1. Theory

Definition: An electroabsorption modulator (EAM) is a reverse-biased p-i-n waveguide whose absorption near the band edge is changed by the electric field in the intrinsic region, so that the transmitted power of a continuous-wave laser follows the drive voltage. In InP technology the absorber is a stack of strained InGaAsP or AlGaInAs quantum wells (Piprek et al.; Chiu et al.; the Glasgow study).
Quantum-confined Stark effect: A field along the growth direction pulls the electron and the hole to opposite walls of the well, lowers the transition energy (a red shift) and reduces the overlap of the wave functions; the exciton survives to about $1 0^{5}$ V/cm because confinement inhibits field ionization (Miller et al.). Compared with the Franz-Keldysh effect of bulk material, the quantum-well effect gives a higher extinction ratio (Wikipedia); the bulk and germanium case is planned as the next page of this folder.
Why the wells matter: Thick wells shift more per volt, because the shift grows as $L_{w}^{4}$ , but thin wells absorb more strongly, so the well width is a compromise (UCSB abstract). A shift of about 15 meV at 100 $kV/cm$ for a 10.4 nm well with assumed masses is 29 nm at 1550 nm (computed from the infinite-well formula, which is a lower estimate for finite barriers).
Transmission: $T = γ^{2} (1 - R)^{2} exp (- α_{m} L)$ ; the extinction ratio is $4.343 ΓΔ α L$ in dB and scales with the length, while the capacitance and so the speed scale with it too (computed: 0.267 to $0.286 dB$ per $μ m$ for 70 to 150 $μ m$ , Glasgow study).
Integration: The modulator is made on the same chip as a DFB laser (EML), by intermixing of the quantum wells to widen the gap locally (Glasgow study) or by butt-joint regrowth (background knowledge); the hybrid waveguide of Mitsubishi uses a buried-heterostructure laser and a high-mesa modulator.
Chirp: The absorption change is accompanied by an index change, and the chirp parameter $α_{H}$ changes sign with the bias; values of $- 0.78$ to $1.58$ over a PAM4 swing and of $0$ to $1.08$ at a linear bias are used in 802.3df (Johnson).
Limits: The bias must follow the temperature (about 0.5 $nm/K$ for the edge against 0.1 $nm/K$ for the laser, Mitsubishi), the insertion loss is several dB, the photocurrent heats the junction, and the dispersion of the fibre limits the reach at the edge wavelengths of 802.3df.

2. Working Principle

Bias: The modulator is held at a reverse bias of about $- 1$ to $- 2.4$ V, where the field in the wells is about 100 $kV/cm$ (Piprek et al.) and the laser wavelength sits on the long-wavelength side of the exciton peak.
Absorption: A small voltage change moves the absorption edge across the laser wavelength; the transmission falls by $20 dB$ /V or more (Piprek et al.; Chiu et al.: more than 30 $dB/V$ in the first volt).
Drive: With a swing of 0.5 to 1 V peak-to-peak the output goes between two or four power levels (Mitsubishi: 1.0 V; Johnson: 0.5 to 0.7 V); a swing of 2 V is used for OOK in Ozolins et al.
Speed: The lumped modulator is limited by $1/ (2 π R C)$ ; a short, narrow ridge (70 $μ m$ , 2 $μ m$ : 41 GHz in simulation) or a traveling-wave electrode (more than 100 GHz at 180 $μ m$ , Ozolins et al.) raises the limit.
Chirp and dispersion: The remaining phase modulation broadens the pulse in dispersive fibre; the bias is chosen so that the chirp has the sign of the fibre dispersion that helps (Johnson: -1.75 V for positive and -1.25 V for negative dispersion).
Temperature: The laser and the absorption edge move at different rates, and the bias is moved to restore the detuning (Mitsubishi, 5 to 85 $^{\circ} C$ ).

3. Applications

100 Gbit/s per lane CWDM links: The Mitsubishi EML runs at 53.125 Gbaud PAM4 over four CWDM lanes of 20 nm spacing with a bandwidth of 53 GHz at 5 $^{\circ} C$ and 48 GHz at 85 $^{\circ} C$ .
200 Gbit/s per lane links: A 112 Gbaud PAM4 EML chip with four CWDM wavelengths gives 800 Gbit/s with four chips and 1.6 Tbit/s with eight (Mitsubishi press release).
High-speed experiments: A DFB laser with a traveling-wave modulator reaches 116 Gbit/s OOK and PAM4 and 105 Gbit/s 8PAM (Ozolins et al.).
Telecom and data-center transmitters: External modulation of a DFB laser on one chip is the standard photonic-integrated transmitter, with lower chirp than direct modulation (Wikipedia).
Analog and linear links: The bias of maximum linearity (0.7 V) differs from that of maximum efficiency (0.95 V) in the simulated device (Piprek et al.).

