

Figure 4.13 Calculation results for the two-dimensional ray tracing

Figure 4.14 Calculation results for rays near the image plane. (a) Calculation results and (b) graph

Figure 4.15 Wave-front aberration

Figure 4.16 A calculation example for the phase (difference) on the image plane

Figure 4.17 Wave-front aberration calculation

Figure 4.18 Calculation results for the wave-front aberration

Figure 4.19 Calculation table for the aspheric lens ray tracing

Figure 4.20 Calculation table for the VBA two-dimensional ray tracing

Figure 4.21 A bi-convex lens

Figure 4.22 Beam focusing using a bi-convex lens

Figure 4.23 Ray tracing results for a bi-convex lens

Figure 4.24 A plano-convex lens

Figure 4.25 Beam focusing using a plano-convex lens

Figure 4.26 Ray tracing results for a plano-convex lens

Figure 4.27 A plano-convex lens

Figure 4.28 Beam focusing using a plano-convex lens

Figure 4.29 Ray tracing results for a plano-convex lens

Figure 4.30 Two planar plates

Figure 4.31 Ray tracing results for two planar plates

Figure 4.32 An achromatic lens

Figure 4.33 Combination of two lenses

Figure 4.34 Beam focusing using an achromatic lens

Figure 4.35 Ray tracing results for an achromatic lens

Figure 4.36 An aplanatic lens system

Figure 4.37 An aplanatic lens system (a combination of two lenses)

Figure 4.38 Beam focusing using an aplanatic lens system

Figure 4.39 Ray tracing results for an aplanatic lens system

Figure 4.40 A beam expander

Figure 4.41 Ray tracing results for a beam expander

Figure 4.42 A laser optical system

Figure 4.43 Ray tracing results for a laser optical system

Figure 4.44 An imaging system using a plano-convex lens

Figure 4.45 Ray tracing results for an imaging system (a plano-convex lens)

Figure 4.46 Specifications of the two achromatic lenses

Figure 4.47 Combination of two lenses

Figure 4.48 Imaging using two achromatic lenses

Figure 4.49 Ray tracing results for an imaging system (two achromatic lenses)

Figure 4.50 Beam focusing using an aspheric lens

Figure 4.51 Ray tracing results for an aspheric lens

Figure 4.52 A laser collimator using an aspheric lens

Figure 4.53 Ray tracing results for a laser collimator using an aspheric lens

Figure 4.54 An achromatic lens for focusing a scanned beam

Figure 4.55 A laser beam scanning system

Figure 4.56 Ray tracing results for a laser beam scanning system calculated directly on Excel cells

Figure 4.57 Ray tracing results for a laser beam scanning system calculated by the VBA program

Figure 4.58 How to prevent scanning line variation in the

y

-direction. (The Figure lies on the

y–z

plane)

Figure 4.59 Beam focusing using an achromatic lens set at a tilt

Figure 4.60 Ray tracing results for an achromatic lens set at a tilt

Chapter 5: Three-Dimensional Ray Tracing

Figure 5.1 Ray refraction on a boundary surface

Figure 5.2 Three-dimensional expression of incident angle

θ

a

and refraction angle

θ

b

Figure 5.3 Cylindrical surface

Figure 5.4 (a) Cylindrical lens (rotated by

φ

) and (b) incident beam (rotated by −

φ

)

Figure 5.5 Rotation of a ray

Figure 5.6 Ray tracing for a plane surface medium

Figure 5.7 Incident and refracted ray vector on a plane boundary surface

Figure 5.8 Vector representation of ray slopes. (a) Incident ray and refracted ray projected onto the lens pupil, (b) vector representation of the refracted ray slope, and (c) vector representation of the refracted ray slope (general case)

Figure 5.9 Ray tracing for an aberration-free lens

Figure 5.10 Virtual object point and virtual image point for an aberration-free lens

Figure 5.11 (a) Incident ray slope vector and (b) incident ray height at the lens

Figure 5.12 Parallel incident rays onto an aberration-free lens (cross-sectional view in the direction of )

Figure 5.13 (a) Exiting ray slope vector and (b) exiting ray height at the lens

Figure 5.14 Parallel beam exiting from an aberration-free lens (cross-sectional view in the direction of )

