1. 1 CHAPTER 4 Stimulated Emission Devices LASERS

2. 2 Ali Javan and his associates William Bennett Jr. and Donald Herriott at Bell Labs were first to successfully demonstrate a continuous wave (cw) helium-neon laser operation (1960-1962).

3. 3 4.1 STIMULATED EMISSION AND PHOTON AMPLIFICATION

4. 4 Absorption, spontaneous emission and stimulated emission Absorption: E 1 + h  E 2 Emission: E 2  E 1 + h –Spontaneous emission: The electron undergoes the downward transition by itself spontaneously. –Stimulated emission: The downward transition is induced by another photon.

5. 5 Spontaneous Emission The electron falls down from level E 2 to E 1 and emits a photon h = E 2 – E 1 in a random direction. The transition is spontaneous provided that the state with E 1 is not already occupied.

6. 6 Stimulated emission An incoming photon of energy h = E 2 – E 1 stimulates the whole emission process by inducing the electron at E 2 to transit down to E 1 Emitted photon and incoming photon –in phase –same direction –same polarization –same energy One incoming photon results in two outgoing photons which are in phase.  photon amplification

7. 7 Population inversion To obtain stimulated emission, the incoming photon should not be absorbed by another atom at E 1. We must have the majority of the atoms at the energy level E 2. When there are more atoms at E 2 than at E 1 we then have what is called a population inversion. With only two levels we can never achieve population inversion because, in the steady state, the incoming photon flux cause as many upward excitations as downward stimulated emissions.

8. 8 The principle of the laser a)Atoms in the ground state are pumped up to the energy level E 3 by incoming photons of energy h 13 = E 3 –E 1. b)Atoms at E 3 rapidly decay to the metastable state at energy level E 2 by emitting photons or emitting lattice vibrations; h 32 = E 3 –E 2. Three energy level system

9. 9 The principle of the laser c)As the states at E 2 are long-lived, they quickly become populated and there is a population inversion between E 2 and E 1. d)A random photon (from a spontaneous decay) of energy h 21 = E 2 –E 1 can initiate stimulated emission. Photons from this stimulated emission can themselves further stimulate emissions leading to an avalanche of stimulated emissions and coherent photons being emitted. The emission from E 2 to E 1 is called the lasing emission.

10. 10 LASER A system for photon amplification LASER  Light Amplification by Stimulated Emission of Radiation

11. 11 4.2 STIMULATED EMISSION RATE AND EINSTEIN COEFFICIENTS

12. 12 Einstein B 12 coefficient N 1 : the number of atoms per unit volume with energy E 1 N 2 : the number of atoms per unit volume with energy E 2 R 12 : the rate of transition from E 1 to E 2 by absorption  R 12 = B 12 N 1  (h ) –B 12 : the Einstein coefficient for absorption. –  (h ) : the photon energy density per unit frequency

13. 13 Einstein A 21, B 21 coefficients R 21 : the rate of transition from E 2 to E 1 by spontaneous and stimulated emission  R 12 = A 21 N 2 + B 21 N 2  (h ) –A 21 : the Einstein coefficient for spontaneous emission –B 12 : the Einstein coefficient for stimulated emission

14. 14 In thermal equilibrium (no external excitation) There is no net change with time in the population at E 1 anf E 2  R 12 = R 21 Boltzmann statistics demands

15. 15 Plank’s black body radiation distribution law In thermal equilibrium, radiation from the atoms must give rise to an equilibrium photon energy density,  eq (h ), that is given by

16. 16

17. 17 The ratio of stimulated emission to adsorption  For stimulated photon emission to exceed photon absorption, we need to achieve population inversion, that is N 2 > N 1.

18. 18 The ratio of stimulated to spontaneous emission  For stimulated photon emission to far exceed spontaneous emission, we must have a large photon concentration which is achieved by building an optical cavity to contain the photons.

19. 19 Population inversion Boltzmann statistics N 2 > N 1  T < 0, i.e. a negative absolute temperature! The laser principle is based on non- thermal equilibrium.

20. 20 4.3 OPTICAL FIBER AMPLIFIERS

21. 21 Optical amplifier A light signal that is traveling along an optical fiber over a long distance suffers marked attenuation. It becomes necessary to regenerate the light signal at certain intervals for long haul communications over several thousand kilometers. We can amplify the signal directly by using an optical amplifier

22. 22 Erbium doped fiber amplifier (EDFA) Host fiber core material: Glass based on SiO 2 -GeO (or other glass forming oxide such as Al 2 O 3 ) Dopants in the core region: Er 3+ ions (or other rare earth ions such as neodymium (Nd 3+ ) ions). It is easily fused to a single mode long distance fiber by a technique called splicing.

23. 23 Energy diagram for Er 3+ in glass fiber medium Er 3+ ions are pumped (by a laser diode ~980 nm) from E 1 to E 3. Er 3+ ions non-radiatively decay rapidly from E 3 to a long-live (  ~10ms) level E 2.  Population inversion between E 2 and E 1. Signal photons at 1550 nm (0.80 eV) give rise to stimulated transitions from E 2 to E 1. from a laser diode long-lived level

24. 24 Optical gain N 1 : the number of Er 3+ ions at E 1 N 2 : the number of Er 3+ ions at E 2 Absorption rate ( E 1 to E 2 )  N 1 Stimulated emission rate (E 2 to E 1 )  N 2 Net optical gain G op where K is a constant that depends on the pumping intensity.

25. 25 Schematic diagram of an EDFA The Er 3+ -doped fiber is pumped by feeding the light from a laser pump diode, through a coupler, into the Er 3+ -doped fiber. Optical isolators inserted at the entry and exit end allow only the signals at 1550 nm to pass in one direction and prevent the 980 nm light from propagation back or forward into the communication system.