Quantum-Confined Stark Effect and Absorption

Name	Formula	Related Formulae	Variables/Constants
Second-order quantum-confined Stark shift in a well	$Δ E_{1} = - C_{s} \frac{m ^{*} e ^{2} F ^{2} L _{w}^{4}}{ℏ ^{2}}, C_{s} = \frac{1}{24 π ^{2}} (\frac{15}{π ^{2}} - 1) = 2.19 \times 1 0^{- 3}$	$Δ E_{t r} = - C_{s} \frac{( m _{e}^{} + m _{hh}^{} ) e ^{2} F ^{2} L _{w}^{4}}{ℏ ^{2}}$ $Δ λ = \frac{λ ^{2}}{h c} ∣ Δ E_{t r} ∣$	$F : f i e l d a l o n g t h e g r o w t h d i r ec t i o n$ $L_{w} : w e l l w i d t h$ $m^{} : e f f ec t i v e ma ss o f t h e c a r r i er$ $e : e l e m e n t a r y c ha r g e$ $ℏ : r e d u ce d P l an c k co n s t an t, t h e f or m u l a i s seco n d - or d er p er t u r ba t i o n t h eor y o f an in f ini t e w e l l (ba c k g r o u n d k n o w l e d g e) an d u n d er es t ima t es t h e s hi f t o f a w e l l w i t h f ini t e ba r r i er s, b ec a u se t h e w a v e f u n c t i o n l e ak s in t o t h e ba r r i er an d t h e e f f ec t i v e w e l l i s w i d er (m y r e a d in g, see t h e co m p u t e d n o t e), t h e s hi f t g r o w s a s F^{2} an d L_{w}^{4}, so t hi c k w e l l s a r e f i e l d se n s i t i v e an d t hin w e l l s ab sor b m or e s t r o n g l y (U C S B ab s t r a c t), M i l l er e t a l . : f or a f i e l d p er p e n d i c u l a r t o t h e l a y er s t h e e x c i t o n p e ak s m o v e t o l o w er e n er g y b y u p t o 2.5 t im es t h e z er o - f i e l d e x c i t o n bin d in g e n er g y, e x c i t o n s s t a y r eso l v e d u p t o ab o u t 100 kV/cm b ec a u se co n f in e m e n t inhibi t s f i e l d i o ni z a t i o n, an d f or b o t h f i e l d or i e n t a t i o n s e x c i t o n s b r o a d e n an d d i s a pp e a r a t ab o u t 10 kV/cm (a s s u mma r i se d f r o m t h e ab s t r a c t p a g e)$ $(co m p u t e d w i t h a f ini t e - d i f f er e n ce so l u t i o n o f t h e in f ini t e w e l l : - 1.51 m e V a t 100 kV/cm, 10.4 nm an d m^{} = 0.045 m_{0} a g ain s t - 1.52 m e V f r o m t h e f or m u l a; f or a ss u m e d ma sses m_{e}^{} = 0.045 m_{0} an d m_{hh}^{} = 0.40 m_{0} t h e t r an s i t i o n s hi f t i s - 15.0 m e V, t ha t i s 29 nm a t 1550 nm (a ss u m e d ma sses); f or a 10.4 nm w e l l w i t h a ss u m e d ban d o f f se t s o f 150 m e V f or t h e e l ec t r o n an d 100 m e V f or t h e h o l e an d e q u a l ma sses in w e l l an d ba r r i er, a f ini t e - d i f f er e n ce so l u t i o n g i v es seco n d - or d er s hi f t s 6.9 an d 2.0 t im es t h e in f ini t e - w e l l v a l u es, an d 3.7 an d 1.6 t im es f or o f f se t s o f 300 an d 200 m e V (co m p u t e d a t 20 kV/cm an d sc a l e d a s F^{2}, so hi g h er - or d er t er m s a r e n o t in c l u d e d))$
Polarization-dependent absorption	$α_{T E} = \frac{1}{4} α_{L H} + \frac{3}{4} α_{H H}$	$α_{T M} = α_{L H}$ $α_{H H} = α_{L H} \Rightarrow polarization insensitive$	$H H : h e a v y - h o l e t r an s i t i o n$ $L H : l i g h t - h o l e t r an s i t i o n, P i p r e k e t a l . g i v e t h ese w e i g h t s f or a q u an t u m - w e l l w a v e g u i d e, t h e Gl a s g o w s t u d y u ses t h e p o l a r i z a t i o n f a c t or s 1.5 f or H H an d 0.5 f or L H in i t s ab sor pt i o n m o d e l, t h e w e l l s o f P i p r e k e t a l . an d C hi u e t a l . c a r r y 0.37 an d 0.35 % t e n s i l e s t r ain, w hi c h b r in g s t h e l i g h t - h o l e t r an s i t i o n t o w a r d s t h e h e a v y - h o l e o n e (ba c k g r o u n d k n o w l e d g e), C hi u e t a l . m e a s u r e er r or - f r ee 10 G bi t / s o p er a t i o n o n b o t h T E an d T M w i t h 1 V p e ak - t o - p e ak$ $(co m p u t e d : 1.5 an d 0.5 d i v i d e d b y t h e i r s u m 2 a r e 0.75 an d 0.25, t h e s am e w e i g h t s a s 3/4 an d 1/4; t h e 10 G bi t / s se n s i t i v i t i es o f C hi u e t a l . a r e p r in t e d a s 32.5 d B m f or T E an d 33.3 d B m f or T M, p r o bab l y w i t h a min u s s i g n l os t, an d t h e T E p e na l t y i s s t a t e d a s 0.8 d B)$
Transmission of the modulator	$T (λ, V) = γ^{2} (1 - R)^{2} exp [- α_{m} (λ, V) L]$	$α_{m} = Γ α_{Q W} + α_{i}$ $T_{d B} = 10 lo g_{10} T$	$γ : f ib r e co u pl in g coe f f i c i e n t$ $R : f a ce t r e f l ec t an ce$ $α_{m} : m o d a l ab sor pt i o n$ $Γ : co n f in e m e n t o f t h e g u i d e d m o d e in t h e w e l l s$ $α_{Q W} : ab sor pt i o n o f t h e w e l l ma t er ia l$ $α_{i} : o t h er l oss, P i p r e k e t a l . : Γ = 0.2 f or t e n 10.4 nm w e l l s, γ (1 - R) = - 3 d B p er f a ce t, L = 300 μ m, t r an s mi ss i o n ab o u t - 17 d B a t 0 V an d ab o u t - 44 d B a t - 1.2 V a t 1550 nm, r es i d u a l in t er ban d l oss o f 29 c m^{- 1} pl u s 1.5 c m^{- 1} f r o m h o l es, in t er na l f i e l d ab o u t 100 kV/cm a t - 1 V t o - 2 V$ $(co m p u t e d : 44 - 17 = 27 d B; t h e tw o f a ce t s t ak e 6 d B, so 11 d B r e main, w hi c h i s α_{m} L = 2.53 an d α_{m} = 84 c m^{- 1}, w h er e a s 30.5 c m^{- 1} w o u l d g i v e o n l y 3.97 d B; Γ sc a l es w i t h t h e t hi c k n ess o f t h e w e l l s, 0.2 t im es 30/104 = 0.058, c l ose t o t h e 5 % o f t h e Gl a s g o w s t u d y)$