Figure 5.16 (a) A tilted lens (represented using incident ray coordinate system) and (b) a tilted lens (represented using lens axis coordinate system)

Figure 5.15 (a) Lens tilt orientation

φ

and (b) lens tilt orientation (for rays passing through the lens)

Figure 5.17 (a) Lens tilt orientation = 0° and (b) lens tilt orientation (restore the beam orientation to the original setting)

Figure 5.18 Rotation of the ray vector around the

x

-axis (counterclockwise through an angle −

α

)

Figure 5.19 A calculation table for the three-dimensional ray tracing

Figure 5.20 Calculation example for spot diagrams on the image plane

Figure 5.21 Calculation example for a phase (difference) on the image plane

Figure 5.23 Calculation example for a wave-front aberration

Figure 5.22 Wave-front aberration calculation

Figure 5.24 A calculation table for the three-dimensional ray tracing

Figure 5.25 (a) Beam focusing using a plano-convex lens and (b) object height

Figure 5.26 Ray tracing results for a plano-convex lens

Figure 5.27 (a) Beam focusing using an achromatic lens and (b) object height

Figure 5.28 Ray tracing results for an achromatic lens

Figure 5.29 (a) Beam expander using two cylindrical lenses and (b) object height

Figure 5.30 Ray tracing results for a beam expander

Figure 5.31 (a) Beam expander using two misaligned cylindrical lenses and (b) object height

Figure 5.32 Ray tracing results for a misaligned beam expander

Figure 5.33 (a) Laser beam collimator, expander, and focusing and (b) incident ray slope

Figure 5.34 Ray tracing results for a laser optical system

Figure 5.35 (a) Laser beam expander and focusing and (b) incident ray height

Figure 5.36 Principal points of a simple lens and the combination of two lenses

Figure 5.37 Ray tracing results for a beam expander and focusing

Figure 5.38 (a) Beam focusing using an achromatic lens which is set at a tilt and (b) object height

Figure 5.39 Ray tracing results for a beam expander and focusing

Chapter 6: Mathematical Formulae for Describing Wave Motion

Figure 6.1 (a) Wave on a rope. (b) Moving reference coordinates

Figure 6.2 (a) Progression of wave . (b) Wave at

Figure 6.3 How

U

varies with

x

and

t

Figure 6.4 Argand diagram

Figure 6.5 (a) Superposition of light waves. (b) Irradiance after superposition

Figure 6.6 (a) Superposing waves by the method of vector addition. (b) Irradiance after superposition

Figure 6.7 (a) Superposition of light waves (phase difference π/2). (b) Superposition of light waves (phase difference π). (c) Irradiance after superposition

Chapter 7: Calculations for Focusing Gaussian Beams

Figure 7.1 Gaussian beam

Figure 7.2 Electric field and irradiance distribution of a Gaussian beam

Figure 7.3 Wave-front sphere

Figure 7.4 Gaussian beam divergence

Figure 7.5 Wave-front curvature radius of a Gaussian beam

Figure 7.6 Beam waist distance

Figure 7.7 Gaussian beam focusing using a lens

Figure 7.8 Conceptual relationship between real and ideal beams. (a) If the real beam is everywhere larger than the ideal beam by a factor of

M,

then the divergence angle will also be larger by the same factor. (b) If both beams have the same divergence angle, but the real beam has a larger waist, then it will be larger by a factor of

M

2

. (c) If both beams have the same size waist, but the real beam diverges faster, then it will diverge faster by a factor of

M

2

Figure 7.9 Focusing a parallel beam

Figure 7.10 (a) Incident beam radius and (b) focused beam radius

Figure 7.11 (a) Configuration A and (b) configuration B

Figure 7.12 Radius of the exiting beam

Figure 7.13 (a) Beam radius and (b) wave-front curvature radius of the exiting beam

Figure 7.14 Configuration of the laser system

Figure 7.15 Calculation for the Gaussian beam radius exiting the collimator

Figure 7.16 Configuration of the laser system

Figure 7.17 Generating a line spot using a beam expander

Figure 7.18 Generating a line spot without using a high magnification ratio expander

Figure 7.19 Two Gaussian beams whose

M

2

factors are 1 and 1.3

Figure 7.20 (a) Beam radii and (b) wave-front curvature radii of two Gaussian beams with and

