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EE 485, Winter 2004, Lih Y. Lin

Chapter 2

- Improves on Ray Optics by including phenomena such as interference and . - Limitations: (1) Cannot provide a complete picture of reflection and refraction at the boundaries between dielectric materials. (2) Cannot explain optical phenomena that require vector formalism, such as .

2.1 Postulates of Wave Optics The c speed in a medium: c = 0 (2.1-1) * * n Wave function u(r,t) [position r = (x, y, z) , time t] satisfies 1 ∂ 2u Wave equation ∇2u − = 0 (2.1-2) c2 ∂t 2 Wave* equation* * is linear. Principle of superposition applies: = + u(r,t) u1(r,t) u2 (r,t) Wave equation approximately applicable to media with position-dependent refractive indices, provided that the variation* is slow* within distances of a wavelength → Locally homogeneous, n = n(r), c = c(r).

Intensity, , and energy Optical intensity: optical power per unit (/cm2) * * I(r,t) = 2 u 2 (r,t) (2.1-3) : average over a period >> 1/. Optical power (watts) flowing into an area A normal to the propagation direction: * P(t) = ∫ I(r,t)dA (2.1-4) A Optical energy (joules) collected in a given time interval T is ∫ P(t)dt . T

2.2 Monochromatic * * * u(r,t) = a(r) cos[]2πft + ϕ(r ) (2.2-1)

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A. Complex Representation and the Helmholtz Equation Complex wavefunction * * * * U (r,t) = a(r) exp[ jϕ(r)]exp( j2πft) = U (r)e jωt (2.2-2,5) * U (r): complex amplitude 1 ∂ 2U ∇2U − = 0 (2.2-4) c2 ∂t 2

The Helmholtz equation and wavenumber Substituting (2.2-5) into (2.2-4) obtains: * (∇2 + k 2 )U (r) = 0 (2.2-7) k = ω : wavenumber (2.2-8) c

Optical intensity * * I(r) = U (r) 2 (2.2-10) not a function of time.

Wavefronts * The wavefronts are the surfaces of equal phase, ϕ(r) = constant.

B. Elementary Waves The plane wave Complex amplitude: * * * U (r) = Aexp(− jk ⋅ r ) = Aexp[− j(k x + k y + k z) (2.2-11) * x y z k*: wavevector → direction of propagation k = k = wavenumber

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2π c Wavelength λ = = (2.2-12) k f c is called the phase of the wave.

In a medium of n, f is the same, c λ c = 0 , λ = 0 , k = nk (2.2-14) n n 0

The spherical wave * A − U (r) = e jkr (2.2-15) r * A 2 I(r) = r 2

The paraboloidal wave Fresnel approximation of the spherical wave.* * Examine a spherical wave originating at r = 0 at points r = (x, y, z) sufficiently close to the z axis but far from the origin, so that (x2 + y 2 ) << z . Use paraxial approximation, Taylor series expansion, and Fresnel approximation: * A  x2 + y 2  U (r) = exp(− jkz)exp− jk  (2.2-16) z  2z  First phase term: planar wave. phase term: paraboloidal wave.

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• Validity of Fresnel approximation Fresnel approximation valid for points (x, y) lying within a circle of radius a centered about the z axis at position z, if a satisfies a 4 << 4z3λ , or N θ 2 F m <<1 (2.2-17) 4 θ = where m a / z is the maximum angle, and a 2 N = Fresnel number (2.2-18) F λz

2.4 Simple Optical Components A. Reflection and Refraction Laws of reflection and refraction can be verified by wave optics. Please read the textbook.

B. Transmission through Optical Components (Ignore reflection and absorption. Main emphasis on phase shift and associated wavefront bending.)

Transmission through a transparent plate • Normal incidence

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Complex amplitude transmittance = − t(x, y) exp( jnk0d) (2.4-3) d → The plate introduces a phase shift nk d = 2π 0 λ

• Oblique incidence = []− θ + θ t(x, y) exp jnk0 (d cos 1 xsin 1 )

Thin transparent plate of varying thickness Transmittance = (Transmittance in air) × (Transmittance in plate) t(x, y) = exp[][]− jk ()d − d(x, y) exp − jnk d(x, y) 0 0 0 (2.4-4) = − []− − exp( jk0d0 )exp j(n 1)k0d(x, y)

Thin lens Utilizing Eq. (2.4-4) results in ª x2 + y 2 º t(x, y) = h exp« jk » (2.4-6) 0 ¬ 0 2 f ¼ = − where h0 exp( jnk0d0 ) : constant phase factor R f = : focus length of the lens n −1

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Diffraction grating An optical component periodically modulates the phase or the amplitude of the incident wave. It can be made of a transparent plate with periodically varying thickness or periodically graded refractive index.