Extinction, Efficiency and Loss

Name	Formula	Related Formulae	Variables/Constants
Static extinction ratio	$E R = 10 lo g_{10} \frac{T ( V _{0} )}{T ( V _{1} )} = 4.343 Γ [α_{Q W} (V_{1}) - α_{Q W} (V_{0})] L$	$E R \propto L$ $E R_{Gl a s g o w} = 10 lo g_{10} exp (Γ α L) (as printed)$	$V_{0} : l o w - l oss bia s$ $V_{1} : hi g h - ab sor pt i o n bia s, 4.343 = 10 lo g_{10} e, P i p r e k e t a l . : 27 d B b e tw ee n 0 V an d - 1.2 V, C hi u e t a l . : 40 d B f or 0 t o 2 V an d 57 d B in t o t a l f or 1 t o 4.5 V (330 μ m l o n g), O z o l in s e t a l . : s t a t i c e x t in c t i o n r a t i o 20 t o 35 d B o v er t h e bia s r an g e (ab s t r a c t : u p t o 30 d B), Gl a s g o w s t u d y (s im u l a t i o n) : 40 d B f or 150 μ m, 27 d B f or 100 μ m an d 20 d B f or 70 μ m, t h e p r in t e d f or m u l a an d s i g n s o f t ha t p a p er a r e in co n s i s t e n t (i t s ab s t r a c t g i v es 2.5 V an d i t s t e x t 2.4 V)$ $(co m p u t e d : 40/150, 27/100 an d 20/70 a r e 0.267, 0.270 an d 0.286 d B p er μ m, n e a r l y l in e a r in l e n g t h; w i t h Γ = 0.05 t h e e f f ec t i v e ab sor pt i o n c han g e w o u l d b e 12 280, 12430 an d 13160 c m^{- 1})$
Modulation efficiency and optimum bias	$η_{m} = \frac{d E R}{d V} = \frac{10}{l n 10} Γ L \frac{d α _{Q W}}{d V} [dB/V]$	$V_{e f f} : max [T \frac{d T}{d V}]$ $V_{l in} : \frac{d ^{2} T}{d V ^{2}} + \frac{d ^{3} T}{d V ^{3}} \approx 0$	$P i p r e k e t a l . : m or e t han 20 dB/V (U C S B ab s t r a c t), s t ee p es t s l o p e o f T (V) n e a r 1 V, ma x im u m m o d u l a t i o n e f f i c i e n cy a t ab o u t 0.95 V an d ma x im u m l in e a r i t y a t ab o u t 0.7 V a t 1550 nm (e f f i c i e n cy an d l in e a r i t y c ann o t b o t h b e ma x imi z e d), C hi u e t a l . : m or e t han 30 dB/V f or 0 t o 1 V, 1 V p e ak - t o - p e ak e n o ug h f or 10 G bi t / s, m o d u l a t i o n e f f i c i e n cy i s t h e s l o p e o f t h e t r an s mi ss i o n in d B, so i t d e p e n d s o n t h e bia s$ $(co m p u t e d : 27 d B o v er 1.2 V i s 22.5 dB/V o n a v er a g e; 40 d B o v er 2 V i s 20 dB/V o n a v er a g e, so w i t h m or e t han 30 d B in t h e f i r s t v o l t a t m os t 10 d B a r e l e f t f or t h e seco n d o n e)$
Insertion loss	$I L = 4.343 Γ α_{Q W} (V_{0}) L + α_{i} L + I L_{co u pl in g}$	$I L_{c hi p} = I L - I L_{co u pl in g}$	$C hi u e t a l . : ab o u t 12 d B f ib r e t o f ib r e a t 1 V f or w a r d bia s pl u s 5 t o 6 d B o f t o t a l co u pl in g l oss, w i t h n o p o w er s a t u r a t i o n u p t o 14 d B m o f in p u t, Gl a s g o w s t u d y : 7.6 d B f or t h e a s - g r o w n m o d u l a t or (s im u l a t e d, 7.97 d B m e a s u r e d in e a r l i er w or k) an d 0.11 d B a f t er in t er mi x in g a t 0 V, ab sor pt i o n a t 1.55 μ m o f 400 c m^{- 1} a s g r o w n an d 2 c m^{- 1} in t er mi x e d a t z er o f i e l d, b u i l t - in f i e l d 59 kV/cm a t 0 V$ $(co m p u t e d w i t h Γ = 0.05 an d 150 μ m : 1.30 d B f or 400 c m^{- 1} an d 0.0065 d B f or 2 c m^{- 1}, a g ain s t 7.6 d B an d 0.11 d B p r in t e d; 7.6 d B w o u l d n ee d 2330 c m^{- 1}, w hi c h i s pl a u s ib l e a t t h e 59 kV/cm o f t h e b u i l t - in f i e l d b u t i s n o t s t a t e d)$
Dynamic extinction ratio and optical modulation amplitude	$E R_{d y n} = 10 lo g_{10} \frac{P _{1}}{P _{0}}$	$O M A = P_{1} - P_{0}$ $P_{1} = O M A \frac{r}{r - 1}, P_{0} = \frac{O M A}{r - 1}, r = 1 0^{E R_{d y n} /10}$	$P_{1}, P_{0} : o pt i c a l p o w er a t t h e hi g h an d l o w l e v e l$ $r : p o w er r a t i o, M i t s u bi s hi 100 G bi t / s E M L (53.125 G ba u d P A M 4, 1.0 V p e ak - t o - p e ak) : e x t in c t i o n r a t i o 4.7 t o 5.3 d B a t 5^{\circ} C an d 7.0 t o 7.3 d B a t 85^{\circ} C a g ain s t a t a r g e t o f a t l e a s t 3.5 d B, O M A a t t h e c hi p f a ce t 8.0 t o 8.6 d B m an d 5.3 t o 7.6 d B m a g ain s t a t l e a s t 5 d B m, T D E C Q 2.4 t o 2.7 d B an d 1.4 t o 2.5 d B a g ain s t a t m os t 3.4 d B, t h e e x t r a c t d oes n o t s a y w hi c h l e v e l d e f ini t i o n t h e O M A u ses$ $(co m p u t e d f or 4.8 d B an d 8.6 d B m, t ak in g O M A a s t h e d i f f er e n ce o f t h e o u t er l e v e l s (a ss u m e d) : r = 3.02, O M A = 7.2 mW, P_{1} = 10.8 mW an d P_{0} = 3.6 mW)$

