ECE 484 Semiconductor Lasers

Transcription

1 ECE 484 Semiconductor Lasers Dr. Lukas Chrostowski Department of Electrical and Computer Engineering University of British Columbia January, 2013 Module Learning Objectives: Understand the importance of the in lasers Calculate the Fabry-Perot transmission and reflection spectrum Determine the Fabry-Perot modes Calculate the Fabry-Perot laser threshold Readings: Yariv Sections , 4.1, (6.1) Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 1

2 Introduction to Semiconductor Lasers A diagram of a semiconductor laser: Injection current (PUMP) Partially reflective surface (CAVITY) Partially reflective surface (CAVITY) ~0.1µm Laser output (infrared) Active layer (e.g. GaAs) (GAIN MEDIUM) Ohmic contacts Confining layers (e.g. AlGaAs) Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 2

3 Wave Equation Starting from Maxwell s equations, the wave equation for the electric field vector, E, in a homogeneous and isotropic media is: 2 E µɛ 2 t 2 E = 0 The solution is a monochromatic electromagnetic plane wave E = E oue i(ωt k r) A is the amplitude of the wave. ω is the frequency. k is the wavevector with a propagation constant k = k ; u is perpendicular to the direction of propagation. The magnitude of the wavevector is β = k = ω µɛ The phase velocity is: ν = ω k = 1 µɛ = 1 n µ 0ɛ 0 where n is the index of refraction. The wavelength in the material is: λ = 2π k Readings: Yariv Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 3

4 Energy of an Electromagnetic Wave The time-averaged energy flux (Poynting s theorem) is S = 1 2η E0 2 u 3 = k 2ωµ E 2 The time-averaged energy density is U = 1 2 ɛ E0 2 = 1 2 ɛ E 2 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 4

5 Reflections at Dielectric Interfaces Consider the plane wave E, normally incident on a dielectric interface: E re n 1 n 2 te The reflection coefficient, r, is r 12 = n1 n2 n 1 + n 2 Note that if light was originating in the n 2 medium, the reflection coefficient would be of opposite sign, i.e. r 21 = r 12. The power reflection coefficient is R = r 2. The power transmission coefficient is T = t 12t 21. Conservation of energy tells us that T + R = 1 The transmission coefficient, t, is t 12 = 2n1 n 1 + n 2 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 5

6 Reflections at Dielectric Interfaces Example What is the power reflection coefficient for a field inside a semiconductor laser (n = 3.6), incident on an air interface (n = 1)? Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 6

7 Fabry-Perot model The Fabry-perot etalon is the most basic optical resonator. It consists of a plane-parallel plate of thickness l and index n that is immersed in a medium of index n. r = reflection coefficient for waves from n toward n. r = reflection coefficient for waves from n toward n. t, t = transmission coefficients Assume normal incidence, θ = 0 Round-trip phase shift: δ = k = 2π n 2l λ M.H. Ayliffe/D.V. Plant E i tt e i δ 2 E i tr 2 t e i δ 2 e iδ Example For a thickness l = 300nm, with n = 1, at what wavelength do we see constructive interference at the output? Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 7

8 Fabry-Perot Model - Multiple Beam Interference UBC ECE Lasers Light is incident from the top, E i, transmitted through bottom, E t. E i tt e i δ 2 E i tr 2 t e i δ 2 e iδ E i tr 4 t e i δ 2 e i2δ E i tr 6 t e i δ 2 e i3δ We can express the transmitted field as: E t = E i tt e i δ 2 +Ei tt (r ) 2 e iδ e i δ 2 +Ei tt (r ) 4 e i2δ e i δ 2 +Ei tt (r ) 6 e i3δ e i δ M.H. Ayliffe/D.V. Plant Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 8