The diffraction grating shown above converts an incident plane wave of λ << Λ θ wavelength , traveling at a small angle i with respect to the z axis, into several plane waves at small angles λ θ = θ + q (2.4-9) q i Λ with the z axis. q = 0, ±1, ±2, … : diffraction order In general, without paraxial approximation λ sinθ = sinθ + q (2.4-10) q i Λ

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C. Graded-index Optical Components Instead of varying thickness, varying refractive index. = []− Varying thickness: t(x, y) exp jn0k0d(x, y) = []− Varying refractive index: t(x, y) exp jn(x, y)k0d0 (2.4-11)

2.5 Interference Linearity of the wave equation → Superposition of the wavefunctions But not superposition of the optical intensity because of interference. (Consider waves of the same frequency in this section.)

A. Interference of* Two Waves* * = + U (r) U1(r) U 2 (r) jϕ jϕ ≡ 1 ≡ 2 U1 I1e , U 2 I 2 e Intensity interference equation: I = U 2 = I + I + 2 I I cosϕ 1 2 1 2 (2.5-4,5) ϕ ≡ ϕ −ϕ 2 1

Spatial redistribution of optical intensity. Power conservation still holds.

Interferometer Waves superimpose with delay d → ϕ = 2π ()d λ

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  d  I = 2I 1+ cos2π  (2.5-6) 0   λ 

Three important examples of intereferometers: Mach-Zehnder interferometer, Michelson interferometer, Sagnac interferometer.

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d nd nfd Intensity I is a very sensitive function of ϕ = 2π = 2π = 2π λ λ 0 c0

→ Can measure small variation of d, n, λ0, or f

Interference of two oblique plane waves U = I exp(− jkz) 1 0 = []− θ + θ U 2 I0 exp jk(z cos xsin ) At z = 0, ϕ = kxsinθ , = []+ θ → I 2I0 1 cos(kxsin ) (2.5-7) Period of interference pattern Λ = λ sinθ

B. Multiple- M waves of equal amplitude and equal phase difference = []− ϕ = U m I0 exp j(m 1) , m 1, 2, ..., M (2.5-9) M = U ∑U m m=1 2 2 sin ()Mϕ 2 I = U = I (2.5-10) 0 sin 2 ()ϕ 2 In the graph of I as a function of ϕ , the number of minor peaks between the main peaks = (M - 1).

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Infinite number of waves of progressively smaller amplitudes and equal phase difference = (m−1) j(m−1)ϕ = U m I0 r e , m 1, 2, 3, ... r < 1 ∞ I = ∑ = 0 U U m jϕ (2.5-13) m=1 1− re 2 I I = U = max (2.5-15) 1+ ()()2F π 2 sin 2 ϕ / 2 I where I ≡ 0 (2.5-16) max (1− r)2 π r F ≡ : Finesse (2.5-17) 1− r

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When r approaches 1, I max can be very large! → Principle of optical resonators, lasers. • Physical meaning of F Consider values of ϕ near the ϕ = 0 resonance peak I I ≈ max (2.5-18) 1+ ()F π 2ϕ 2 Full width at half maximum (FWHM) 2π ∆ϕ = (2.5-19) F → Finesse is a measure of the sharpness of the interference function.

2.6 Polychromatic Light Can be expressed as the sum of monochromatic waves over frequency.

The pulsed plane wave ∞ * 1 − jkz jωt U (r,t) = ∫ Aω e e dω 2π 0 (2.6-8,9) z = a(t − ) c ∞ 1 jωt a(t) = ∫ Aω e dω (2.6-10) π 2 0 is a function with arbitrary shape

σ If a(t) is of finite duration t in time → σ Pulse width in space = c t

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Pulse width in frequency domain (spectral bandwidth) depends on pulse shape, see Appendix A. E.g., for Gaussian function in space, its is still Gaussian in 1 frequency domain. σ = 1 ps, cσ = 0.3 mm, σ = = 80 GHz. t t f πσ 4 t

Interference (beating) between two monochromatic waves j2πf t j2πf t = 1 + 2 U (t) I1e I 2 e (2.6-11)

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Utilizing the interference equation (2.5-4), = 2 = + + []π ()− I U I1 I 2 2 I1I 2 cos 2 f1 f2 t (2.6-12) − ≡ f1 f2 Beat frequency

Interference of M monochromatic waves Consider an odd number M = 2L + 1 waves, each with intensity I0 and = + fq f0 qf F , q = -L, …, 0, … L << centered about f0 and spaced by f F f0 . L = []π ()+ U (t) I0 ∑exp j2 f0 qf F t (2.6-13) q=−L Utilizing Eq. (2.5-10), 2 2 sin ()Mπt T I = U = I F (2.6-14) 0 2 ()π sin t TF ≡ 2 → A periodic sequence of pulses with period TF 1/ f F , peak intensity M I0 .

Application: Mode-locked laser for generating short laser pulses.

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