E x Direction of Propagation. y B y

Transcription

1 x E x Direction of Propagation k z z y B y An electromagnetic wave is a travelling wave which has time varying electric and magnetic fields which are perpendicular to each other and the direction of propagation, z.

2 z E B E E and B have constant phase in this xy plane; a wavefront k Propagation E x E x = E o sin(ωt 뻢 z) z A plane EM wave travelling along z, has the same E x (or B y ) at any point in a given xy plane. All electric field vectors in a given xy plane are therefore in phase. The xy planes are of infinite extent in the x and y directions.

3 y Direction of propagation k O θ r r E(r,t) z A travelling plane EM wave along a direction k

4 Wave fronts (constant phase surfaces) Wave fronts k Wave fronts λ P λ k λ O P r E z A perfect plane wave A perfect spherical wave A divergent beam (a) (b) (c) Examples of possible EM waves

5 Wave fronts y (b) x 2w o O θ z Beam axis Gaussian Intensity (c) (a) (a) Wavefronts of a Gaussian light beam. (b) Light intensity across beam cross section. (c) Light irradiance (intensity) vs. radial distance r from beam axis (z). 2w r

6 ω + δω ω? δω E max E max δk δω Wave packet ω Two slightly different wavelength waves travelling in the same direction result in a wave packet that has an amplitude variation which travels at the group velocity.

7 N g n Wavelength (nm) Refractive index n and the group index N g of pure SiO 2 (silica) glass as a function of wavelength.

8 vδt z E Area A k Propagation direction B A plane EM wave travelling along k crosses an area A at right angles to the direction of propagation. In time Δt, the energy in the cylindrical volume AvΔt (shown dashed) flows through A.

9 A t Refracted Light λ t k t B t y O z θ t A B θ t n 2 θ i θ r θ i n 1 θ r A A B λ k i λ k r A i B r Incident Light B i A r Reflected Light A light wave travelling in a medium with a greater refractive index (n 1 > n 2 ) suffers reflection and refraction at the boundary.

10 Transmitted (refracted) light θ t k t n 2 Evanescent wave k i Incident light θ i θ i k r n > n 1 2 θ c θ c θ i >θ c TIR Reflected light (a) (b) (c) Light wave travelling in a more dense medium strikes a less dense medium. Depending on the incidence angle with respect to θ c, which is determined by the ratio of the refractive indices, the wave may be transmitted (refracted) or reflected. (a) θ i < θ c (b) θ i = θ c (c) θ i > θ c and total internal reflection (TIR).

11 y x into paper z E i,// k i θ t θ i θ r E t,// Transmitted wave k t E t, E E i,// r, θ t =90 E t, n 2 Evanescent wave θ i θ r n 1 > n 2 E r, Incident wave E i, E r,// k r Reflected wave (a) θ i < θ c then some of the wave is transmitted into the less dense medium. Some of the wave is reflected. Incident wave E i, E r,// Reflected wave (b) θ i > θ c then the incident wave suffers total internal reflection. However, there is an evanescent wave at the surface of the medium. Light wave travelling in a more dense medium strikes a less dense medium. The plane of incidence is the plane of the paper and is perpendicular to the flat interface between the two media. The electric field is normal to the direction of propagation. It can be resolved into perpendicular ( ) and parallel (//) components

12 Magnitude of reflection coefficients Phase changes in degrees (a) r θ p θ c r // (b) θ p θ c φ // TIR φ Incidence angle, θ i Incidence angle, θ i Internal reflection: (a) Magnitude of the reflection coefficients r // and r vs. angle of incidence θ i for n 1 = 1.44 and n 2 = The critical angle is 44? (b) The corresponding phase changes φ // and φ vs. incidence angle.

13 External reflection θ p r // r Incidence angle, θ i The reflection coefficients r // and r vs. angle of incidence θ i for n 1 = 1.00 and n 2 = 1.44.

14 d n 1 n 2 n 3 A B Surface Antireflection coating Semiconductor of photovoltaic device Illustration of how an antireflection coating reduces the reflected light intensity

15 λ 1 /4 λ 2 /4 Reflectance A B C λ (nm) n 1 n 2 n 1 n λ o Schematic illustration of the principle of the dielectric mirror with many low and high refractive index layers and its reflectance.

16 M 1 M 2 m = 1 A m = 2 Relative intensity 1 υ f R ~ 0.8 R ~ 0.4 δυ m B L m = 8 υ υ m - 1 υ m υ m + 1 (a) (b) (c) Schematic illustration of the Fabry-Perot optical cavity and its properties. (a) Reflected waves interfere. (b) Only standing EM waves, modes, of certain wavelengths are allowed in the cavity. (c) Intensity vs. frequency for various modes. R is mirror reflectance and lower R means higher loss from the cavity.

17 Partially reflecting plates Transmitted light Input light Output light L λ Fabry-Perot etalon λ m - 1 λ m Transmitted light through a Fabry-Perot optical cavity.

