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Laser Modeling

A Numerical Approach with Algebra and Calculus

## Preview

## Book Description

Offering a fresh take on laser engineering, **Laser Modeling: A Numerical Approach with Algebra and Calculus **presents algebraic models and traditional calculus-based methods in tandem to make concepts easier to digest and apply in the real world. Each technique is introduced alongside a practical, solved example based on a commercial laser. Assuming some knowledge of the nature of light, emission of radiation, and basic atomic physics, the text:

- Explains how to formulate an accurate gain threshold equation as well as determine small-signal gain
- Discusses gain saturation and introduces a novel pass-by-pass model for rapid implementation of "what if?" scenarios
- Outlines the calculus-based Rigrod approach in a simplified manner to aid in comprehension
- Considers thermal effects on solid-state lasers and other lasers with new and efficient quasi-three-level materials
- Demonstrates how the convolution method is used to predict the effect of temperature drift on a DPSS system
- Describes the technique and technology of Q-switching and provides a simple model for predicting output power
- Addresses non-linear optics and supplies a simple model for calculating optimal crystal length
- Examines common laser systems, answering basic design questions and summarizing parameters
- Includes downloadable Microsoft
^{®}Excel^{™}spreadsheets, allowing models to be customized for specific lasers

Don’t let the mathematical rigor of solutions get in the way of understanding the concepts.** Laser Modeling: A Numerical Approach with Algebra and Calculus **covers laser theory in an accessible way that can be applied immediately, and numerically, to real laser systems.

## Table of Contents

**Basic Laser Processes**The Laser and Laser Light

Atomic Processes of the Laser

Example: Emission of Thermal Light

Three- and Four-Level Schemes

Example: Achieving Inversion in a Three-Level Laser

Rate Equations

Level Lifetime

Example: Lifetime of HeNe Energy Levels

Laser Gain

Example: Gain in a HeNe Amplifier

Losses in a Laser

Cavity Optics

Example: Stability of a HeNe Cavity

Optical Characteristics (Longitudinal and Transverse Modes)

**Threshold Gain**

Gain and Loss: Achieving Lasing

The Gain Threshold Equation

Example: Threshold Gain of a HeNe Laser

Example: Threshold Gain of a Non-Uniformly Pumped Ruby Laser

Example: Handling Distributed Losses

The Tale of Two Gains: g0 and gth

Application of gth: Determining g0

Example: Determining the Gain of a HeNe Laser

Example: Determining the Gain of a YAG Laser

An Atomic View of Gain: Cross-Section

Example: Calculating the Cross-Section of Transitions

Applications of the Gain Threshold Equation: Designing Laser Optics

Example: Calculating Minimum Reflectivity

Example: Calculating Cavity Optic Reflectivities

Example: Polarization in a HeNe Laser

A Theoretical Prediction of Pumping Threshold

Example: Minimum Pump Power of a YAG Laser

Example: Minimum Pump Power of a Diode Laser

**Gain Saturation**

Gain is Not Constant

A Third Gain Figure: Saturated Gain

Saturation Intensity

Example: Calculating the Saturation Power of a HeNe Transition

Saturated Gain and Intra-Cavity Power

Slope Efficiency

Predicting Output Power

Example: Predicting the Output Power of a HeNe Laser

Minimum Pump Power Revisited

Alternative Notations

A Model for Power Development in a Laser

Example: Modeling Power Buildup in a HeNe Laser

Improving the Model for use with High Gain Lasers

Example: Comparing Models for a Semiconductor Laser

Determining Cavity Decay Parameters

Example: Decay in a HeNe Laser

**Analytical Solutions**

The Rigrod Approach

Example: Predicting Output Power using the Rigrod Approach

Example: Application to a High Gain Laser

Ring Lasers

Example: A Ring Laser Example

Optimal Output Coupling

Example: Predicting Optimal Cavity Optics

**Thermal Issues**

Thermal Populations and Re-absorption Loss

Quasi-Three-Level Systems

Example: Estimating the Thermal Population of LLLs

Quantum Defect Heating

Example: Quantum Defect Calculations

Thermal Populations at Threshold

Example: Minimum Pump Power of a 946nm YAG Laser

Example: Computing Fractional Populations

Thermal Populations in an Operating Laser

Example: Pumping a 946nm Nd:YAG Laser

Thermal Effects on Laser Diodes

Modeling the Effects of Temperature on Laser Diodes (Wavelength)

