Chapter2 Fundamentals of VCSEL and Febry-Perot Resonator

Transcription

1 Chater Fundamentals of VCSEL and Febry-Perot Resonator. Introduction to VCSELs [5-7] What is VCSEL? It is a contracted name of vertical cavity surface emitting laser and it is a kind of semiconductor laser. What s the advantage of VCSEL? What s the alication of VCSEL? Before talking about these, let s review the history of the semiconductor laser. History overview More than forty years have been assed since the solid-state ruby laser and the He-Ne gas laser were successfully made in 960. It was from that time, more efforts have been ut on the research of laser, esecially the ossibility of lasing in semiconductor laser. In 96, first stimulated emission in semiconductor laser was reorted by several grous. Then, the research of the semiconductor laser never sto until now, and the related roducts, such as otical disc layers, laser rinters, and fiber communication links, have layed an imortant role in our daily life. However, the conventional used semiconductor laser, edge emitting laser (EEL), as shown in Fig.(a), still has some roblems, e.g., the initial robe test of such devices is imossible before searating into chis, the monolithic integration of lasers into an otical circuit is limited due to the finite cavity length, and so on. In 977, K. Iga at the Tokyo Institute of Technology, Tokyo, Jaan suggested a vertical cavity surface emitting laser for the urose of overcoming such difficulties as mentioned above [8]. As the name suggests, VCSEL emission occurs erendicular, rather than arallel, to the wafer surface. The cavity is formed by two surfaces of an eitaxial layer, and light outut is taken vertical from one of the mirror surfaces, as deicted in Fig.(b). According to the suggested laser structure, many novel advantages could be done if VCSEL is realized. (a) (b) Figure. Schematic diagram of (a) an EEL and (b) a VCSEL. 4

2 The advantages of VCSEL stated by Iga are as follows [8]. () The laser device is fabricated by a fully monolithic rocess. () A densely acked two-dimensional laser array could be fabricated. (3) The initial robe test could be erformed before searation into chis. (4) Dynamic single longitudinal mode oeration is exected because of its large mode sacing. (5) It is ossible to vertically stack multithin-film functional otical devices on to the VCSEL. (6) A narrow circuit beam is achievable. These benefits of VCSEL encourage the researchers to devote themselves to seeding u the develoment of VCSEL. In 979, Iga demonstrated first VCSEL used.3µm-wavelength GaInAsP/InP material for the active region. In 984, they made a room temerature ulsed oeration GaAs-based device, and the first room temerature continuous wave (CW) oeration GaAs-based VCSEL was fabricated in 987. Until now, Iga has done a lot of research and imrovement on VCSEL. It is worth to say, his outstanding contributions on VCSEL earn him the name of Father of VCSEL. So far, GaAs-based device have been extensively studied and some of the 0.98, 0.85, and 0.78µm wavelength devices are now commercialized into otical systems. Besides, the technique of.3 and.5µm devices has already set u. In the mean while, green-blue-ultraviolet device research has been started. It is believed that the commercial roducts with green-blur-ultraviolet VCSEL will come out in no more future. Tyical structure of VCSEL Tyical structure of VCSEL includes central active region and the to and bottom high reflecting mirror, as shown in Fig.. Besides, the high reflecting mirror is usually in the form of distributed Bragg reflectors (DBRs), comosed of high and low reflection index material quarter wavelength stack, as shown in Fig.. The light is amlified in the microcavity and emission out in the vertical direction. The entirely oeration mechanism of VCSEL will be discussed later in section.5, and the theory of DBRs will also be reorted in section.. DBR active region DBR n H n L n H n L VCSEL DBR Figure. Tyical structures of VCSEL and DBRs. 5