26. 26 4.4 GAS LASERS: THE He-Ne LASER

27. 27 HeNe laser Lasing emission: 632.8 nm  red color The actual stimulated emission occurs from the Ne atoms. He atoms are used to excite the Ne atoms by atomic collisions.

28. 28 HeNe laser The HeNe laser consists of a gaseous mixture of Ne and He atoms in a gas discharge tube. An optical cavity is formed by the end-mirrors so that reflection of photons back into the lasing medium builds up the photon concentration in the cavity.

29. 29 He and Ne atoms Ne –inert gas –ground state: 1s 2 2s 2 2p 6 (represented by 2p 6 ) –excited state: 2p 5 5s 1 He –inert gas –ground state: 1s 2 –exited state: 1s 1 2s 1

30. 30 Excited He atom Dc or RF high voltage  electrical discharge  He atoms become excited by collisions with the drifting electrons: He + e   He* + e  He* –excited He atom (1s 1 2s 1 ) with parallel spins –metastable (long lasting) state He* (1s 1 2s 1 )  He (1s 2 )   l   1  no photon emission or absorption  He* cannot spontaneously emit a photon and decay down to the He (1s 2 ) ground state.  Large number of He* build up during the electrical discharge.

31. 31 He*-Ne collisions He* (1s 1 2s 1 )  He (1s 2 )   E = 20.61 eV Ne* (2p 5 5s 1 )  Ne (1p 6 )   E = 20.66 eV When He* collides with Ne, it transfers its energy to Ne: He* + Ne  He + Ne* With many He*-Ne collisions  population inversion between (2p 5 5s 1 ) and (2p 5 3p 1 ) states

32. 32 The principle of HeNe laser A population inversion between (2p 5 5s 1 ) and (2p 5 3p 1 ) states A spontaneous emission from one Ne* atom falling from (2p 5 5s 1 ) to (2p 5 3p 1 ) gives rise to an avalanche of stimulated emission processes. Lasing emission: = 632.8 nm in the red.

33. 33 Lasing transitions in the HeNe laser (2p 5 5s 1 )  (2p 5 3p 1 )  = 632.8 nm (red)  = 543.5 nm (green) (2p 5 5s 1 )  (2p 5 4p 1 )  = 3.39  m (infrared) To suppress lasing emissions at the unwanted wavelengths, the reflecting mirrors can be made wavelength selective.

34. 34 Lasing transitions in the HeNe laser (2p 5 5s 1 )  (2p 5 3p 1 )  = 632.8 nm (red)  = 543.5 nm (green) (2p 5 5s 1 )  (2p 5 4p 1 )  = 3.39  m (infrared) To suppress lasing emissions at the unwanted wavelengths, the reflecting mirrors can be made wavelength selective.

35. 35 HeNe laser Lasing intensity  as tube length  since more Ne atoms are used in stimulated emission. Lasing intensity  as tube diameter  since Ne atoms in the (2p 5 3s 1 ) states can only return to the ground state by collisions with the walls of tube. He : Ne = 5 : 1 Pressure ~ torrs

36. 36 EXAMPLE 4.4.1 Efficiency of the HeNe laser A typical low-power 5 mW HeNe laser tube operates at a dc voltage 2000V and carries a current of 7 mA. What is the efficient of the laser?

37. 37 EXAMPLE 4.4.1 Efficiency of the HeNe laser Solution Typical HeNe laser efficiencies < 0.1% Laser has high concentration of coherent photon: Beam diameter = 1 mm

38. 38 EXAMPLE 4.4.2 Laser beam divergence A typical He-Ne laser –Diameter of output beam = 1 mm –Divergence = 1 mrad What is the diameter of the beam at a distance of 10 m?

39. 39 EXAMPLE 4.4.2 Laser beam divergence Solution We can assume that the laser beam emanates like a light-cone.

40. 40 4.5 THE OUTPUT SPECTRUM OF A GAS LASER

41. 41 Output radiation from a gas laser The output radiation from a gas laser is not actually at one single well-defined wavelength corresponding to the lasing transition, but covers a spectrum of wavelengths with a central peak.  The result of the broadening of the emitted spectrum by the Doppler effect.

42. 42 Kinetic molecular theory The gas atoms are in random motion with an average kinetic energy  We have to consider the random motion of atoms in the laser tube.

43. 43 Doppler Effect When a gas atom is moving away from the observer 1 = 0 (1  v x /c) When a gas atom is moving towards the observer 2 = 0 (1+ v x /c) where 0 : the source frequency 1, 2 : the detected frequency v x : the relative velocity of the atom along the laser tube (x-axis) c : the speed of light

44. 44 Doppler broadened linewidth Since the atoms are in random motion the observer will detect a range of frequencies due to the Doppler effect  Doppler broadened linewidth

45. 45 Maxwell velocity distribution The velocity of gas atoms obey the Maxwell distribution  The stimulated emission wavelength exhibits a distribution about a central wavelength 0 = c/ 0 P( v x, v y, v z ) d v x d v y d v z = The probability for finding the atom within velocity range of (v x +d v x, v y + d v y, v z + d v z )

46. 46 Optical gain lineshape The variation in the optical gain with the wavelength is called the optical gain lineshape. For the Doppler broadening, this lineshape turns out to be a Gaussian function.  = 2 - 1 ~ 2-5 GHz (for many gas lasers)  ~ 0.02 A for He-Ne laser

47. 47 Frequency linewidth When we consider the Maxwell velocity distribution, we find that the linewidth  1/2 between the half-intensity points ( full width at half maximum FWHM) is given by M : the mass of the lasing atom or molecule