Chapter 8: Diffraction: Theory and Calculations

Figure 8.1 Diffraction wave generated by a small aperture on the surface of a body of water

Figure 8.2 Huygens–Fresnel principle

Figure 8.3 Diffraction at a slit

Figure 8.4 Diffraction from a slit aperture

Figure 8.5 Vector addition in an Argand diagram

Figure 8.6 Diffraction at a rectangular aperture

Figure 8.7 Diffraction at a circular aperture

Figure 8.8 Polar coordinates on the aperture plane

Figure 8.9 (a) Diffraction by an aperture with no lens. (b) Diffraction by an aperture with a focusing lens

Figure 8.10 Diffraction caused by a slit

Figure 8.11 Diffraction irradiance caused by a slit

Figure 8.12 Diffraction irradiance by a slit (calculated by numerical integration)

Figure 8.13 Diffraction caused by a rectangular aperture

Figure 8.14 Diffraction irradiance caused by a rectangular aperture. (a) Three-dimensional irradiance. (b) Irradiance along

X

-axis. (c) Irradiance along

Y

-axis

Figure 8.15 Bessel functions

Figure 8.16 (a) Diffraction caused by a circular aperture and (b) diffraction pattern after exiting a focusing lens

Figure 8.17 Diffraction irradiance caused by a circular aperture. (a) With no lens. (b) With a focusing lens

Chapter 9: Calculations for Gaussian Beam Diffraction

Figure 9.1 Power of a Gaussian beam

Figure 9.2 Power of a Gaussian beam vs. power of a circular beam with uniform irradiance

Figure 9.3 Electric field and irradiance

Figure 9.4 Circular beam and elliptical beam

Figure 9.5 (a) Elliptical Gaussian beam. (b) Irradiance distribution of an elliptical Gaussian beam

Figure 9.6 (a) A one-dimensional Gaussian beam. (The beam maintains a constant value in the

y

-direction.) (b) Irradiance of a one-dimensional Gaussian beam

Figure 9.7 Diffraction wave generated at an aperture

Figure 9.8 Diffraction wave for focusing a beam

Figure 9.9 Diffraction wave for focusing a beam on a defocused plane

Figure 9.10 Coordinate conversion to polar coordinates. (a) Aperture plane (

x

,

y

) =>

h

∠

ψ

. (b) Observation plane (

X

,

Y

) =>

ρ

∠

ϕ

Figure 9.11 Diffraction wave for collimating a beam

Figure 9.12 Diffraction wave for a collimated beam with a defocused setting

Figure 9.13 Complex number representation

Figure 9.14 A calculation table for the two-dimensional R-θ diffraction calculation program

Figure 9.15 A calculation table for the one-dimensional diffraction calculation program

Figure 9.16 An input table for the two-dimensional X-Y diffraction calculation program

Figure 9.17 An output table for the two-dimensional X-Y diffraction calculation program

Figure 9.18 Irradiance distribution of a Gaussian beam ()

Figure 9.19 Focusing a Gaussian beam

Figure 9.20 Settings on the two-dimensional R-θ diffraction calculation table

Figure 9.21 (a) Diffraction field and (b) irradiance of a spot after focusing

Figure 9.22 Focusing a Gaussian beam (by varying the incident beam radius)

Figure 9.23 Focused spot irradiance which is maximized when . (a) Aperture/beam size ratio vs. central irradiance. (b) Focused spot irradiance when

a

/

w

=1.12. (c) Normalized irradiance of the focused spot when

a

/

w

=1.12

Figure 9.24 Focusing a truncated/untruncated Gaussian beam

Figure 9.25 Focused spot irradiance for (a) a truncated and (b) an untruncated Gaussian beam

Figure 9.26 Focusing an elliptical Gaussian beam

Figure 9.27 Focused spot irradiance for an elliptical Gaussian beam. (a) Focused spot irradiance when

w

a

= 0.76 mm,

w

b

= 0.6333 mm. (b) Focused spot irradiance when

w

a

= 1.9 mm,

w

b

= 0.6333 mm

Figure 9.28 Focusing an elliptical Gaussian beam (with a semicircular aperture)

Figure 9.29 Focused spot irradiance after passing through a semicircular aperture.