3 Advantages of VCSEL In the ast, the semiconductor laser market was filled with the roduct which alied the EEL, esecially in otical storage devices (780nm lasers for CDs and 650nm lasers for DVDs). Recent, the semiconductor laser market has been shared by the VCSEL, due to the demand of the otical data link. Besides, the develoment of VCSEL grew u vary quickly these years, and more advantages of VCSEL start to surface. These advantages are as follows. Technical advantages: () Vertical emission from the substrate with circular beam is easy to coule to otical fibers. () Low threshold and high driving current oeration is ossible. (3) Large relaxation frequency rovides high seed modulation caability. (4) Single mode oeration is ossible, and wavelength and threshold are relatively insensitive against temerature variation (5) Long device lifetime and high ower-conversion efficiency. Manufacturing and cost advantages: () Smaller size of laser devices can lower the cost of the wafer. () The initial robe test can be erformed before searating devices into discrete chis. (3) Easy bonding and mounting, chea module and ackage cost. (4) Densely acked and recisely arranged two-dimensional laser arrays can be erformed. (5) It is ossible to vertically stack multithin-film functional otical devices on to the VCSEL. Alications of VCSEL The alication of VCSEL is mainly on otical data transmission, including otical interconnects, arallel data links, and so on. The 300nm and 550nm long wavelength VCSEL should be useful for silica-based fiber links, roviding ultimate transmission caability by taking advantage of single wavelength oeration and massively arallel integration. The 650nm VCSEL is useful for short-distance data links by using mm diameter low loss lastic fibers. Moreover, blue to UV VCSEL should be useful in the high density data storage. Now, more and more alications of VCSEL are under develoment, such as comact disc otical icku modules, rinting heads, otical scanners, otical dislays, rojection systems, and otical sensor. It is believed that those roducts will be made sooner or later. 6

4 . The Theory of DBRs The DBRs are a simlest kind of eriodic structure, which is made u of a number of quarter-wave layers with alternately high- and low- index materials. Therefore, it s necessary to know the theory of quarter-wave layer before discussing the DBRs. Quarter-wave layer [9-30] Consider the simle case of a transarent late of dielectric material having a thickness d and refractive index n f, as shown in Fig.3. Suose that the film is nonabsorbing and that the amlitude-reflection coefficients at the interfaces are so low that only the first two reflected beams (both having undergone only one reflection) need be considered. The reflected rays are arallel on leaving the film and will interference at image lane. θ i D n A C θ θ t t n f d B n Figure.3 Schematic draw of the light reflected from the to and bottom of the thin film. The otical ath difference (P) for the first two reflected beam is given by P nf[( AB) + ( BC)] n( AD) (.) and since ( AB) ( BC) d cos θt (.) nfd P n( AD) cosθt (.3) also ( AD) ( AC)sin θi (.4) Using Snell s Law nf ( AD) ( AC) sin ni θt (.5) ( AC) d tan θt (.6) 7

5 The exression for P now becomes or finally nfd P ( sin θt) cosθt P nfd cosθ (.7) (.8) The corresonding hase difference (δ) associated with the otical ath length difference is then just the roduct of the free-sace roagation number and P, that is, K o P. If the film is immersed in a single medium, the index of refraction can simly be written as n n n. It is noted that no matter n f is greater or smaller than n, there will be a relative hase shift π radians. Therefore, or λf d cosθ t (m + ) 4 4πn f δ ( nf n sin θi) ± π λ 0 (.9) (.0) The interference maximum of reflected light is established when δmπ, in other words, an even multile of π. In that case Eq. (.9) can be rearranged to yield [maxima] d cosθ t (m + ) (m 0,,, ) (.) 4nf λ 0 The interference maximum of reflected light is established when δ(m±)π, in other words, an odd multile of π. In that case Eq. (.9) can be rearranged to yield [minima] d cosθ t m (m 0,,, ) (.) 4nf λ 0 Therefore, for an normal incident light into thin film, the interference maximum of reflected light is established when d λ0/4n f (at m0). Based on the theory, a eriodic structure of alternately high- and low- index quarter-wave layer is useful to be a good reflecting mirror. This eriodic structure is also called Distributed Bragg Reflectors (DBRs). 8

6 Distributed Bragg Reflectors (DBRs) [5-6][3-34] DBRs serve as high reflecting mirror in numerous otoelectronic and hotonic devices such as VCSEL. There are many methods to analyze and design DBRs, and the matrix method is one of the oular one. The calculations of DBRs are entirely described in many otics books, and the derivation is a little too long to write in this thesis. Hence, we ut it in simle to understand DBRs. Consider a distributed Bragg reflector consisting of m airs of two dielectric, lossless materials with high- and low- refractive index n H and n L, as shown in Fig.4. The thickness of the two layers is assumed to be a quarter wave, that is, L λ B /4n H and L λ B /4n L, where theλ B is the Bragg wavelength m L L substrate n o n H n L L en effective reflector n s Figure.4 Schematic diagram of DBRs. Multile reflections at the interface of the DBR and constructive interference of the multile reflected waves increase the reflectivity with increasing number of airs. The reflectivity has a maximum at the Bragg wavelength λ B. The reflectivity of a DBR with m quarter wave airs at the Bragg wavelength is given by ns nl ( ) o H R n n s L n n + ( ) no nh (.3) where the n o and n s are the refractive index of incident medium and substrate. The high-reflectivity or sto band of a DBR deends on the difference in refractive index of the two constituent materials, n (n H -n L ). The sectral width of the sto band is given by λ stoband λb n πneff (.4) 9