48. 48 Laser cavity modes Consider an optical cavity of length L with parallel end mirrors  a Fabry-Perot optical resonator or etalon Only standing waves with certain wavelengths can be maintained within the optical cavity where m is the mode number Modes that exist along the cavity axis are called axial (or longitudinal) modes Laser cavity modes in a gas laser

49. 49 Output spectrum We have spikes of intensity in the output. There is a finite width to the individual intensity spikes which is primarily due to nonidealities of the optical cavity such as thermal fluctuation of L and nonideal mirrors ( R < 100 %) –Typical He-He laser  spike width ~1 MHz –Highly stabilized gas laser  spike width ~ 1 kHz

50. 50 EXAMPLE 4.5.1 Doppler broadened linewidth He-Ne laser –Transition for = 632.8 nm –Gas discharge temperature ~ 127  C –Atomic mass of Ne = 20.2 (g/mol) –Laser tube L = 40 cm Linewidth in the output wavelength spectrum? Mode number m of the central wavelength? Separation between two consecutive modes? How many modes within the linewidth  1/2 of the optical gain curve?

51. 51 EXAMPLE 4.5.1 Doppler broadened linewidth Solution The mass of He atom The central frequency The rms velocity v x along x

52. 52 EXAMPLE 4.5.1 Doppler broadened linewidth Solution (cont.) The rms frequency linewidth The observed FWHM frequency width

53. 53 EXAMPLE 4.5.1 Doppler broadened linewidth Solution (cont.) To get FWHM wavelength width  1/2, differentiate = c/ For = 0 = 632.8 nm, the corresponding mode number m 0 is actual m 0 has to be the closest integer value to 1.264  10 6

54. 54 EXAMPLE 4.5.1 Doppler broadened linewidth Solution (cont.) The separation  m between two consecutive modes The number of modes within the linewidth We can expect at most 4 to 5 modes within the linewidth of the output.

55. 55 EXAMPLE 4.5.1 Doppler broadened linewidth Solution (cont.) Number of laser modes depends on how the cavity modes intersect the optical gain curve.

56. 56 4.6 LASER OSCILLATION CONDITION

57. 57 4.6 LASER OSCILLATION CONDITIONS Energy band Diagrams

58. 58 4.6 LASER OSCILLATION CONDITIONS A. Optical Gain Coefficient g

59. 59 Consider an EM wave propagating in a medium along the x -direction. If the light intensity were decreasing If the light intensity were increasing

60. 60 Optical gain coefficient  The gain coefficient g is defined as the fractional change in the light power (or intensity) per unit distance

61. 61 Optical gain coefficient Optical power –N ph : the concentration of cohenernt photons –h : photon energy In time  t photons travel a distance –n: refractive index The optical gain coefficient

62. 62 We can neglect spontaneous emission which are in random direction and do not, on average contribute to the directional wave.  Stimulated emission  Absorption

63. 63 Lineshape function The emission and absorption processes would be distributed in photon energy or frequency over some frequency interval  ( due to Doppler broadening or broadening of the energy levels E 2 and E 1 ). The optical gain will reflect this distribution The spectral shape of the gain curve is called the lineshape function.

64. 64 General optical gain coefficient  General optical gain coefficient  This equation gives the optical gain of the medium at the center frequency 0 Radiation energy density per unit frequency at h 0

65. 65 4.6 LASER OSCILLATION CONDITIONS B. Threshold Gain g th

66. 66 Power condition for maintaining oscillations Consider an optical cavity with mirrors at the ends. The cavity contains a laser medium so that lasing emissions build up to a steady state. P i : initial optical power P f : final optical power Under steady state condition, oscillations do not build up and do not die out Net round-trip optical gain Power condition for maintaining oscillations

67. 67 Reflections at the faces 1 and 2 reduce the optical power by the reflectances R 1, and R 2 of the faces. As the wave propagates, its power increases as exp( g x) Losses –Absorption by impurities and free carriers –Scattering at defects and inhomogenities  These losses decrease the power as exp(  x) –  : the attenuation or loss coefficient of the medium

68. 68 Threshold optical gain The power P f after one round trip of path length 2L is given by For steady state oscillations G op =P f /P i = 1 must be satisfied g th is the optical gain needed in the medium to achieve a continuous wave lasing emission. Threshold optical gain

69. 69 Threshold population inversion The necessary g th has to be obtained by suitably pumping the medium so that N 2 is sufficiently greater than N 1. This corresponds to a threshold population inversion or N 2  N 1 = (N 2  N 1 ) th Form Threshold population inversion

70. 70 Until the pump rate can bring (N 2  N 1 ) to the threshold value (N 2  N 1 ) th, there would be no coherent radiation output. When the pumping rate exceeds the threshold value, then (N 2  N 1 ) remains clamped at (N 2  N 1 ) th because this controls the optical gain g which must remain at g th. Additional pumping increases the rate of stimulated transitions and hence increases the optical output power P 0.

71. 71 4.6 LASER OSCILLATION CONDITIONS C. Phase Condition and Laser Modes

72. 72 Unless the total phase change after one round trip from E i to E f is a multiple of 2 , the wave E f cannot be identical to the initial wave E i. We need an additional condition: Phase condition for laser oscillations Phase condition for laser oscillations

73. 73 Longitudinal axial modes If k = 2  / is the free space wavevector, only those special k value, denoted as k m, that satisfy  round-trip = m(2  ) can exits as radiation in the cavity  The usual mode condition These modes are controlled by the length L of the optical cavity along its axis and are called longitudinal axial modes.

74. 74 Off-axis transverse mode All practical laser cavity have a finite transverse size, a size perpendicular to the cavity axis. Furthermore, not all cavities have flat reflectors at the ends.  An off-axis mode is able to self-replicate after one round trip.