Example: Predicting the Effect of Diode Wavelength Shift on Vanadate

Thermal Effects on Laser Diodes (Power and Threshold)

Example: Experimentally Determining Characteristic Temperature

Low Power DPSS Design

Scaling DPSS Lasers to High Powers

**Generating Massive Inversions: Q-Switching**

Inversion Buildup

Q-Switch Loss

Example: Minimum Loss of a Q-Switch

AOM Switches

Example: Bragg Angle in a Q-Switch

Example: AOM Deflection

EOM Switches

Example: Determining the Gain of a Laser using an EOM

Example: An Imperfectly-Aligned EOM

Passive Q-Switches

Example: A Passive Q-Switch

A Model for Pulse Power

Example: Output Power of a Q-Switched Laser

Multiple Pulse Output

Example: Predicting Q-Switch Settings for a Double-Pulse Laser

Modeling Flashlamp-Pumped Lasers

Example: Calculating the Time for Peak Inversion

Example: Calibrating the Model

Repetitively-Pulsed Q-Switched Lasers

Giant First Pulse

Ultrafast Lasers: Modelocking

Example: Modelocking Rate and Laser Size

**Non-Linear Optics**

Origins of Non-Linear Effects

Phase Matching

Non-Linear Materials

Practical Conversion Efficiency

Applications to Laser Design

Example: Intra- and Extra-Cavity Intensities

Application to DPSS Design

The Simple Approach

The Rigrod Approach

Example: A Small Green "Laser Pointer" DPSS

**Common Lasers and Parameters**

CW Gas Lasers

The Helium-Neon (HeNe) Gas Laser

Ion Gas Lasers

The Carbon-Dioxide Gas Laser

Pulsed Gas Lasers

TEA CO2 Lasers

Excimer Gas Lasers

Semiconductor (Diode) Lasers

Solid State Lasers

The Ruby Laser

Side-Pumped Nd:YAG Lasers

End-Pumped Nd:YAG Lasers

Other YAG Lasers

Other Solid-State Lasers

## Author(s)

### Biography

**Mark Steven Csele** is a full-time professor at Niagara College, Welland, Ontario, Canada. A physicist and professional engineer, he has taught for over 20 years at levels ranging from two-year technician to four-year undergraduate. Currently, he teaches photonics at Niagara College, which features an array of dedicated laboratories hosting a variety of laser systems. He has authored a previous book on fundamental laser concepts as well as several articles in magazines and trade encyclopedias.

## Reviews

"One of Marc Csele's key strengths is his clear and illustrative style; he grounds the material in everyday words and examples. …the choice and order of the chapters brings the reader along gradually from basic knowledge to practical application in a logical and comfortable way."

––Marc Nantel, Niagara College, Niagara Falls, Ontario, Canada"This is a textbook, written so as to be accessible to undergraduate students. The aim is to introduce the reader to laser science, parallel with the presentation of basic mathematical models used for the description of lasers of various types, and of basic physical properties of those lasers. Accordingly, the mathematical models are classified as algebra-based and calculus-based ones. Particular chapters are dealing with fundamental topics, such as the lasing threshold, gain saturation, thermal effects, Q-switching, and some basic effects of nonlinear optics. Many particular examples are included, which may be used as teaching material. The book also contains a lot of practical material about basic types of existing lasers, such as gas lasers, semiconductor lasers, and solid-state ones."

––Boris A. Malomed (Tel Aviv), fromZentralblatt MATH 1320 – 1