7 where n eff is the effective refractive index of the mirror. It can be calculated by requiring the same otical ath length normal to the layers for the DBR and the effective medium. The effective refractive index is then given by neff ( + ) nh nl (.5) The length of a cavity consisting of two metal mirrors is the hysical distance between the two mirrors. For DBRs, the otical wave enetrates into the reflector by one or several quarter-wave airs. Only a finite number out of the total number of quarter-wave airs are effective in reflecting the otical wave. The effective number of airs seen by the wave electric field is given by meff nh + nl nh nl tanh(m ) nh nl nh + nl (.6) For very thick DBRs (m ) the tanh function aroaches unity and one obtains meff n n H H + nl nl (.7) Also, the enetration deth is given by Len L + L 4r tanh(mr) (.8) where r (n -n )/ (n +n ) is the amlitude reflection coefficient. For a large number of airs (m ), the enetration deth is given by Len L + L 4r L + L 4 nh + nl nh nl (.9) Comarison of Eqs. (.7) and (.9) yields that Len meff ( + L L ) (.0) The factor of (/) in Eq. (.0) is due to the fact that m eff alies to effective number of eriods seen by the electric field whereas Len alies to the otical ower. The otical 0

8 ower is equal to the square of the electric field and hence it enetrates half as far into the mirror. The effective length of a cavity consisting of two DBRs is thus given by the sum of the thickness of the center region lus the two enetration deths into the DBRs..3 Febry-Perot Resonator [5-6][3-34] Background The Fabry-Perot interferometer was first built and analyzed by the French hysicists Fabry and Perot at the University of Marseilles about one century ago and made use of interference of light reflected many times between two colanar lightly-silvered mirrors. It is a high resolution instrument that has been used today in recision measurement and wavelength comarisons in sectroscoy. Recent, it has come into rominence as the Fabry-Perot cavity emloyed in nearly all lasers. The Fabry-Perot cavities can be used to ensure recise tuning of laser frequencies. Theory A schematic draw of a Fabry-Perot cavity with two metallic reflectors with reflectivity R and R is shown in Fig.5. Plane waves roagating inside the cavity can interference constructively and destructively resulting in stable (allowed) and attenuated (disallowed) otical modes, resectively. The allowed frequencies are integer multiles of the mode sacing ν c/nl c, where L c is the length of the cavity, c is the velocity of light in vacuum. Figure.5 A schematic draw of a Fabry-Perot cavity with two metallic reflectors with reflectivity R and R For lossless reflectors, the transmittance through the two reflectors is given by T -R, and T -R. Taking into account multile reflections inside the cavity, the transmittance through a Fabry-Perot cavity can be exressed in terms of a geometric series. The transmitted light intensity (transmittance) is then given by

9 T T T + RR RR cos φ (.) where ψ is the hase change of the otical wave for a single ass between the two reflectors. The ψ can also be exressed in term of wavelength and frequency by using nlc nlcυ φ π π λ c (.) The maxima of the transmittance occur if the condition of constructive interference is fulfilled, that is, if ψ 0, π,...,. Insertion of these values into Eq. (.) yields the transmittance maxima as T max T T + ( RR ) (.3) For asymmetric cavities (R R ), it is T max <. For symmetric cavities (R R ), the transmittance maxima are unity, T max. Near ψ 0, π,..., the cosine term in Eq. (.) can be exanded into a ower series (cosψ~- ψ ). One obtains T T T ( RR) RR 4ϕ + (.4) Equation (.4) indicates that near the maxima, the transmittance can be aroximated by a Lorentzian function. The transmittance T in Eq. (.4) has a maximum at ψ 0. The transmittance decreases to half of the maximum value at φ / ( R R R / ) /(4 R ). For high values of R and R (i.e., R ~ and R ~), it is φ / ( / )( R R )..4 The Finesse and the Quality Factor of Resonant Cavity [5-6][3-34] Since the theory of Fabry-Perot cavity has been exlained, we can talk about the finesse and the quality factor of resonant cavity. The cavity finesse, F, is defined as the ratio of the transmittance eak seeration ( ψ) to the transmittance full-width at half-maximum (δψ): φ π π F δφ φ R R / (.5)