75. 75 Off-axis transverse mode Such a mode would be non-axial. Its properties would be determined not only by the off-axis round-trip distance and but also by the transverse size of the cavity. The greater the transverse size, the more of these off-axis modes can exist.

76. 76 Transverse modes A mode represents particular electric field pattern in the cavity that can replicate itself after one round trip. A mode with a certain field pattern at a reflector can propagate to the other reflector and back again return the same field pattern.  Transverse modes or transverse electric and magnetic (TEM) modes.

77. 77 TEM modes Each allowed mode corresponds to a distinct spatial field distribution at a reflector. TEM p,q,m –p: the number of nodes in the field distribution along the transverse direction y –q: the number of nodes in the field distribution along the transverse direction z –m: the number of nodes along the cavity axis x (usually m is very large (~ 10 6 in gas lasers) and is not written.)

78. 78 TEM 00 mode The lowest order mode. Gaussian intensity distribution across the beam cross section everywhere inside and outside cavity. It has the lowest divergence angle. Many laser designs optimize on TEM 00 while suppressing other modes. Such a design usually requires restrictions in the transverse size of the cavity.

79. 79 EXAMPLE 4.6.1 Threshold population inversion for the He-Ne laser Show that the threshold population inversion  N th = (N 2 - N 1 ) th can be written as – 0 = peak emission frequency (at peak of output spectrum) – n = refractive index –  sp = 1/A 21 = mean time for spontaneous transition –  = optical gain bandwidth (frequency-linewidth of the optical gain linewidth)

80. 80 EXAMPLE 4.6.1 Threshold population inversion for the He-Ne laser (cont.) He-Ne laser – 0 = 632.8 nm –L = 50 cm –R 1 = 100% –R 2 = 90% –linewidth  = 1.5 GHz –loss coefficient   0.05 m -1 –  sp = 1/A 21  300 ns –n  1 What is the threshold population inversion ?

81. 81 EXAMPLE 4.6.1 Threshold population inversion for the He-Ne laser Solution The relationship between Einstein A, B coefficients is The threshold population inversion is then

82. 82 EXAMPLE 4.6.1 Threshold population inversion for the He-Ne laser Solution (cont.) For He-Ne laser –The emission frequency –The threshold gain

83. 83 EXAMPLE 4.6.1 Threshold population inversion for the He-Ne laser Solution (cont.) –The threshold population inversion This is the threshold population inversion for Ne atoms in configuration 2p 5 5s 1 and 2p 5 3p 1 N2N2 N1N1

84. 84 4.7 PRINCIPLE OF THE LASER DIODE

85. 85 Degenerately doped pn junction Fermi level: –E Fp in the p-side is in the VB –E Fn in the n-side is in the CB No bias  Fermi level is continuous across the diode  E Fp = E Fn Depletion region (SCL) is very narrow. Built-in potential barrier eV 0 prevents electron (hole) diffusion from n + (p + )-side to p + (n + )-side.

86. 86 Forward bias If eV =  E F > E g  The applied voltage diminishes the built-in potential barrier to almost zero  Electrons from n + -side and holes from p + -side flow into the SCL.  SCL region is no longer depleted.

87. 87 Population inversion In the SCL, there are more electrons in the CB at energies near E c than electrons in the VB near E v.  There is a population inversion between energies near E c and those near E v around the junction.

88. 88 Inversion layer The population inversion region is a layer along the junction and is called the inversion layer or the active region. The region where there is population inversion has an optical gain because an incoming photon is more likely to cause stimulated emission than being absorbed.

89. 89 At low temperature (T  0 K) The states between E c and E Fn are filled with electrons and those between E Fp and E v are empty. E g < hv (photon energy) < E Fn – E Fp  Stimulated emission hv (photon energy) > E Fn – E Fp  Absorption

90. 90 At T > 0 As the temperature increases  the Fermi-Dirac function spreads the energy distribution of electron in CB to above E Fn and hole below E Fp in the VB  a reduction in optical gain The optical gain depends on E Fn – E Fp which depends on the applied voltage and hence on the diode current.

91. 91 Injection pumping The population inversion between energies near E c and those near E v is achieved by the injection of carriers across the junction under a sufficiently large forward bias. The pumping mechanism is the forward diode and the pumping energy is supplied by external battery. This type of pumping is called injection pumping.

92. 92 Homojunction laser diode The pn junction uses the same direct bandgap semiconductor material. The ends of the crystal are cleaved to be flat and optically polished to provide reflection and hence form an optical cavity. Photons that are reflected from the cleaved surface stimulate more photons of the same frequency and so on.

93. 93 Modes in optical cavity The wavelength of the radiation that can builds up in the cavity is determined by the length L of the cavity Each radiation satisfying the above relationship is essentially a resonant frequency of the cavity, that is a mode of the cavity.

94. 94 Threshold current Lasing radiation is only obtained when the optical gain in the medium can overcome the photon losses from the cavity, which require the diode current I exceed a threshold value –I th : threshold current

95. 95 Characteristics of a laser diode I < I th : Spontaneous emission I > I th : Stimulated emission Above I th, the light intensity becomes coherent radiation consisting of cavity wavelength (or modes) and increases steeply with current. The number of modes in the output spectrum and their relative strengths depend on the diode current.

96. 96 Main problem with the homojunction laser diode The threshold current density J th is too high for practical uses. Ex. for GaAs at room temperature J th ~ 500 A mm -2  GaAs homojunction laser can only be operated continuously at very low temperature. J th can be reduced by orders of magnitude by using heterostructured semiconductor laser diodes.