9 Fabry-Perot Model - Multiple Beam Interference The transmitted field, E t, simplifies to: E t =E i tt e iδ/2 [ 1 + (r ) 2 e iδ + (r ) 4 e i2δ +... ] [ tt e iδ/2 ] =E i 1 (r ) 2 e iδ (infinite geometric series) [ ] T e iδ/2 =E i 1 Re iδ The normalized transmission is: E t = T eiδ/2 E i 1 Re iδ And the power transmitted (light intensity): I t I i = E te t E i E i = T 2 (1 Re iδ )(1 Re iδ ) = (1 R) 2 (1 R) 2 + 4R sin 2 (δ/2) Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 9

10 Fabry-Perot Model - Multiple Beam Interference When sin 2 (δ/2) = 0 I t /I i = 1 (i.e. all of the light is transmitted no reflection). Since sin 2 (δ/2) is periodic, there are many solutions which yield 100% transmission. These wavelengths are the Fabry-Perot modes, or Fabry-Perot resonant frequencies. This occurs for sin 2 (δ/2) = 0 δ/2 = mπ = 2πnl, (m = 1, 2, 3...) λ m The solutions are λ m, or in frequency, ν m = c = m c λ m 2nl As we will see later laser will oscillate at one of these wavelengths Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 10

11 Fabry-Perot Model - Transmission, Free Spectral Range Transmission δ / 2π The free spectral range (FSR) is the spacing between adjacent modes of the filter, and is: (R=0.3) ν = ν m ν m 1 = c 2nl Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 11

12 Fabry-Perot Model - Transmission, Linewidth 1 Transmission λ 1/ Wavelength [um] The linewidth ( λ 1 ) of the resonator is defined as the full-width 2 half-max (FWHM) of the transmission spectrum: (R=0.6) λ 1 2 = λ 2 λ 1 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 13

13 Fabry-Perot Model - Transmission Transmission R=0 R=0.3 R=0.7 R= δ / 2π The linewidth of the resonator changes with the reflection coefficient A larger reflection leads to a narrower linewidth Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 14

14 Fabry-Perot Model - Reflection UBC ECE Lasers Light is incident from the top, E i, reflected from the top, E r. E i tt e i δ 2 E i tr 2 t e i δ 2 e iδ E i tr 4 t e i δ 2 e i2δ E i tr 6 t e i δ 2 e i3δ We can express the reflected field as: E r = E i r + E i tt r e iδ + E i tt (r ) 3 e i2δ + E i tt (r ) 5 e i3δ +... M.H. Ayliffe/D.V. Plant Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 15

15 Fabry-Perot Model - Reflection The reflected field, E r, simplifies to: E r =E i r + E i tt r e iδ [ 1 + (r ) 2 e iδ + (r ) 4 e i2δ +... ] [ =E i r + tt r e iδ ] 1 (r ) 2 e iδ (infinite geometric series) [ =E i r T ] reiδ 1 Re iδ, r = r The normalized reflection is: E r = r T reiδ E i 1 Re iδ = r(1 eiδ ) 1 Re iδ And the power reflected (light intensity): I r I i = E re r E i E i = 4R sin 2 (δ/2) (1 R) 2 + 4R sin 2 (δ/2) Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 16

16 Fabry-Perot Model - Reflection Transmission δ / 2π Transmission and reflection spectra are related (in this case, the resonator is lossless). Question: Sketch the reflection spectrum, for R = 0.6. Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 17

17 Fabry-Perot Model - with Loss If we consider loss in the cavity, where the per-pass intensity gain (or loss) is A = a 2 = e αl and the per-pass field amplitude gain (or loss) is a = e α 2 l The transmitted field, E t, becomes: E t =E i tt ae iδ/2 [ 1 + (ar ) 2 e iδ + (ar ) 4 e i2δ +... ] [ tt ae iδ/2 ] =E i 1 (ar ) 2 e iδ (infinite geometric series) [ ] T ae iδ/2 =E i 1 RAe iδ The normalized power transmitted (light intensity) is: I t I i = E te t E i E i = (1 R) 2 A (1 AR) 2 + 4AR sin 2 (δ/2) Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 19