10 Figure.6 shows the transmission attern of a Fabry-Perot cavity in frequency domain. The finesse of the cavity in the frequency is then given by F νfsr/ ν. Intensity ν 0 ν 0 3ν 0 Figure.6 The transmission attern of a Fabry-Perot cavity in frequency domain. The cavity quality factor Q is frequently used and is defined as the ratio of the transmittance eak frequency (ψ) to the eak width (δψ): Q φ δφ nl λ c π R R (.6) Besides the quality factor Q is also equal to λ/δλ, where δλ is the narrow emission linewidth around λ. λ Q (.7) δλ.5 Oeration Mechanism of VCSEL [5-7][35-38] The oeration of a VCSEL, like any other diode lasers, can be understood by observing the flow of carriers into its active region, the generation of hotons due to the recombination of some of these carriers, and the transmission of some of these hotons out of the otical cavity. These dynamics can be described by a set of rate equations, one for the carriers and one for the hotons in each of the otical modes. In fact, the construction of such rate equations rovides a clear definition of the basic laser arameters that we will need in describing the terminal characteristic of the VCSEL. 3

11 Carrier density rate equation The carrier density in the active region is governed by a dynamic rocess. In fact, we can comare the rocess of establishing a certain steady-state carrier density in the active region to that of establishing a certain water level in a reservoir which is being simultaneously filled and drained. This is shown schematically in Fig..7. As we roceed, the various filling (generation) and drain (recombination) terms illustrated will be defined. The current leakage illustrated in Fig..7 contributes to reducing η i and is created by ossible shunt aths around the active region. The carrier leakage, R l, is due to carriers slashing out of the active region (by thermionic emission or lateral diffusion if no lateral confinement exists) before recombining. Thus, this leakage contributes to a loss of carriers in the active region that could otherwise be used to generate light. I / qv electron G gen η i I/ qv current leakage N th R st ~ gv g N hoton R l R nr R nr + R l ~ (AN+CN 3 ) R rec R s + R nr + R l + R st dn rate equation : G gen R rec dt R s ~ BN Figure.7 Reservoir with continuous suly and leakage as an analog to a DH active region with current injection for carrier generation and radiative and nonradiative recombination. For the DH active region, the injected current rovides a generation term, and various radiative and nonradiative recombination rocess as well as carrier leakage rovide recombination terms. Thus, we can write the carrier density rate equation, dn G gen R rec (.8) dt 4

12 where N is the carrier density (electron density), G gen is the rate of injected electrons and R rec is the ratio of recombining electrons er init volume in the active region. Since there areη i I/q electrons er second being injected into the active region, G gen ηi I, where V is the volume qv of the active region. The recombination rocess is comlicated and several mechanisms must be considered. Such as, sontaneous recombination rate, R s ~ BN, nonradiative recombination rate, R nr, carrier leakage rate, R l, (R nr + R l AN+CN 3 ), and stimulated recombination rate, R st. Thus we can write R rec R s + R nr + R l +R st. Besides, N/τ R s + R nr + R l, where τ is the carrier lifetime. Therefore, the carrier density rate equation could be exressed as carrier density rate equation dn dt ηii qv N R τ st (.9) Photon density rate equation Now, we describe a rate equation for the hoton density, N, which includes the hoton generation and loss terms. The hoton generation rocess includes sontaneous recombination (R s ) and stimulated recombination (R st ), and the main hoton generation term of laser above threshold is R st. Every time an electron-hole airs is stimulated to recombine, another hoton is generated. Since, the cavity volume occuied by hotons, V, is usually larger than the active region volume occuied by electrons, V, the hoton density generation rate will be [V/V ]R st not just R st. This electron-hoton overla factor, V/V, is generally referred to as the confinement factor, Γ. Sometimes it is convenient to introduce an effective thickness (d eff ), width (w eff ), and length (L eff ) that contains the hotons. That is, V d eff w eff L eff. Then, if the active region has dimensions, d, w, and L a, the confinement factor can be exressed as, ΓΓ x Γ y Γ z, whereγ x d/d eff, Γ y w/w eff, Γ z L a /L eff. Photon loss occurs within the cavity due to otical absortion and scatting out of the mode, and it also occurs at the outut couling mirror where a ortion of the resonant mode is usually coule to some outut medium. These net losses can be characterized by a hoton (or cavity) lifetime, τ. Hence, the hoton density rate equation takes the form hoton density rate equation dn dt N ΓRst + Γβ s Rs (.30) τ where β s is the sontaneous emission factor. As to R st, it reresents the hoton-stimulated net electron-hole recombination which generates more hotons. This is a gain rocess for 5