97. 97 4.8 HETEROSTRUCTURE LASER DIODES

98. 98 Reduction of threshold current I th Requires Improving the rate of stimulated emission Improving the efficiency of the optical cavity We need both Carrier confinement Photon confinement

99. 99 Carrier confinement We can confine the injected electrons and holes to a narrow region around the junction.  Narrowing of the active region  Less current is needed to establish the necessary concentration of carriers for population inversion.

100. 100 Photon confinement We can build a dielectric waveguide around the optical gain region  To increase the photon concentration and hence the probability of stimulated emission.

101. 101 Double heterostructure (DH) device A DH device is based on two junctions between different semiconductor materials with different bandgaps. –E g (AlGaAs)  2 eV –E g (GaAs)  1.4 eV p-GaAs region : thin layer ~ 0.1 – 0.2  m  the active layer in which lasing recombination takes place.

102. 102 Carrier confinement p-GaAs and p-AlGaAs –heavily p-type doped  degenerate with E F in the VB When a sufficiently large forward bias is applied –E c of n-AlGaAs moves above E c of p-GaAs  A large injection of electrons in the CB of n-AlGaAs into p-GaAs  These electrons are confined to the CB of p-GaAs since there is a barrier  E c between p-GaAs and p-AlGaAs due to the change in the bandgap.

103. 103 p-GaAs is a thin layer  The concentration of injected electrons in the p-GaAs layer can be increased quickly even with moderate increases in forward current.  This effectively reduces the threshold current for population inversion or optical gain.

104. 104 Photon confinement E g (AlGaAs) > E g (GaAs)  Index n (AlGaAs) < Index n(GaAs)  The change in the refractive index defines an optical dielectric waveguide that confines the photons to the active region of the optical cavity.  This increase in the photon concentration increases the rate of stimulated emission.

105. 105 Double heterostructure laser diode The p and n-AlGaAs layers provide carrier and optical confinement in the vertical direction by forming heterojunctions with p-GaAs. The advantage of the AlGaAs/GaAs heterojunction  only a small lattice mismatch between the two crystal stuuctre and hence negligible strain induced interfacial defects (e.g. dislocations) in the device.

106. 106 Contacting layer There is an additional p-GaAs layer, call contacting layer, next to p-AlGaAs. It can be seen that the electrodes are attached to the GaAs semiconductor materials rather than AlGaAs. This choice allows for better contacting and avoids Schottky junctions which limit the current.

107. 107 Stripe geometry Stripe geometry or stripe contact on p-GaAs J is greatest along the central path 1, and decreases away from path1, towards 2 or 3. The current is confined to flow within paths 2 and 3. The current density paths through the active layer where J is greater than the threshold value J th define the active region where population inversion and hence optical gain takes place.

108. 108 Gain guided lasers The width of the active region, or the optical gain region, is defined by the current density from the stripe contact. Optical gain is highest where the current density is greatest. Such lasers are called gain guided.

109. 109 Advantages to using a stripe geometry The reduced contact area also reduces the threshold current I th –Typical strip widths W ~ few  m  I th ~ tens of mA. The reduced emission area makes light coupling to optical fibers easier.

110. 110 Reducing the reflection losses from the rear crystal facet n (GaAs) = 3.7  Reflectance ~ 0.33 Dielectric mirror at the rear facet  Reflectance  1  Optical gain   I th 

111. 111 Buried DH laser diode The active layer, p-GaAs, is bound vertically and laterally by a wider bandgap semiconductor, AlGaAs, which has lower refractive index.  The photons are confined to the active or optical gain region which increases the rate of stimulated emission and hence efficiency of the diode.

112. 112 Index guided LDs The optical power is confined to the waveguide defined by the refractive index variation, these diodes are called index guided. If the buried heterostructure has the right dimension compared with the wavelength of the radiation then only the fundamental mode can exist in the waveguide structure  single mode laser diode.

113. 113 EXAMPLE 4.8.1 Modes in a laser and the optical cavity length AlGaAs based heterostructure laser diode –Optical cavity L = 200  m –Peak radiation = 900 nm –Refractive index of GaAs n  3.7 What is the mode integer m of the peak radiation and the separation between the modes of the cavity? If the optical gain has a FWHM wavelength width  1/2  6 nm, how many modes are there within the bandwidth? How many modes are there if L = 20  m?

114. 114 EXAMPLE 4.8.1 Modes in a laser and the optical cavity length Solution The wavelength separation  m between modes m and m+1 is

115. 115 EXAMPLE 4.8.1 Modes in a laser and the optical cavity length Solution (cont.)  There will be 10 modes within the bandwidth. If L=20  m,  There will be one mode that corresponds to about 900 nm.

116. 116 EXAMPLE 4.8.1 Modes in a laser and the optical cavity length Solution (cont.)  Reducing the cavity length suppresses higher modes.

117. 117 4.9 ELEMENTARY LASER DIODE CHARACTERISTICS

118. 118 Factors of the output spectrum of a laser diode (LD) The nature of the optical resonator used to build the laser oscillations The optical gain curve (lineshape) of the active medium.

119. 119 Laser cavity Fabry-Perot cavity Length L determines the longitudinal mode separation. Width W and Height H determine the transverse modes (or lateral modes).

120. 120 If the transverse dimensions ( W and H ) are sufficiently small, only the lowest transverse mode, TEM 00 mode, will exit. This TEM 00 mode will have longitudinal modes whose separation depends on L.

121. 121 Beam divergence The emerging laser beam exhibits divergence. This is due to diffraction of the waves at the cavity ends. The smallest aperture ( H ) causes the greatest diffraction.

122. 122 Output spectra from LDs The spectrum is either multimode or single mode depending on the optical resonator structure and pumping current level. Index guided LDs –Low output power  multimode –High output power  single mode Gain guided LDs –tend to remain multimode even at high diode currents.