Chirp and Dispersion

Name	Formula	Related Formulae	Variables/Constants
Chirp parameter and frequency chirp	$α_{H} = 2 P \frac{d ϕ}{d P}$	$Δ ν (t) = \frac{α _{H}}{4 π} \frac{1}{P} \frac{d P}{d t}$ $α_{H} = \frac{4 π}{λ} \frac{δ n}{δ α} (as printed by the Glasgow study)$	$ϕ : o pt i c a l p ha se$ $n : r e f r a c t i v e in d e x$ $α : p o w er ab sor pt i o n coe f f i c i e n t, t h e d e f ini t i o n an d t h e f r e q u e n cy c hi r p a r e ba c k g r o u n d k n o w l e d g e an d t h e s i g n o f t h e p r in t e d f or m d e p e n d s o n t h e p ha se co n v e n t i o n, t h e Gl a s g o w s t u d y o b t ain s t h e in d e x c han g e f r o m t h e ab sor pt i o n b y a K r am er s - K r o ni g in t e g r a l, i t s v a l u es : α_{H} ab o v e 4 a t z er o f i e l d (in t er mi x e d) a t 1.55 μ m an d f a l l in g t o 0 n e a r 60 kV/cm f or t h e a s - g r o w n m o d u l a t or, P i p r e k e t a l . : l a ser c hi r p f a c t or s a r e 4 t o 6, t h e m o d u l a t or i s e x p ec t e d n e a r 2, n e g a t i v e - c hi r p cr osso v er n e a r 1.52 μ m, J o hn so n (802.3 df, 112.5 G ba u d P A M 4) : α_{H} (V_{0}, V_{b}, V_{3}) = 0, 0.6, 1.08 a t V_{b} = - 1.45 V an d V_{pp} = 0.6 V, a t - 1.75 V an d 0.5 V t h e f o u r l e v e l s g i v e - 0.78, - 0.05, 0.26 an d 0.51, a t - 1.25 V an d 0.7 V t h ey g i v e 0.33, 0.72, 1.06 an d 1.58$ $(t h e s i g n o f α_{H} m o v es w i t h t h e bia s, so t h e s am e d e v i ce c an ha v e p os i t i v e or n e g a t i v e c hi r p; n o c hi r p v a l u e w a s r e p or t e d b y C hi u e t a l . or O z o l in s e t a l .)$
Chromatic dispersion of a standard fibre	$D (λ) = \frac{S _{0}}{4} (λ - \frac{λ _{0}^{4}}{λ ^{3}})$	$Δ τ = D L_{f} Δ λ$	$S_{0} : d i s p er s i o n s l o p e a t t h e z er o - d i s p er s i o n w a v e l e n g t h λ_{0}$ $L_{f} : f ib r e l e n g t h (ba c k g r o u n d k n o w l e d g e f or t h e f or m u l a), J o hn so n (802.3 df) u ses z er o - d i s p er s i o n w a v e l e n g t h s o f 1324 nm an d 1300 nm an d l a ser w a v e l e n g t h s o f 1264.5 nm an d 1337.5 nm, d i s p er s i o n o f 6.7 an d 13.4 p s / nm a t 2 an d 4 k m a t 1337.5 nm, an d - 11.9 an d - 23.7 p s / nm a t 1264.5 nm, T D E C Q a t 1337.5 nm an d V_{b} = - 1.45 V i s 2.69 d B a t 2 k m an d 13.2 d B a t 4 k m, a t V_{b} = - 1.75 V i t i s 2.31 d B an d 3.58 d B, a t 1264.5 nm i t i s 1.61 d B an d 9.97 d B a t - 1.45 V an d 1.12 d B an d 4.91 d B a t - 1.25 V$ $(co m p u t e d w i t h S_{0} = 0.092 t o 0.093 p s / (n m^{2} k m) (a ss u m e d, v a l u es o f a s t an d a r d s in g l e - m o d e f ib r e f r o m ba c k g r o u n d k n o w l e d g e) : D (1337.5 nm, λ_{0} = 1300 nm) = + 3.31 t o + 3.34 p s / (nm k m), t h e p r in t e d 6.7/2 = 3.35; D (1264.5 nm, λ_{0} = 1324 nm) = - 5.87 t o - 5.94, t h e p r in t e d - 11.9/2 = - 5.95)$