18 Fabry-Perot Model - with Loss Transmission δ 1/2 δ 1/2 A=1 A=0.9 A=0.5 A=0.1 δ δ 0 1/2 1/ δ / 2π Transmission spectrum amplitude decreases with increasing losses. Linewidth broadens. R=0.8 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 20

19 Fabry-Perot Model - with Loss Transmission, Reflection δ / 2π Reflection spectrum dip decreases with increasing losses. R=0.8 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 21

20 Fabry-Perot - Linewidth The full-width half-max linewidth is found by finding the half-power point, δ 1 2 : T (δ 1 2 ) = 1 2 T (0) (1 R) 2 A (1 AR) 2 + 4AR sin 2 ( δ ) The FWHM is: δ 1 = 2δ AR 1 = AR since: δ = 2π ωn2l n2l = λ c ω 1 2 = 21 AR AR = 1 (1 R) 2 A 2 (1 AR) 2 1 AR δ 1 = 2 sin AR 1 AR AR c 2nl, ν 1 = 1 AR 2 π c AR 2nl Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 22

21 Fabry-Perot Model - Finesse The finesse (F) of a resonator is defined as the ratio of the FSR to resonance linewidth: F = F SR λ 1 2 = c 2nl 1 AR π c AR 2nl = π AR 1 AR = ν ν 1 2 This is a measure of the sharpness of a resonance relative to the mode spacing. Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 23

22 Fabry-Perot Model - Q The resonator Quality factor, Q, is a measure of the sharpness of the filter relative to the central frequency, and defined as (E is stored energy): Q = ω E de/dt Note: This expression works very well to estimate Q from FDTD simulations. (Plot the intensity versus time.) For this, we need to calculate the total losses, both from the mirrors and propagation losses (absorption, scattering). Per-pass intensity loss = AR Distributed total loss, per unit length, α tot : A tot = AR = e αl R = e αtotl α tot = α 1 l ln R Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 24

23 Fabry-Perot Model - Photon Lifetime Total loss in cavity = Mirror loss + absorption, α tot = α m + α. The loss α tot [cm 1 ] describes the loss over distance, P (z) = P (0)e αtotz However, we would like to model the time behaviour of a resonator, dp dt = dp dz dz dt dp dz = α totp, dz dt = multiplying, get dp dt = α totp c n = P τ p = R loss where τ p = [ (α + α m ) c n] 1 is the photon lifetime (1 / Rate of photon decay = average time spent in the cavity) Q: Calculate the photon lifetime for α tot = 10cm 1, n = 3. Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 25 c n

24 Fabry-Perot Model - Q The decay of the energy in the cavity is: So Q is: Q is also defined as: de dt = E τ p Q = ω E de/dt = ωτ p Q = ω ω 1 2 The two definitions give equal results for small-enough losses, i.e. large Q values. This occurs when ln R 1 R R. The field decays to 1/e in time τ p. Thus, Q is the number of optical field oscillations before the field decays to 1/e. Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 26

25 Fabry-Perot Model - Physical Interpretations Finesse: F = π AR 1 AR This can be related to the total loss α via ln R. F π αl We can find the field decaying to 1/e when: 1 e = e 1 = e αl2n where N is the number of round trips in the resonator. αl2n = 1, π 2N = 1, F N = F 2π Hence, finesse/2π describes the average number of oscillations the field lives in the cavity. Ref: Optical Microresonators, F. Heebner, et. al., Springer Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 27

26 Ring Resonator Experiment Double-bus ring resonator with radius = 30µm, coupler length = 15µm, coupler spacing = 250nm. Optical microscope images from: Raha Vafaei, EECE 571U, 2009 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 28

27 Ring Resonator Experiment SEM images from: Nicolas Rouger, Raha Vafaei, EECE 571U, 2009 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 29