Figure 9.30 Focusing a uniform irradiance beam (with a ring aperture)

Figure 9.31 Focused spot irradiance after passing through a ring/full-circle aperture. (a) Ring aperture. (b) Full-circle aperture. (c) Ring/full-circle aperture (along

X

-axis)

Figure 9.32 Double diffraction for a truncated Gaussian beam

Figure 9.33 Settings on the two-dimensional R-θ diffraction calculation table

Figure 9.34 Calculated irradiances at planes A, B, and C. (a) Planes A and B. (b) Plane C

Figure 9.35 Focusing a one-dimensional Gaussian beam

Figure 9.36 Settings on the one-dimensional diffraction calculation table

Figure 9.37 Diffraction irradiance for a one-dimensional Gaussian beam

Figure 9.38 Focusing a two-dimensional Gaussian beam

Figure 9.39 Two-dimensional diffraction irradiance combining

x

- and

y

- one-dimensional diffraction. (a) Combining

x

- and

y

- one-dimensional diffraction. (b) Calculated by

X-Y

two-dimensional diffraction. (c) (a)−(b)

Figure 9.40 Laser optical system consisting of a beam expander and a focusing lens

Figure 9.41 Settings on the two-dimensional VBA ray tracing calculation table

Figure 9.42 Ray tracing results with wave-front aberration. (a) Ray tracing results. (b) Wave-front aberration

Figure 9.43 Settings on the two-dimensional R-θ diffraction calculation table

Figure 9.44 Diffraction irradiance is obtained by shifting the image plane. (a) Image plane shift vs. central irradiance. (b) Irradiance calculated by diffraction and Gaussian beam calculation

Figure 9.45 Collimating an elliptical Gaussian beam

Figure 9.46 Diffraction irradiance after collimation. (a) Irradiance depicted in three dimensions. (b) Normalized irradiance along the

X

-axis. (c) Normalized irradiance along the

Y

-axis

Appendix A: Paraxial Theory: A Detailed Account

Figure A.1 Combination of two lenses

Figure A.2 A simple lens, viewed as a combination of two surfaces

Figure A.3 Relationship between the two focal lengths

Figure A.4 Imaging by a single surface sphere

Appendix C: Equations for Plane Waves, Spherical Waves, and Gaussian Beams

Figure C.1 A plane wave propagating along the

k

-axis. (a) Propagating along the

z

-axis. (b) Propagating along an arbitrary direction in three-dimensional space

Figure C.2 Vector

k

in

x–y–z

space

Figure C.3 The geometry of spherical coordinates

Figure C.4 Amplitude of a spherical wave

Figure C.5 Cylindrical coordinate system

Figure C.6 Coordinates of a sphere

Appendix D: Numerical Integration Methods

Figure D.1 Trapezoidal rule

Figure D.2 Simpson's rule

Appendix E: Fresnel Diffraction and Fraunhofer Diffraction

Figure E.1 Diffraction irradiance patterns observed at various distances from the aperture

Figure E.2 Diffraction from an aperture

Figure E.3 Inclination factor

Appendix F: Wave-Front Conversion by a Lens

Figure F.1 Wave-front conversion by a convex lens

Figure F.2 Ray height (

x

,

y

)

List of Tables

Chapter 1: Geometrical Optics

Table 1.1 Calculations for

r

,

t

,

R,

and

T

Chapter 4: Two-Dimensional Ray Tracing

Table 4.1 Example values for the aspheric surface coefficients

Chapter 9: Calculations for Gaussian Beam Diffraction

Table 9.1 Electric field and transmission coefficients at the aperture

Table 9.2 Settings for the sheet name and cell address (row number and column number)

OPTICAL DESIGN USING EXCEL®

PRACTICAL CALCULATIONS FOR LASER OPTICAL SYSTEMS

Hiroshi Nakajima

This edition first published 2015

© 2015 John Wiley & Sons Singapore Pte. Ltd.