Speed, Drive and Integration

Name	Formula	Related Formulae	Variables/Constants
Capacitance of the junction	$C_{j} = \frac{ε _{0} ε _{r} L w}{d}$	$f_{R C} = \frac{1}{2 π R _{t o t} ( C _{j} + C _{p} )}$	$L : m o d u l a t or l e n g t h$ $w : r i d g e w i d t h$ $d : t hi c k n ess o f t h e in t r in s i c r e g i o n$ $ε_{r} : r e l a t i v e p er mi tt i v i t y o f I n P (a ss u m e d 12.5)$ $C_{p} : p a d an d p a r a s i t i c c a p a c i t an ce$ $R_{t o t} : ser i es pl u s l o a d r es i s t an ce, P i p r e k e t a l . an d C hi u e t a l . : i - r e g i o n ab o u t 300 nm, r i d g e 2 μ m, l e n g t h s 300 an d 330 μ m, Gl a s g o w s t u d y (m e a s u r e d o n t h e a s - g r o w n m o d u l a t or a t - 1.6 V) : C_{j} = 0.17 pF, C_{0} = 0.09 pF, C_{p} = 0.14 pF, R_{j} = 10 k Ω, R_{s} = 24 Ω, L_{o} = 0.4 n H, R_{o} = 50 Ω, 150 μ m l o n g w i t h a 2.5 μ m r i d g e$ $(co m p u t e d w i t h d = 300 nm (a ss u m e d f or t h e Gl a s g o w d e v i ce) : 0.22 pF (300 μ m), 0.24 pF (330 μ m) an d 0.14 pF (150 μ m), c l ose t o t h e m e a s u r e d 0.17 pF)$
Lumped-element bandwidth	$f_{3 d B} = \frac{1}{2 π R C}$	$f_{3 d B} \propto \frac{1}{L} for fixed R, w, d$	$R : e f f ec t i v e r es i s t an ce see n b y t h e j u n c t i o n (ser i es r es i s t an ce pl u s t h e so u r ce an d l o a d s ha r e)$ $C : t o t a l c a p a c i t an ce, ban d w i d t h s r e p or t e d : P i p r e k e t a l . m or e t han 25 G H z (f or t h e e a r l i er f ab r i c a t e d d e v i ces) an d C hi u e t a l . ab o u t 20 G H z (1.5 d B r o l l - o f f a t 20 G H z, bia s 0.4 V; t h e d i s t r ib u t e d m o d e l g i v es ab o u t 40 G H z), Gl a s g o w s t u d y 22 G H z f or 150 μ m an d 41 G H z f or 70 μ m w i t h a 2 μ m r i d g e (s im u l a t i o n), C hin . P h y s . L e tt . ab s t r a c t : m or e t han 20 G H z an d m or e t han 20 d B a t 3 V, M i t s u bi s hi 53 G H z a t 5^{\circ} C an d 48 G H z a t 85^{\circ} C (ab o u t 35 G H z n ee d e d f or 100 G bi t / s), O z o l in s e t a l . : m or e t han 100 G H z, 2 d B r i ppl e, 180 μ m t r a v e l in g - w a v e m o d u l a t or$ $(co m p u t e d : R = 24 Ω an d C = 0.17 + 0.14 = 0.31 pF g i v e 21.4 G H z, n e a r t h e 22 G H z p r in t e d (m y r e a d in g); 24 Ω an d 0.17 pF g i v e 39 G H z; 74 Ω an d 0.17 pF g i v e 12.7 G H z; a s in g l e p o l e w i t h 1.5 d B a t 20 G H z ha s i t s 3 d B p o in t a t 31 G H z w h e n b o t h l e v e l s a r e r e a d in t h e s am e co n v e n t i o n; i f t h e 1.5 d B w er e an o pt i c a l (10 l o g) f i g u r e, 20 G H z w o u l d a l r e a d y b e t h e e l ec t r i c a l 3 d B p o in t an d t h e o pt i c a l 3 d B p o in t w o u l d b e 34.6 G H z; t h e e a r l i er 46 G H z d i d n o t f o l l o w f r o m e i t h er r e a d in g, a u d i t cor r ec t i o n)$
Drive swing and operating point	$V_{b} \approx V_{e f f}, V_{pp} ≲ Δ V_{l in e a r} (my reading)$	$E R_{d y n} = 10 lo g_{10} \frac{T ( ∣ V _{b} ∣ - V _{pp} /2 )}{T ( ∣ V _{b} ∣ + V _{pp} /2 )}$	$V_{b} : bia s (r e v er se, w r i tt e n a s a ma g ni t u d e in t h e d y nami c r a t i o)$ $V_{pp} : p e ak - t o - p e ak s w in g, M i t s u bi s hi : V_{pp} = 1.0 V, ce n t er bia s - 2.3 t o - 2.4 V a t 5^{\circ} C an d - 1.0 t o - 1.1 V a t 85^{\circ} C, J o hn so n (802.3 df) : V_{b} = - 1.45 V w i t h V_{pp} = 0.6 V f or a l in e a r bia s, - 1.75 V w i t h 0.5 V an d - 1.25 V w i t h 0.7 V, O z o l in s e t a l . : bia s - 2 V, 2 V p e ak - t o - p e ak f or O O K an d 8 P A M an d 1 V f or 4 P A M (116 G bi t / s O O K an d 4 P A M, 105 G bi t / s 8 P A M), C hi u e t a l . : 1 V p e ak - t o - p e ak f or 10 G bi t / s$ $(t h e d r i v e s w in g i s 0.5 t o 2 V in t h ese so u r ces, se t b y t h e s t ee p p a r t o f T (V) r a t h er t han b y a ha l f - w a v e v o l t a g e)$
Temperature detuning of an electroabsorption modulated laser	$Δ λ_{d e t} = (\frac{d λ _{E A}}{d T} - \frac{d λ _{D F B}}{d T}) Δ T$	$Δ V_{b} restores Δ λ_{d e t}$	$λ_{E A} : ab sor pt i o n e d g e$ $λ_{D F B} : l a ser w a v e l e n g t h, M i t s u bi s hi : l a ser ab o u t 0.1 nm/K an d ab sor pt i o n e d g e ab o u t 0.5 nm/K, l an es a t 5^{\circ} C an d 85^{\circ} C o f 1267.5 an d 1274.4, 1287.4 an d 1294.6, 1307.5 an d 1314.6, 1327.4 an d 1334.4 nm, C W D M s p a c in g 20 nm, l a ser c u r r e n t 30 m A a t 5^{\circ} C an d 120 m A a t 85^{\circ} C$ $(co m p u t e d : t h e l an es m o v e b y 0.086 t o 0.090 nm/K, in l in e w i t h 0.1; (0.5 - 0.1) t im es 80 K i s 32 nm o f e x t r a d e t u nin g, co m p e n s a t e d b y m o v in g t h e bia s f r o m ab o u t - 2.4 V t o ab o u t - 1.1 V, t ha t i s 1.3 V; t h e e x t in c t i o n r a t i o r i ses f r o m ab o u t 5 d B t o ab o u t 7 d B an d t h e ban d w i d t h f a l l s f r o m 53 t o 48 G H z)$
Output power of the integrated laser and modulator, and photocurrent	$P_{o u t} = η_{s} (I - I_{t h}) T_{E A M}$	$I_{p h} = R P_{ab s}, P_{d i ss} = ∣ V_{b} ∣ I_{p h}$	$η_{s} : s l o p e e f f i c i e n cy o f t h e l a ser$ $I_{t h} : t h r es h o l d c u r r e n t$ $T_{E A M} : t r an s mi ss i o n o f t h e m o d u l a t or$ $R : r es p o n s i v i t y o f t h e m o d u l a t or a s a d e t ec t or$ $P_{ab s} : ab sor b e d p o w er, O z o l in s e t a l . : t h r es h o l d ab o u t 25 m A, s l o p e e f f i c i e n cy 0.04 W / A, ab o u t 2 mW a t 80 m A, a v er a g e o u tp u t - 2 d B m, 180 μ m m o d u l a t or w i t h 12 w e l l s o f ab o u t 9 nm, D F B 440 μ m w i t h 7 w e l l s o f 7 nm, M i t s u bi s hi : b u r i e d - h e t er os t r u c t u r e l a ser w i t h a hi g h - m es a m o d u l a t or an d a s p o t - s i z e co n v er t er (t h e 200 G bi t / s, 112 G ba u d P A M 4 c hi p u ses t hi s h y b r i d w a v e g u i d e), t h e d i ss i p a t i o n i s a ba c k g r o u n d - k n o w l e d g e r e l a t i o n$ $(co m p u t e d : (80 - 25) m A t im es 0.04 W / A = 2.2 mW; - 2 d B m i s 0.63 mW, w hi c h i s 0.29 or - 5.4 d B o f t h e l a ser p o w er; f or 2 mW in c i d e n t, ha l f ab sor b e d, R = 1 A / W an d 2 V o f r e v er se bia s (a ss u m e d) t h e p h o t oc u r r e n t i s 1 m A an d t h e d i ss i p a t i o n 2 mW)$