28 Ring Resonator Experiment 16 Double-bus ring resonator with radius = 30µm, coupler length = 15µm, coupler spacing = 250nm. Measured through port spectrum: Power v Wavelength for RR15 T=25C [d] 16 Power v Wavelength for RR15 T=25C [a] Power [dbm] Power [dbm] Wavelength [nm] Wavelength [nm] from: Raha Vafaei, Nadia Makan, EECE 571U, 2009 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 30

29 Q estimation techniques (double-bus RR) Q estimation techniques, for varied loss (α = 1 is loss-less): 0 Drop, Through Phase Linewidth, δ 1/2 10 δ 1/2 δ 1/ δ / 2π Theory (approx) Drop port, 3 db Through port, 3 db Resonator loss, α Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 31

30 Bragg Grating Fabry-Perot Experiment I Bragg gratings: design (top), electron microscope images (bottom). Fabricated via CMOS 193 nm lithography. Uniform Gratings!#$%&$ Phase-Shift!"#$%&$!"#$%&$ from: X.Wang, et. al, IEEE Photon. Conf., 2011, pp Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 32

31 Bragg Grating Fabry-Perot Experiment Bragg gratings: design (top), electron microscope images (bottom). Fabricated via CMOS 193 nm lithography. 0 5 Transmission (db) ~50 pm Wavelength (nm) from: X.Wang, et. al, IEEE Photon. Conf., 2011, pp Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 33

32 Fabry-Perot Applications Dichroic filters can design any spectral response; light is reflected rather than absorbed; long lifetime. 3D movies slightly different RGB filters for left/right eyes dolby-stakes-its-claim-in-3d-movie-tech htm Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 34

33 Fabry-Perot Laser A laser is constructed using a. For each pass in the cavity, the field is modified (phase and amplitude) by the factor r e ikl. Fabry-Perot Fabry-Perot Pump (Energy Source) r e ikl E t r e ik l E E i E i t E i E i t In the case of the laser, energy is provided, and modifies the propagation constant, k k. Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 35

34 Fabry-Perot Laser The propagation constant is modified by providing optical gain. k = k + k + i γ 2 iα 2 where k is the original propagation constant k is the change in propagation constant (due to active atoms) γ is the optical gain why factor of 1 α is the optical loss Note: All these parameters are wavelength dependant. Optical gain, γ(λ), is typically approximately a Gaussian function, with a bandwidth of 100nm in semiconductors. 2? Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 36

35 Fabry-Perot Laser Similar to an RF oscillator, a laser will begin to oscillate when the round-trip provides unity gain. From the diagram, the round trip gain is: (two reflections, and a path length of 2l) (r ) 2 e ik 2l =1 r 2 e i(k+ k+i γ 2 i α 2 )2l =1 r 2 e i(k+ k)2l e ( γ 2 α 2 )2l =1 The real part is: r 2 e ( γ 2 α 2 )2l = 1. This is the amplitude condition, where the field returns with the same amplitude after a round trip. The imaginary part is: e i(k+ k)2l = 0. This is the phase condition, where the field must return with the same phase (or a multiple of 2π). 2mπ = (k + k) 2l, (m = 1, 2, 3...) Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 37

36 Laser Threshold The amplitude condition can be solved to provide the minimum gain required for the laser to operate. This is called the threshold gain, γ t. r 2 e ( γ 2 α 2 )2l =1, R = r 2 ( γ ln(r) + 2 α ) 2l =1 2 γ t =α 1 l ln(r) Thus, the laser will turn on only when the gain, γ o γ th E-Field Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 38

37 Laser Threshold Example What is the threshold gain for the following laser? index of refraction of gain region, n = 3.5 length, l = 200µm loss, α = 0. Active region (with! and ") Output beam Cleaved facet (R1=33%) SIDE VIEW Cleaved facet (R2=33%) Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 39