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About the Author

Hiroshi Nakajima graduated from Doshisha University (Kyoto, Japan) with a degree in electronics. From 1967 to 2002, he worked for Hitachi Electronics Engineering Co., Ltd, as an engineer. He was responsible for developing automated manufacturing systems for a variety of electrical products, including the following: automatic adjustment systems for color picture tubes (CPTs), optical inspection systems for the silicon wafer surfaces of integrated circuits (ICs), and optical inspection systems for hard disk (HD) surfaces. From 2003 to 2012, he worked for Hitachi Electronics Engineering Co. and Hitachi High-Technologies Corporation as a technical consultant. During this time, he was responsible for developing optical inspection systems for HD surfaces. The author's specialty is developing machines which are capable of detecting microscopic surface defects (less than 0.1 µm in diameter) using laser optical systems. This book is largely based on his practical experience of working with laser optical systems.

Preface

Allow me to introduce myself. My name is Hiroshi Nakajima, and I am an electronic and optical engineer in Japan. During my job training, I had to study optics. It was tougher than I had anticipated. It took me a great deal of effort to acquire an understanding of optics and the various technologies that it employs. While I was studying optics, I always wondered if there was an easier way to master the subject. That is what motivated me to write this book.

There are lots of excellent books about theoretical and applied optics. However, there are very few books on the market that explain the practical calculation methods used for designing optical systems, with the aid of fully worked-out examples. This book is intended for the non-specialist reader, whose grasp of the mathematical complexities of optical theory is somewhat limited. For ease of comprehension, the technical explanations have been made as clear and simple as possible. The book provides a detailed explanation of the design of optical systems, starting from the basic theoretical principles and then moving on to practical, real-life examples, with supporting calculations included. By the time the reader finishes this book on optics, he/she should have a fairly complete understanding of the subject.

One of the distinguishing features of this book is that it contains a detailed treatment of optical laser systems, which generate a laser-focused beam and then use it to irradiate an object, thereby obtaining images through processes such as reflection, refraction, and diffraction. From a theoretical standpoint, light emitted from a laser is the simplest kind of light. Starting with lasers is an ideal way of acquiring an understanding of general optics, with a minimum of effort.

This book employs various calculation methods to describe the focusing of a laser beam on a spot. After exploring the basic principles of geometrical optics and simple optical systems, the reader will be provided with multiple worked examples of two- and three-dimensional ray tracing techniques. The more complex phenomena of interference and diffraction require a mathematical grasp of diffraction theory, which treats light as a wave. Next, we shall derive the equations for Gaussian beam focusing, a very simple and practical method for performing laser optical calculations. Finally, the reader will be introduced to some basic techniques that make use of diffraction calculations in order to solve applied problems relating to the wave motion of light.

Some of the calculations in this book (ray trace calculations and Gaussian beam focusing calculations) are performed directly on Excel spreadsheets, which makes them easy to understand, while others (ray trace calculations and diffraction calculations) employ VBA programming, where users can perform complex calculations by inputting simple settings. Readers can either choose to perform the ray trace calculations directly on Excel spreadsheets, or use VBA programs instead. Additionally, all of the calculations have been recorded on the Excel files available on the book's companion web site. I sincerely hope that my book will help readers obtain an understanding of what is needed to design optical systems – especially laser systems.

Lastly, I would like to thank Dr Vincent J. Torley for patiently proofreading my English.

Hiroshi Nakajima

OUTLINE OF CONTENTS

Chapter 1 – Geometrical Optics

In this chapter, the reader is introduced to the basic principles of geometrical optics. The fundamental characteristics of light are discussed, including Fermat's principle. Starting from these basic principles, we can derive the principles of paraxial theory, the five Seidel aberrations and the sine condition, which together form the basis of geometrical optics.

Chapter 2 – Examples of Simple Optical Design Using Paraxial Theory

The basic principles for designing optical systems are discussed in this chapter, for various types of lenses. Examples of simple optical systems are also provided. Additionally, various technical considerations relating to the design of laser optical systems are discussed.

Chapter 3 – Ray Tracing Applications of Paraxial Theory

In this brief chapter, we examine a method for performing ray tracings, using paraxial theory. Problems with practical applications of this method are also included.

Chapter 4 – Two-Dimensional Ray Tracing

Here, we discuss a method for performing ray tracing calculations in two dimensions, using the laws of refraction. We show how ray tracing calculations can be performed for rays traveling from the object to the image, for a variety of optical systems. We can calculate the ray aberration occurring on the image plane, as well as the wave-front aberration of an optical system. As this book is intended to give the reader an understanding of the practical applications of optical systems, this chapter includes fully worked solutions to real-life problems relating to several different types of lenses, including biconvex lenses, plano-convex (concave) lenses, cylindrical lenses, meniscus lenses, achromatic lenses, and aspheric lenses. We can also use aberration-free lenses in ray tracing, as a substitute for lenses whose specifications we do not know in detail.

