PHY 452/562 Lasers and Modern Optics. Prof. Eden Figueroa. Lecture 2: Electromagnetic waves I

Transcription

1 Lasers and Modern Optics Prof. Eden Figueroa Lecture 2: Electromagnetic waves I August 29 th 2013

2 Organizational issues. Waves and light. A reminder on electromagnetic theory. Propagation of light. PHY 452/562

3 Organizational issues 1) Replacement lectures: Option 1: Wednesday Sept. 4th and Sept. 11 th 13:00 14:20 2) Schedule for Introductory Lasers Lab (about 2 hours work + 1 page report).

4 Bhuphaibool,Tanaboon Biemann,TaraJ ean Cheung,Carl Wayne Dragan,J ames Ev ans,stefan Davis Gasparik,Jessica Theresa Krupin,Matthew John Lebre cht,michael Kenneth N am,jung hyun Nguyen,Phuong Ngoc Rind,Samuel Philip John Wen,Yan Chen, Yuning Monday Tue sday Wedne sday Thur sday Fr iday PHY 452/562 Lecture Roster Monday Tuesday Wednesday Thursday Friday Bhuphaibool,Tanaboon Biemann,TaraJean Cheung,Carl Wayne Dragan,James Evans,Stefan Davis Gasparik,Jessica Theresa Krupin,Matthew John Lebrecht,Michael Kenneth Nam,Junghyun Nguyen,Phuong Ngoc Rind,Samuel Philip John Wen,Yan Chen, Yuning

5 Discussion: What is a wave? In general, a wave is a disturbance that travels in space. Light is an electromagnetic wave. The disturbance is electric and magnetic field, propagating at the speed of light. Waves transport energy.

6 Describing waves PHY 452/562 If the wave repeat itself it is called a periodic wave. Suppose that at x=0 the disturbance is given by: Since the wave travels at speed v, the motion of point x at time t has to be the same as the motion of x=0 at earlier time t x/v. Substituting this into the function above and noting that cos( θ) = cos(θ) :

7 Wave Nomenclature A: Amplitude : Period f = 1/T : Frequency ω = 2πf: Angular frequency λ: Wavelength K = 1/ λ: Wave number k = 2π/ λ:wavevector magnitude

8 How fast is the wave traveling? The phase velocity is the wavelength / period: v = Since f = 1/ : v = f In terms of the k-vector, k = 2 and the angular frequency, = 2 this is: v = / k The wave moves one wavelength,, in one period,. Discussion: What is the electric field oscillation period of laser light at 795nm? How can we detect light then?

9 The 1 dimensional Wave Equation Partial time and space derivatives of the sinusoidal wave: Combining these, we obtain the one dimensional wave equation:

10 Waves using complex numbers The electric field of a light wave can be written:

11 Define E(x,t) to be the complex field The resulting complex amplitude is: The complex form of the field is then:

12 The 3D wave equation PHY 452/562 A wave can propagate in any direction in space. So we must allow the space derivative to be three dimensional: Which has a solution:

13 Wave front: The set of points in space reached by a wave or vibration at the same instant as the wave travels through a medium. Wave fronts generally form a continuous line or surface.

14 Wave front: Surfaces of constant phase. If we define and use we can get

15 Discussion: Are there plane waves in nature? It would have infinite energy. It doesn t exist. A laser beam is more localized. We can approximate a laser beam as a plane wave in z times a Gaussian in x and y:

16 Laser pulses If the beam is localized in space, we can also localize it in time by multiplying by a Gaussian.

17 Paraxial wave equation We are interested in laser beams, which are very directional, similar to plane waves. Therefore, we write the field in the following form: The function varies slowly, controlling the diameter and strength of a laser beam as it propagates.

18 Using the following hints: We can obtain: This equation is horribly hard to solve, thus we use: (the field varies only gradually along the z axis)

19 Paraxial equation: The term paraxial is used because all of the light must travel nearly parallel to the z axis in order for the beam to have a sufficiently slow z dependence. Laser beams typically obey this approximation very well. A very important class of solutions to the paraxial wave equation is the Gaussian Beams.