38 Summary The optical cavity (e.g. Fabry-Perot) is important for lasers because: The cavity modes determine the possible lasing wavelengths The cavity losses (mirror loss, internal losses) determine the laser threshold Other laser optical cavities are possible Ring resonators (waveguides, optical fiber ring laser) Photonic crystals a a ARTICLES b 1 µm Nature Physics 2, (2006) Intensity (a.u.) Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers λ (nm) 40 Intensity (a.u.) λ (nm)

39 in SOI waveguides n eff Material dispersion is usually the dominant component of the total dispersion in a waveguide, and usually small (e.g. in SiO 2, n varies by less than 1% in the range of λ = 1µm 2µm) However, because of the high index contrast and the small geometries in the SOI waveguides, waveguide dispersion is very significant, and the effective index variation is important for modelling SOI devices (n eff is the effective index seen by the optical mode). Effective Index width=375 nm width=450 nm width=510 nm width=540 nm width=570 nm width=600 nm Wavelength (nm) Ref: Robi Boeck, Simulation and Fabrication of Microring Add-Drop Filters, EECE 571U 2009 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 41

40 Fabry-Perot with FSR ν The free spectral range (FSR), ω = ω 2 ω 1, is the spacing between adjacent modes of the filter, δ 2, δ 1 : 2π = δ 2 δ 1 = β 2 L β 1 L, β = n ω c Assume n varies linearly with ω, hence n = n 0 + ω dn 2 dω, where ω = ω 2 ω 1, n 0 = n(ω 0 ), ω 0 = ω2+ω1 2πc L = n 2ω 2 n 1 ω 1 = ω 2 ( ) dn = ω n 0 + ω 0 dω 2 ( n 0 + ω 2 ) ( dn ω 1 n 0 ω dω 2 2πc ω = L ( ) n + ω dn = 2πc Ln dω g c ν = L ( ) n + ν dn = c Ln dν g Use group index, n g, instead of n eff, for calculating FSR. ) dn dω Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 42

41 Fabry-Perot with FSR λ The free spectral range (FSR), λ, is the spacing between adjacent modes of the filter, δ m+1, δ m, and is found by: 2π = δ m+1 δ m = β m+1 L β m L β m+1 β m = β dβ dλ ( ) 1 dβ λ β = 2π ( dβ dλ L dλ β = 2πn λ, dβ dλ = 2π λ λ = λ, (Approx. that β varies linearly with λ) ) 1 dn dλ + 2πn( λ 2 ) = λ 2 L ( n λ dn dλ ν c λ λ 2 = c Ln g ) = λ2 Ln g Use group index, n g, instead of n eff, for calculating FSR. 2π ( λ 2 n dn ) dλ λ Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 43

42 in SOI waveguides n g n g can be found from experimental measurements or mode simulations of the effective index: n g = n eff λ dn eff dλ Using this expression on the n eff vs. λ data on slide 41, we obtain the group index, for different waveguide dimensions: Group Index Measured Modesolver: width=375 nm Modesolver: width=450 nm Modesolver: width=510 nm 2D FDTD: width=375 nm, gap=290 nm 2D FDTD: width=500 nm gap=165 nm Wavelength (nm) Ref: Robi Boeck, Simulation and Fabrication of Microring Add-Drop Filters, EECE 571U 2009 Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 44

43 Fabry-Perot with - Linewidth From slide 22, AR δ 1 = 21 2 AR We need to consider both frequency (wavelength) change and index change over the linewidth. since: δ = ω c 2nl dδ dω = 2l c ( n + ω dn dω So, δ 1 2 = 2l c n g ω 1 2 ω 1 2 = 21 AR AR ) = 2l c n g c 2n g l, ν 1 = 1 AR 2 π AR Use group index, n g, for calculating linewidth, Q, etc. c 2n g l Dr. Lukas Chrostowski ECE 484 Semiconductor Lasers 45