Chapter 5 – Three-Dimensional Ray Tracing

Using the method of ray tracing discussed in Chapter 4, we can expand the number of dimensions from two to three, and thereby obtain a method for performing three-dimensional ray tracing calculations. Using this method, we can then generate spot diagrams on the image plane. Two- and three-dimensional ray tracing calculations can be performed on Excel cells directly, or on VBA programs. Support is provided for the reader, whether he/she chooses Excel or VBA.

Chapter 6 – Mathematical Formulae for Describing Wave Motion

Chapters 1–5 relate to geometrical optics. However, geometrical optics alone cannot provide us with a complete analysis of the behavior of light in all situations. In particular, it does not tell us how to design optical systems relating to the phenomena of interference and diffraction. In these cases, we need a mathematical theory which enables us to deal with light as a wave. In this chapter, we study the fundamental characteristics of light, the wave equations, and the mathematical expressions used to describe the amplitude and phase of waves, including Argand diagrams. (For the benefit of those readers who may be interested in further study, a detailed derivation of the equations for plane waves, spherical waves, and Gaussian beams is contained in the Appendices.)

Chapter 7 – Calculations for Focusing Gaussian Beams

By combining the lens formula in Chapter 1 and the Gaussian beam equations in this chapter, we can derive focusing equations for a Gaussian beam. In this chapter, we discuss the characteristics of Gaussian beams and derive the focusing equations for a Gaussian beam. The chapter also contains a treatment of the M-squared factor, which can be used to calculate the focused spot size of an actual laser beam whose beam quality is a little worse than the ideal Gaussian beam because of being superimposed with higher order lateral modes.

Chapter 8 – Diffraction: Theory and Calculations

In this chapter, we examine the theory and calculation methods of diffraction for various apertures, including a slit, a rectangular aperture, and a circular aperture.

Chapter 9 – Calculations for Gaussian Beam Diffraction

Here, we discuss methods for calculating diffraction irradiance, especially for Gaussian beams, as they are the laser beams which are generally used in real-life situations. In this book, the diffraction calculations are performed by numerical integration methods using the VBA program. The VBA program includes one-dimensional diffraction calculations (for the case of diffraction from a slit), two-dimensional R-θ diffraction calculations (diffraction from a circular aperture) and two-dimensional x-y diffraction calculations (diffraction from an aperture with an arbitrary shape). The numerous examples at the end of the chapter provide the reader with solutions to practical problems with real-life applications, which the reader is likely to encounter in an everyday context.

Appendices

During the preparation of this book, some of the more detailed theoretical explanations were edited from the text, for the sake of ease of comprehension. However, some of these detailed explanations have been included in the Appendices for the benefit of those seeking a deeper understanding of optics, and for anyone who may be interested in undertaking further studies in the future.

Chapter 1Geometrical Optics

1.1 Characteristics of Lasers

This book is about optical calculation methods and the principles for applying these methods to actual optical devices. Most of these devices use lasers, so we will begin by briefly examining the characteristics of lasers. The following exposition will be especially beneficial for readers whose understanding of lasers is rather limited.

The term LASER is an acronym for “Light Amplification by Stimulated Emission of Radiation.” Lasers have special characteristics that distinguish them from most light-emitting devices: a narrow, low-divergence beam (sharp directivity), and a very narrow wavelength spectrum (monochromaticity). These features of lasers make them ideally suited for the generation of high intensity beams.

The history of lasers goes back about half a century to 1960, when the first working laser was demonstrated. Lasers are now widely used in a variety of fields, including optical storage (e.g., CD drives and DVD drives), fiber-optic communication, manufacturing (especially for cutting, bending, welding, and marking materials), scientific measurement, and medicine. This diversity of application is due to the following four properties of lasers, which give them a commercial and scientific edge over other light-emitting devices.

1.