18 y B Virtual reflecting plane n 2 Penetration depth, δ z A θ i Δz θ r n 1 > n 2 Incident light Reflected light The reflected light beam in total internal reflection appears to have been laterally shifted by an amount Δz at the interface.

19 y C n 1 B A θ i θ r n 2 n 1 > n 2 d z Incident light Reflected light When medium B is thin (thickness d is small), the field penetrates to the BC interface and gives rise to an attenuated wave in medium C. The effect is the tunnelling of the incident beam in A through B to C.

20 Reflected light B = Low refractive index transparent film ( n 2 ) Reflected (a) n2 n n 1 1 TIR n1 θ i > θ c Transmitted Incident A light Glass prism C FTIR θ i > θ c A Incident light (b) (a) A light incident at the long face of a glass prism suffers TIR; the prism deflects the light. (b) Two prisms separated by a thin low refractive index film forming a beam-splitter cube. The incident beam is split into two beams by FTIR.

21 (a) P Time Q Amplitude Amplitude υ ο υ (b) Field Δt Time Space l = cδt υ ο Δυ = 1/Δt υ (c) P Q Time Amplitude υ (a) A sine wave is perfectly coherent and contains a well-defined frequency υ o. (b) A finite wave train lasts for a duration Δt and has a length l. Its frequency spectrum extends over Δυ = 1/Δt. It has a coherence time Δt and a coherence length l. (c) White light exhibits practically no coherence.

22 (a) No interference A Interference Δt No interference B Source Time (b) P Q c Spatially coherent source (c) c An incoherent beam Space (a) Two waves can only interfere over the time interval Δt. (b) Spatial coherence involves comparing the coherence of waves emitted from different locations on the source. (c) An incoherent beam.

23 Light intensity pattern Diffracted beam Incident light wave Circular aperture A light beam incident on a small circular aperture becomes diffracted and its light intensity pattern after passing through the aperture is a diffraction pattern with circular bright rings (called Airy rings). If the screen is far away from the aperture, this would be a Fraunhofer diffraction pattern.

24 Incident plane wave A secondary wave source New wavefront Another new wavefront (diffracted) θ z (a) (b) (a) Huygens-Fresnel principles states that each point in the aperture becomes a source of secondary waves (spherical waves). The spherical wavefronts are separated by λ. The new wavefront is the envelope of the all these spherical wavefronts. (b) Another possible wavefront occurs at an angle θ to the z-direction which is a diffracted wave.

25 Incident light wave δy Y Incident light wave R = Large y Screen y y θ A a θ b c z δy θ ysinθ z θ = 0 θ Light intensity (a) (b) (a) The aperture is divided into N number of point sources each occupying δy with amplitude δy. (b) The intensity distribution in the received light at the screen far away from the aperture: the diffraction pattern

26 a b The rectangular aperture of dimensions a b on the left gives the diffraction pattern on the right.

27 y S 1 A 2 S 1 Δθ s Δθ S 2 A 1 S 2 L I Screen Resolution of imaging systems is limited by diffraction effects. As points S 1 and S 2 get closer, eventually the Airy disks overlap so much that the resolution is lost.

28 y Incident light wave a y One possible diffracted beam Single slit diffraction envelope m = 2 Second-order m = 1 First-order m = 0 Zero-order d θ dsinθ z m = -1 m = -2 First-order Second-order Diffraction grating Intensity (a) (b) (a) A diffraction grating with N slits in an opaque scree. (b) The diffracted light pattern. There are distinct beams in certain directions (schematic)

29 Incident light wave First-order Zero-order First-order Incident light wave m = 1 First-order m = 0 Zero-order m = -1 First-orde (a) Transmission grating (b) Reflection grating (a) Ruled periodic parallel scratches on a glass serve as a transmission grating. (b) A reflection grating. An incident light beam results in various "diffracted" beams. The zero-order diffracted beam is the normal reflected beam with an angle of reflection equal to the angle of incidence.

30 Normal to face Normal to grating plane First order γ d γ Blazed (echelette) grating.

31 Spherical mirror L Optical cavity Spherical mirror A F 2θ B R R Wave front Two confocal spherical mirrors reflect waves to and from each other. F is the focal point and R is the radius. The optical cavity contains a Gaussian beam

32 A 0 A 1 A 2 A 3 n 1 B 1 B 3 B 5 n 2 B 2 B 4 B 6 n 3 C 1 C 2 C 3 Thin film coating of refractive index n 2 on a semiconductor device

33 L FP etalon Screen Input light k θ Fabry-Perot etalon Output light θ θ Broad monochromatic source Lens θ Screen Fabry-Perot optical resonator and the Fabry-Perot interferometer (schematic)

34 Laser light Prism d = Adjustable coupling gap Air n 2 n 1 Thin layer Thin layer Glass substrate n 3 Glass substrate (a) (b) (a) Light propagation along an optical guide. (b) Coupling of laser light into a thin layer - optical guide - using a prism. The light propagates along the thin layer.