13 hotons. It is given by dn dt gen R st N t v gn g (.30) where v g is the grou velocity and g is the gain er unit length. Now, we rewrite the carrier and hoton density rate equations carrier density rate equation dn dt ηi I qv N vggn τ (.3) hoton density rate equation dn dt Γv gn g N + Γβ s Rs τ (.3) Threshold gain In order for a mode of the laser to reach threshold, the gain in the active section must be increased to the oint when all the roagation and mirror losses are comensated. As illustrated in Fig..8, most laser cavities can be divided into two general sections: an active section of length L a and a assive section of length L. The threshold gain is given by Γ g th α i + ln L RR (.33) where α i is the average internal loss which is defined by (α ia L a + α i L )/L, and R and R is the reflectivity of to and bottom mirror of the laser cavity, resectively. For convenience the mirror loss term is sometimes abbreviated as, α m (/L) ln(/r R ). Noting that the cavity life time (hoton decay rate) is given by the otical loss in the cavity, /τ /τ i + /τ m v g (α i +α m ). Thus, the threshold gain in the steady state can be exressed with following equation Γ g th α + α i m v τ g α + ln i L RR (.34) 6

14 L a LL a +L L eff L a Figure.8 Schematic diagram of VCSEL Outut ower versus driving current The characteristic of outut ower versus driving current (L-I characteristic) in a laser diode can be realized by using the rate equation Eq. (.3) and (.3). Consider the below threshold (almost threshold) steady-state (dn/dt 0) carrier rate equation, the Eq. (.3) is ηi I th given by ( R + R + R ) qv N th s nr l. While the driving current is above the threshold th τ (I>Ith), the carrier rate equation will be above threshold carrier density rate equation dn dt ( I I ηi qv th ) v gn g (.35) From Eq. (.35), the steady-state hoton density above threshold where g g th can be calculated as steady state hoton density N i ( I Ith ) η (.36) qv g V g th The otical energy stored in the cavity, E os, is constructed by multilying the hoton density, N, by the energy er hoton, hν, and the cavity volume, V. That is E os N hνv. Then, we multile this by the energy loss rate through the mirrors, v g α m /τ m, to get the otical ower outut from the mirrors, P 0 v g α m N hνv. By using Eq. (.34) and (.36), and ΓV/V, we can write the outut ower as the following equation 7

15 outut ower P O α m hν η i ( I I th ) (.37) α i + α m q Now, by defining η α i m η d, the Eq. (.37) can be simlified as α iα m P O hν η d ( I I th ) (I > I th ) (.38) q Thus, the η d can be exressed as differential quantum efficiency q dpo η d h (I > I (.39) ν di th ) In fact, η d is the differential quantum efficiency, defined as number of hotons out er electron. Besides, dp o /di is defined as the sloe efficiency, S d, equal to the ratio of outut ower and injection current. Figure.9 shows the illustration of outut ower vs. current for a diode laser. below threshold only sontaneous emission is imortant; above threshold the stimulated emission ower increase linearly with the injection current, while the sontaneous emission is clamed at its threshold value. P o sontaneous emission section laser oscillation section g th α i + ln L R R P P o / hν ηd I / q sontaneous emission I S d P o I I th I Figure.9 Illustration of outut ower vs. current for a diode laser. 8

16 Temerature deendent threshold current The threshold current I th is relative to three temerature deendent factors: N tr, g o, and αi, where N tr α T, g o α /T, andαi α T. The transarency carrier density is increased and the gain arameter is reduced because injected carriers sread over a wider range in energy with high temerature. The increased internal loss results from the required higher carrier densities for threshold. Since, both the gain and the internal loss variations result in an exonential temerature deendence of the threshold current, while the linear deendence of the transarency carrier density is not significant over small temerature ranges. Thus, the threshold current can be aroximately modeled by T T ' I th( T ) Ith( T ') ex (.40) TO where I th (T) is the threshold at T, and T o is called the characteristic temerature. Both temeratures are given in degrees Kelvin, K. The larger the characteristic temerature is, the smaller the temerature deendence. 9