123. 123 Temperature dependence of the LD output As the temperature increases, the threshold current increases steeply, typically as the exponential of the absolute temperature.

124. 124 Mode hopping Single mode LDs The peak emission wavelength 0 exhibits “jumps” at certain temperatures.  At new operating temperature, another mode fulfills the laser oscillation condition.

125. 125 Mode hopping Between mode hops, 0 increases slowly with T due to the slight increase in n and L. If mode hope are undesirable, then the device structure must be such to keep the modes sufficiently separated.

126. 126 Multimode LDs Gain guided LDs  The output spectrum has many modes so that 0 vs. T behavior tends to follow the changes in the bandgap rather than the cavity properties.

127. 127 Slope efficiency  slope –P o : the output optical power –I : the diode current –I th : the threshold current Slope efficiency determines the output optical power in terms of the diode current above the threshold current. Slope efficiency depends on the LD structure as well as the semiconductor packaging. Typically,  slope  1 W/A

128. 128 Conversion efficiency The conversion efficiency gauges the overall efficiency of the conversion from the input of electrical power to the output of optical power. In some modern LDs this maybe as high as 30- 40 %.

129. 129 EXAMPLE 4.9.1 Laser output wavelength variations The refractive index n of GaAs has a temperature dependence Estimate the change in the emitted wavelength 870 nm per degree in temperature between mode hops.

130. 130 EXAMPLE 4.9.1 Laser output wavelength variations Solution Index n will also depend on the optical gain of the medium and hence its temperature dependence is likely to be somewhat higher than the dn/dt value we used.

131. 131 4.10 STEADY STATE SEMICONDUCTOR RATE EQUATIONS

132. 132 Active layer rate equation Consider a DH LD under forward bias. Under steady state operation: d : thickness, L : length, W : width, I : current, n : the injected electron concentration, N ph : the coherent photon concentration in the active layer,  sp : the average time for spontaneous recombination, C : a constnt (depends on B 21 ) Active layer rate equation

133. 133 Active layer rate equation As the current increases and provides more pumping, N ph increases (helped by the optical cavity), and eventually the stimulated term dominates the spontaneous term. The output light power P 0 is proportional to N ph.

134. 134 Rate of stimulated emissions Consider the coherent photon concentration N ph in the cavity. Under steady state condition:  ph : the average time for a photon to be lost from the cavity due to transmission through the end-faces, scattering and absorption in the semiconductor.

135. 135 Threshold concentration n th : threshold electron concentration I th : threshold current n th and I th  the condition when the stimulated emission just overcomes the spontaneous emission and the total loss mechanisms in  ph. This occurs when injected n reaches n th Threshold concentration

136. 136 Threshold current When I > I th, the output optical power increases sharply with current  N ph = 0 when I = I th Threshold current

137. 137 Threshold current  I th decreases with d, L and W  low I th for the heterostructure and stripe geometry lasers.

138. 138 Above threshold, form with n clamped at n th, Using and defining J = I/WL we can find Coherent photon concentration

139. 139 Output optical power  t = nL/c : the time for photons crossing the laser cavity length L (1  R): the fraction of the radiation power that will escape. (1/2) N ph  Only half of the photons in the cavity would be moving towards the output face of the crystal

140. 140 Laser diode equation

141. 141

142. 142 4.11 LIGHT EMITTERS FOR OPTICAL FIBER COMMUNICATIONS

143. 143 Light emitters for optical communications LEDs –simpler to drive –more economic –have a longer lifetime –provide the necessary output power –wider output spectrum  Short haul application (e.g. local networks) Laser diodes –narrow linewidth –high output power –higher signal bandwidth capability  Long-haul and wide bandwidth communications

144. 144

145. 145

146. 146 Rise time The speed of response of an emitter is generally described by a rise time. Rise time is the time it takes for the output optical power to rise from 10% to 90% in response to a step current input. LED: Rise time  5-20 ns Laser diode: Rise time < 1 ns  Laser diodes are used whenever wide bandwidths are required.

147. 147 4.12 SINGLE FREQUENCY SOLID STATE LASERS

148. 148 Active layer rate equation Consider a DH LD under forward bias. Under steady state operation: d : thickness, L : length, W : width, I : current, n : the injected electron concentration, N ph : the coherent photon concentration in the active layer,  sp : the average time for spontaneous recombination, C : a constnt (depends on B 21 ) Active layer rate equation

149. 149 Active layer rate equation As the current increases and provides more pumping, N ph increases (helped by the optical cavity), and eventually the stimulated term dominates the spontaneous term. The output light power P 0 is proportional to N ph.

150. 150 Rate of stimulated emissions Consider the coherent photon concentration N ph in the cavity. Under steady state condition:  ph : the average time for a photon to be lost from the cavity due to transmission through the end-faces, scattering and absorption in the semiconductor.

151. 151  ph : the average time for a photon to be lost from the cavity due to transmission through the end-faces, scattering and absorption in the semiconductor.  t : the total attenuation coefficient representing all the loss mechanisms.

152. 152 Threshold concentration n th : threshold electron concentration I th : threshold current n th and I th  the condition when the stimulated emission just overcomes the spontaneous emission and the total loss mechanisms in  ph. This occurs when injected n reaches n th Threshold concentration

153. 153 Threshold current When I > I th, the output optical power increases sharply with current  N ph = 0 when I = I th Threshold current

154. 154 Threshold current  I th decrease with d, L and W  low I th for the heterostructure and stripe geometry lasers.