Important Questions

Ques: Is the extinction ratio of an EAM $27 dB$ , $40 dB$ , 20 to $35 dB$ or $5 dB$ ?

These are different quantities. Piprek et al. give $27 dB$ between 0 and $- 1.2$ V, Chiu et al. $40 dB$ for 0 to 2 V and $57 dB$ for 1 to 4.5 V, and Ozolins et al. a static ratio of 20 to $35 dB$ across the bias range (up to $30 dB$ in the abstract). The Mitsubishi module has 4.7 to $5.3 dB$ at 5 $^{\circ} C$ and 7.0 to $7.3 dB$ at 85 $^{\circ} C$ , because the drive is only 1.0 V peak-to-peak and the PAM4 levels sit on the steep part of the curve. A static ratio is the best case of the whole bias range, whereas the dynamic ratio is what a receiver sees. The target of $3.5 dB$ is therefore reasonable, and the $5 dB$ figure is the more likely one for a system (my reading). Reference: Piprek et al., two-dimensional simulation of an InGaAsP/InP quantum-well electroabsorption modulator (UCSB, file piprek02pw.pdf; title not recorded; read through a fetch-tool summary); Chiu et al., InGaAsP multiquantum-well traveling-wave electroabsorption modulator, IEEE Photonics Technology Letters (2002) (UCSB file chiu02ptl.pdf; title not recorded; read through a fetch-tool summary); Ozolins et al., J. Lightwave Technol. (2017) (file 201701_JLT_Ozolins.pdf; title not recorded), DFB laser with traveling-wave EAM (KTH, Syntune, u2t); Mitsubishi Electric Advance, 100-Gbit/s CWDM EML article (Advance no. 184, 2023, TR2; read through a fetch-tool summary).

Ques: Is the efficiency 20 or 30 $dB/V$ , and is it constant?