Monochromaticity

: Radiation that has a very narrow frequency band (or wavelength band) is said to possess the property of

monochromaticity

. Because the band is so narrow, the radiation can be regarded as having a single frequency (or alternatively, a single wavelength). Laser light typically has a very narrow frequency band (or wavelength band), which makes it an ideal example of the property of monochromaticity. Sunlight, by contrast, has a very broad band spectrum, with multiple frequencies and wavelengths.

2.

Beam directivity

: The term

directivity

, when used in relation to lasers, refers to the directional properties of the electromagnetic radiation they emit. A laser beam exhibits a very sharp (narrow) directivity, allowing it to propagate in a straight line with almost no expansion. By contrast, normal light sources such as flashlights and car headlights have broader directivity than lasers, so their beams cannot travel as far.

3.

Coherence

: If split laser beams which were emitted from the same source are superimposed, a fringe pattern will appear. This fringe pattern is never observed in an isolated beam. We refer to this phenomenon as

interference

caused by the wave characteristics of light. If these two beams traveling onto the same plane are superimposed while they are

in phase

(with their crests and troughs lined up), the resultant beam will appear brighter, but if these waves are superimposed with a 180° phase difference (i.e., if crests and troughs are superimposed), the beam will appear dark and the resultant amplitude will be zero. A laser can easily generate interference patterns because of its

coherence

(uniformity of phase). Sunlight cannot readily generate interference patterns, due to its incoherence: because its coherence time and coherence length are very short, and the phases of the superimposed waves will not come into phase very easily. Sunlight will only exhibit coherence over a very short interval both of time and space.

4.

High concentration of energy (high intensity)

: A sheet of paper can be burnt simply by focusing sunlight on it, using a convex lens. A laser is much more concentrated: it can even weld two pieces of steel together. This is not merely due to the high power of the laser, but also because of the extremely high

intensity

of the laser beam, where the light energy is narrowly focused. It is relatively easy to concentrate a laser beam on a small target. A high intensity beam can be generated very easily using a laser.

All of these characteristics of lasers can best be summed up by the word “coherent.” A laser is both temporally coherent and spatially coherent. What this means is that a laser has a uniform phase over time at an arbitrary point in space, and it also has a uniform phase in space at an arbitrary point in time. Thus a laser has a uniform phase both in time and in space. A laser radiates a single-wavelength (more precisely, a very narrow wavelength spectrum) beam with a constant phase and it can propagate in a specific direction. An electric light, by contrast, radiates a multitude of different wavelengths at various phases and in all directions. Thus it is both spatially and temporally incoherent.

1.2 The Three Fundamental Characteristics of Light Which Form the Basis of Geometrical Optics

To understand geometrical optics, which is the most basic form of optics, we need to first study the fundamental characteristics of light – including light emitted by lasers. The following properties of light are confirmed by everyday experience:

1.

Light rays travel in straight lines in a uniform medium.

2.

Light rays are independent of one another.

3.

Light rays can be reflected and refracted: they change their direction at a boundary between different media, in accordance with the laws of reflection and refraction.

The whole science of geometrical optics can be derived from these three characteristics of light.

1.2.1 Light Rays Travel in Straight Lines

Many common optical phenomena attest to the fact that light rays travel in straight lines. For instance, when the sun is shining outdoors, a tree casts a shadow whose shape is identical with its own. Without using any lenses, we can construct a pinhole camera that can capture the image of an object, simply by making a pinhole in one of the walls of a black box as shown in Figure 1.1.

Figure 1.1 Pinhole camera

1.2.2 Light Rays Act Independently of One Another

Figure 1.2 illustrates the independent action of light rays using the example of three spotlights whose light is of different colors: red, blue, and green. When we irradiate the same area on a white sheet of paper with these three spotlights, we perceive the light as “white.” However, if we replace the paper with a mirror, and then project the reflected light onto another sheet of paper, the three separate beams of light reappear in their original colors of red, blue, and green. The fact that these three beams reappear in their original colors demonstrates that light rays coming from different sources act independently of one another after being reflected by the mirror. Likewise, the fact that the light from the three spotlights appears white when they are all focused on the same area of paper can be explained in terms of the constituent light wavelengths (red, blue, and green) reaching our eyes and acting on our retinas independently. It is the superposition of waves which causes us to perceive them as white.

Figure 1.2 Independent action of light rays

1.2.3 Reflection of Light Rays

As shown in Figure 1.3