155. 155 Above threshold, form with n clamped at n th, Using and defining J = I/WL we can find Coherent photon concentration

156. 156 Output optical power  t = nL/c : the time for photons crossing the laser cavity length L (1  R): the fraction of the radiation power that will escape. (1/2) N ph  Only half of the photons in the cavity would be moving towards the output face of the crystal

157. 157 Laser diode equation

158. 158

159. 159 4.11 LIGHT EMITTERS FOR OPTICAL FIBER COMMUNICATIONS

160. 160 Light emitters for optical communications LEDs –simpler to drive –more economic –have a longer lifetime –provide the necessary output power –wider output spectrum  Short haul application (e.g. local networks) Laser diodes –narrow linewidth –high output power –higher signal bandwidth capability  Long-haul and wide bandwidth communications

161. 161

162. 162

163. 163 Rise time The speed of response of an emitter is generally described by a rise time. Rise time is the time it takes for the output optical power to rise from 10% to 90% in response to a step current input. LED: Rise time  5-20 ns Laser diode: Rise time < 1 ns  Laser diodes are used whenever wide bandwidths are required.

164. 164 4.12 SINGLE FREQUENCY SOLID STATE LASERS

165. 165 Methods of ensuring a single mode of radiation in the laser cavity Using frequency selective dielectric mirrors at the cleaved surface.  Distributed Bragg reflector (DBR) laser Using a corrugated guiding layer next to the active layer.  Distributed feedback (DFB) laser Using two different optical cavities which are coupled.  Cleaved-coupled-cavity (C 3 ) laser

166. 166 Distributed Bragg reflector (DBR) The DBR is a mirror that has been designed like a reflection type diffraction grating. It has a periodic corrugated structure. Partial reflection from the corrugations interface constructively to give a reflected wave only when the wavelength corresponds to twice the corrugation periodicity.

167. 167 Distributed Bragg reflector (DBR)  : the period of corrugation 2  = the optical path difference between waves A and B A and B interfere constructively if 2  is a multiple of the wavelength within the medium. Each of these wavelength is called a Bragg wavelength B : –n : the refractive index of the corrugated material –q : the diffraction order The DBR has a high reflectance around B but low reflectance away from B  Only that particular cavity mode, within the optical gain curve, that is close to B can lase and exist in the output.

168. 168 Distributed feedback (DFB) laser In the normal laser, the crystal faces provides the necessary optical feedback into the cavity to build up the photon concentration. In the distributed feedback (DFB) laser, there is a corrugated layer, call the guiding layer, next to the active layer; radiation spreads from the active layer to the guiding layer.

169. 169 Distributed feedback (DFB) laser These corrugations in the refractive index act as optical feedback over the length of the cavity by producing partial reflections.  Optical feedback is distributed over the cavity length.

170. 170 Allowed DFB lasing modes Traveling waves are reflected partially and periodically as they propagate. The medium alters the wave-amplitudes via optical gain. The allowed DFB modes:

171. 171 Allowed DFB lasing modes The relative threshold gain for higher modes is so large that only the m = 0 mode effectively lases. A perfect symmetric device has two equally spaced modes placed around B. In reality, either inevitable asymmetry introduced by the fabrication process, or asymmetry introduced on purpose, leads the only one of the modes to appear. Typically, L >>   B >> B 2 /(2n L )  0  B

172. 172 Cleaved-coupled-cavity (C 3 ) laser In the cleaved-coupled-cavity (C 3 ) laser device, two different laser optical cavities L and D (different lengths) are coupled. The two lasers are pumped by different currents. Only those waves that can exist as modes in both cavities are now allowed because the system has been coupled. The wide separation between the modes results in a single mode operation.

173. 173 EXAMPLE 4.12.1 DFB Laser DFB laser –Corrugation period  = 0.22  m –Grating length L= 400  m –Effective refractive index of medium n  3.5 Assuming a first order grating, calculate –the Bragg wavelength –the mode wavelengths and their separation

174. 174 EXAMPLE 4.12.1 DFB Laser Solution The Bragg wavelength is given by The symmetric DFB laser wavelengths about B are The m = 0 mode wavelengths are The two are separated by 0.0017  m (or 1.7 nm). Due to asymmetry, only one mode will appear in the output and for most practical purpose wavelength can be taken as B.

175. 175 4.13 QUANTUM WELL DEVICES

176. 176 Quantum well device A typical quantum well (QW) device has an ultra thin (typically, d < 50 nm ) narrow bandgap semiconductor sandwiched between two wider bandgap semiconductors. A QW device is a heterostructure device. Ex. AlGaAs/GaAs/AlGaAs system.

177. 177 Quantum well device  E c and  E v, the discontinuities in E c and E v at the interfaces, depend on the semiconductor materials and their doping. GaAs/AlGaAs heterostructure:  E c : the potential energy barrier  The conduction electrons are confined in the x -direction but free in the yz plane.  Two dimensional electron gas confined in the x -direction.

178. 178 Energy in a quantum well device D y, D z >> d  the minimum energy E 1  n = 1 E 2  n = 2 E 3  n = 3

179. 179 Density of states g( E ) g( E ) = number of quantum states per unit volume In the 2D electron gas, g( E ) is constant at E 1 until E 2 where it increases as a step and remains constant until E 3 where again it increases as a step by the same amount and at every value of E n. In the bulk semiconductor, g( E )  E 1/2

180. 180 Single quantum well (SQW) lasers g( E ) is finite and substantial at E 1  A large concentration of electrons in the CB can easily occur at E 1. Similarly, a large concentration of holes in the VB can easily occur at E 1. Under a forward bias:  Electrons are injected into the GaAs layer (active layer).  The electron concentration at E 1 increases rapidly with current.  Population inversion occurs quickly without the need of large current. 