It is not constant. Piprek et al. quote more than 20 $dB/V$ and a steepest slope of the transmission near 1 V, and their $27 dB$ over 1.2 V is 22.5 $dB/V$ on average (computed). Chiu et al. quote more than 30 $dB/V$ for 0 to 1 V and $40 dB$ for 0 to 2 V, which is 20 $dB/V$ on average (computed). If the first volt gives at least $30 dB$ , the second gives at most $10 dB$ (computed). The slope is therefore the highest where the exciton edge crosses the laser wavelength and falls at larger bias because the absorption saturates. The two figures are consistent and refer to different bias ranges; the quoted values describe the best slope and the averages describe the usable swing (my reading). A drive of 1 V peak-to-peak placed on the steep part is the usual choice. Reference: Piprek et al., two-dimensional simulation of an InGaAsP/InP quantum-well electroabsorption modulator (UCSB, file piprek02pw.pdf; title not recorded; read through a fetch-tool summary); UCSB abstract page of the same InGaAsP/InP modulator simulation (electrochem.org file 0657.pdf); Chiu et al., InGaAsP multiquantum-well traveling-wave electroabsorption modulator, IEEE Photonics Technology Letters (2002) (UCSB file chiu02ptl.pdf; title not recorded; read through a fetch-tool summary).

Ques: Is the zero-bias loss of the 300 $μ m$ modulator 29 cm $^{- 1}$ or 84 cm $^{- 1}$ ?

Piprek et al. state a residual interband loss of 29 $c m^{- 1}$ plus 1.5 $c m^{- 1}$ from holes, and a facet loss of $3 dB$ each and a transmission of about $- 17$ dB at 0 V. The two facets take $6 dB$ , which leaves $11 dB$ over 300 $μ m$ , an absorption of 84 $c m^{- 1}$ (computed). The quoted 30.5 $c m^{- 1}$ would give only $3.97 dB$ (computed), so the total transmission would be about $- 10$ dB, not $- 17$ dB. If the $3 dB$ was the total for both facets the residual would be $14 dB$ and 107 $c m^{- 1}$ (computed). The transmission is the simulated and compared quantity, so I take 84 $c m^{- 1}$ as the more likely loss at 0 V and read 29 $c m^{- 1}$ as one part of it, perhaps the tail loss at a longer wavelength (my reading). The extract was a summary, not the paper. Reference: Piprek et al., two-dimensional simulation of an InGaAsP/InP quantum-well electroabsorption modulator (UCSB, file piprek02pw.pdf; title not recorded; read through a fetch-tool summary).

Ques: Why does the Glasgow insertion loss of $7.6 dB$ not follow from 400 cm $^{- 1}$ ?

The paper gives an absorption of 400 $c m^{- 1}$ at zero field and 2 $c m^{- 1}$ after intermixing, a confinement factor of 5 %, a 150 $μ m$ modulator, and insertion losses of $7.6 dB$ and $0.11 dB$ at 0 V. With $4.343 Γ α L$ the two absorptions give $1.30 dB$ and $0.0065 dB$ (computed), and $7.6 dB$ would need 2330 $c m^{- 1}$ (computed). The loss at 0 V is evaluated at the built-in field of 59 $kV/cm$ , not at zero field, where the exciton has shifted towards the laser wavelength, so a larger absorption is possible (my reading). The extinction ratios are consistent across lengths (0.267 to $0.286 dB$ per $μ m$ , computed), and the abstract gives 2.5 V where the text gives 2.4 V; I take 2.4 V as more likely, because the figures use it (my reading). Reference: Glasgow simulation study of an AlGaInAs/InP electroabsorption modulated laser with quantum-well intermixing (eprints.gla.ac.uk/274428; title not recorded; read through a fetch-tool summary).

Ques: Which bandwidth is realistic: 20, 22, 41, 53 or more than 100 GHz?

All are real, for different devices. Piprek et al. quote more than 25 GHz and Chiu et al. about 20 GHz for 300 and 330 $μ m$ modulators, Glasgow 22 GHz for 150 $μ m$ and 41 GHz for 70 $μ m$ , Mitsubishi 53 GHz (5 $^{\circ} C$ ) and 48 GHz (85 $^{\circ} C$ ), and Ozolins et al. more than 100 GHz with $2 dB$ ripple for a 180 $μ m$ traveling-wave modulator. The Glasgow circuit gives $1/ (2 π R C) = 21.4$ GHz for $R = 24$ $Ω$ and $C = 0.31$ $pF$ (computed), close to the 22 GHz printed, so the pad capacitance of 0.14 $pF$ is almost half of the total (my reading). The usable figure for 100 Gbit/s is at least 35 GHz (Mitsubishi), reached by a short chip and low parasitics. Reference: Piprek et al., two-dimensional simulation of an InGaAsP/InP quantum-well electroabsorption modulator (UCSB, file piprek02pw.pdf; title not recorded; read through a fetch-tool summary); Chiu et al., InGaAsP multiquantum-well traveling-wave electroabsorption modulator, IEEE Photonics Technology Letters (2002) (UCSB file chiu02ptl.pdf; title not recorded; read through a fetch-tool summary); Glasgow simulation study of an AlGaInAs/InP electroabsorption modulated laser with quantum-well intermixing (eprints.gla.ac.uk/274428; title not recorded; read through a fetch-tool summary); Optimization of 1.3-um InGaAsP/InP Electro-Absorption Modulator, Chin. Phys. Lett. 32(8), 084203 (2015) (abstract page only); Mitsubishi Electric Advance, 100-Gbit/s CWDM EML article (Advance no. 184, 2023, TR2; read through a fetch-tool summary); Ozolins et al., J. Lightwave Technol. (2017) (file 201701_JLT_Ozolins.pdf; title not recorded), DFB laser with traveling-wave EAM (KTH, Syntune, u2t).

Ques: Is the chirp parameter 0, 2 or above 4?