181. 181 Advantages of the SQW lasers The threshold current is markedly reduced. –SQW laser: I th ~ 0.5  1 mA –Double heterostructure laser: I th ~ 10  50 mA Majority of electrons are at and near E 1 and holes are at and near E 1  The range of emitted photon energies are very close to E 1  E 1  The linewidth in the output spectrum is substantially narrower than that in bulk semiconductor laser.

182. 182 Multiple quantum well (MQW) lasers

183. 183 EXAMPLE 4.13.1 A GaAs quantum well GaAs quantum well –Effective mass of electron in GaAs m e * = 0.07 m e –Effective mass of hole in GaAs m h * = 0.50 m e –Thickness d = 10 nm Calculate –First two electron energy levels –First hole energy level –The change in the emission wavelength with respect to bulk GaAs ( E g = 1.42 eV)

184. 184 EXAMPLE 4.13.1 A GaAs quantum well Solution The electron energy levels with respect to E c in GaAs The hole energy levels below E v

185. 185 EXAMPLE 4.13.1 A GaAs quantum well Solution (cont.) The wavelength of emission from bulk GaAs with E g = 1.42 eV The wavelength of emission from GaAs QW The difference is

186. 186 4.14 VERTICAL CAVITY SURFACE EMITTING LASERS (VCSELs)

187. 187 Vertical cavity surface laser (VCSELs) The optical cavity axis is along the direction of current flow rather than perpendicular to the current flow as in conventional laser diodes. The active region length is very short (< 0.1  m) compared with the lateral dimensions  High reflectance end mirrors (~ 90%) are needed because the short cavity length L reduced the optical gain of the active layer.

188. 188 Vertical cavity surface laser (VCSELs) Dielectric mirrors (~20-30 layers)  distributed Bragg reflector (DBR).  It has a high degree of wavelength selective reflectance at the required free-space wavelength

189. 189 4.15 OPTICAL LASER AMPLIFIERS

190. 190 Laser amplifiers A semiconductor structure can also be used as an optical amplifier that amplifies light waves passing through its active region. The wavelength of radiation to be amplified must fall within the optical gain bandwidth of the laser. Such a device would not be a laser oscillator, emitting lasing emission without an input, but an optical amplifier with input and output ports for light entry. (b)

191. 191 Traveling wave semiconductor laser amplifier The ends of the optical cavity have antireflection (AR) coating.  The optical cavity does not act as an efficient optical resonator, a condition for laser-oscillations. Light from an optical fiber is coupled into the active region of laser structure. As the radiation propagates through the active layer, optically guided by this layer, it becomes amplified by the induced stimulated emissions, and leaves the optical cavity with higher intensity.

192. 192 Fabry-Perot laser amplifier It is operated below the threshold current for laser oscillations.  The active region has an optical gain but not sufficient to sustain a self-lasing output. Light passing through such an active region will be amplified by stimulated emissions. Optical frequencies around the resonant frequencies will experience higher gain than those away from resonant frequencies. Pump current

193. 193 4.16 HOLOGRAPHY

194. 194 Dennis Gabor (1900 - 1979), inventor of holography, is standing next to his holographic portrait. Professor Gabor was a Hungarian born British physicist who published his holography invention in Nature in 1948 while he as at Thomson-Houston Co. Ltd, at a time when coherent light from lasers was not yet available. He was subsequently a professor of applied electron physics at Imperial College, University of London.

195. 195 Holography A technique of reproducing three dimensional optical images of an object by using a highly coherent radiation from a laser source.

196. 196 Principle of holography Reference beam Reflected wave from the cat. E cat will have both amplitude and phase variations that represent the cat’s surface.

197. 197 Reflected wave E cat If we were looking at the cat, our eyes would register the wavefront of the reflected wave E cat. Moving our head around we would capture different portions of the reflected wave and we would see the cat as a three dimensional object.

198. 198 The reflected wave from the cat (Ecat) is made to interfere with the reference wave (Eref) at the photographic plate and give rise to a complicated interference pattern that depends on the magnitude and phase variation in Ecat. The recorded interference pattern in the photographic film is called a hologram. It contains all the information necessary to reconstruct the wavefronts Ecat reflected from the cat and hence produce a three dimensional image.

199. 199 Principle of holography Reference beam The recorded interference pattern in the photographic film Diffracted beam

200. 200 Virtual and real image One diffracted beam is an exact replica of the original wavefront E cat from the cat. The observer sees this wavefront as if the waves were reflected from the original cat and registers a three dimensional image of the cat.  Virtual image There is a second image, called real image, which is of lower quality.

201. 201 We can qualitatively think of a diffracted beam from one locality in the hologram as being determined by the local separation d between interference pattern produced by Ecat, the whole diffracted beam depends on Ecat and the diffracted beam wavefront is an exact scaled replica of E cat.

202. 202 Suppose that the photographic plate is in the xy plane. The reference wave The reflected wave from the cat When E ref and E cat interference, the “brightness” of the photographic image depends on the intensity and hence on

203. 203 The pattern on the photographic plat –UU * : the intensity of the reflected wave –U r U r * : the intensity of the reference wave –U r *U and U r U* contain the information on the magnitude and phase of U(x,y)

204. 204 When we illuminate the hologram, I (x,y), with the reference beam U r, the transmitted light wave has a complex magnitude U t (x,y)

205. 205 a = U r (UU*+U r U r *) : the through beam bU(x,y): the diffracted wave  a scaled version of U(x,y) and represents the original wavefront amplitude from the cat (includes the magnitude and phase information)  wavefront reconstruction  the virtual image cU*(x,y): the diffracted wave  Its amplitude is the complex conjugate of U(x,y)  the real image  the conjugate image The complex magnitude of transmitted wave