The values refer to different fields and wavelengths. The Glasgow study finds $α_{H}$ above 4 at zero field at 1.55 $μ m$ and, for the as-grown modulator, a fall to 0 near 60 $kV/cm$ . Piprek et al. expect about 2 for the modulator, against laser chirp factors of 4 to 6, and a negative-chirp crossover near 1.52 $μ m$ . Johnson (802.3df) uses $α_{H}$ of 0, 0.6 and 1.08 over a PAM4 swing at $- 1.45$ V, and $- 0.78$ to 0.51 at $- 1.75$ V. The sign and size follow the bias and the detuning of the laser from the edge, so a single number is not a property of the device. For links the values at the operating bias, between about $- 1$ and $+ 1.6$ , are the relevant ones, and above 4 refers to the weak-field tail (my reading). Reference: Glasgow simulation study of an AlGaInAs/InP electroabsorption modulated laser with quantum-well intermixing (eprints.gla.ac.uk/274428; title not recorded; read through a fetch-tool summary); Piprek et al., two-dimensional simulation of an InGaAsP/InP quantum-well electroabsorption modulator (UCSB, file piprek02pw.pdf; title not recorded; read through a fetch-tool summary); Johnson, IEEE 802.3df presentation of 11 October 2022 (johnson_3df_01a_221011).

Ques: How much does the temperature change the detuning, and why must the bias follow?

Mitsubishi states about 0.1 $nm/K$ for the laser and 0.5 $nm/K$ for the absorption edge. The four lanes move by 6.9, 7.2, 7.1 and 7.0 nm between 5 and 85 $^{\circ} C$ , or 0.086 to 0.090 $nm/K$ (computed), in line with the laser. The extra detuning is $0.4 \times 80 = 32$ nm (computed), which would make the modulator transparent if the bias were fixed. The centre bias is moved from about $- 2.4$ V to about $- 1.1$ V (1.3 V, computed); the extinction ratio rises from 4.7 to $5.3 dB$ to 7.0 to $7.3 dB$ and the bandwidth falls from 53 to 48 GHz. The rise of the ratio is more likely a result of the chosen bias than of the temperature itself (my reading). Reference: Mitsubishi Electric Advance, 100-Gbit/s CWDM EML article (Advance no. 184, 2023, TR2; read through a fetch-tool summary).

Ques: Do the 802.3df dispersion numbers follow from the fibre formula?

Yes. With the slope $S_{0} = 0.092$ to 0.093 ps/(nm $^{2}$ km) and a zero-dispersion wavelength of 1300 nm the formula gives $+ 3.31$ to $+ 3.34$ ps/(nm km) at 1337.5 nm, and the printed 6.7 ps/nm over 2 km is 3.35 ps/(nm km). At 1264.5 nm with 1324 nm it gives $- 5.87$ to $- 5.94$ , and the printed $- 11.9/2 = - 5.95$ (all computed). The slides list zero-dispersion wavelengths of 1324 nm and 1300 nm, which are the two extremes, one for each lane edge. The TDECQ then depends on the bias: at 1337.5 nm and 4 km it is $13.2 dB$ at $- 1.45$ V and $3.58 dB$ at $- 1.75$ V, so the chirp sign matters more than the length (Johnson). The assignment of the two limits to the two lanes is my reading. Reference: Johnson, IEEE 802.3df presentation of 11 October 2022 (johnson_3df_01a_221011).

Ques: Why do the drive swings differ from 0.5 V to 2 V?

They are chosen from the transmission curve, not from a half-wave voltage. Johnson uses 0.6 V at $- 1.45$ V, 0.5 V at $- 1.75$ V and 0.7 V at $- 1.25$ V for 112.5 Gbaud PAM4, Mitsubishi 1.0 V peak-to-peak with a centre bias of $- 2.3$ to $- 2.4$ V (5 $^{\circ} C$ ), Ozolins et al. 1 V for 4PAM and 2 V for OOK and 8PAM at $- 2$ V, and Chiu et al. 1 V for 10 Gbit/s. A smaller swing needs a steeper part of the curve, which sits at a different bias for each material and wavelength detuning, and the modulator saturates when the swing is large. The sources differ in length, detuning and format, so no value is typical; for PAM4 about 0.5 to 1 V is the usual range in these sources (my reading). Reference: Johnson, IEEE 802.3df presentation of 11 October 2022 (johnson_3df_01a_221011); Mitsubishi Electric Advance, 100-Gbit/s CWDM EML article (Advance no. 184, 2023, TR2; read through a fetch-tool summary); Ozolins et al., J. Lightwave Technol. (2017) (file 201701_JLT_Ozolins.pdf; title not recorded), DFB laser with traveling-wave EAM (KTH, Syntune, u2t); Chiu et al., InGaAsP multiquantum-well traveling-wave electroabsorption modulator, IEEE Photonics Technology Letters (2002) (UCSB file chiu02ptl.pdf; title not recorded; read through a fetch-tool summary).

Ques: Is the polarization weighting 3/4 and 1/4 or 1.5 and 0.5?

Both describe the same weights. Piprek et al. write $α_{T E} = α_{L H} /4 + 3 α_{H H} /4$ and $α_{T M} = α_{L H}$ , and the Glasgow study uses factors of 1.5 for the heavy-hole and 0.5 for the light-hole transition. Divided by their sum of 2 these are 0.75 and 0.25 (computed), so the two forms agree once the normalization is stated. The consequence is that a polarization-insensitive modulator needs $α_{H H} = α_{L H}$ , which tensile strain in the wells helps to reach (0.35 to 0.37 % in the devices described; background knowledge for the effect). Chiu et al. report TE and TM sensitivities of 32.5 and $33.3 dBm$ with the minus signs missing from the extract, a TE penalty of $0.8 dB$ , and error-free operation on both polarizations. The reading as negative dBm is the more likely one (my reading). Reference: Piprek et al., two-dimensional simulation of an InGaAsP/InP quantum-well electroabsorption modulator (UCSB, file piprek02pw.pdf; title not recorded; read through a fetch-tool summary); Glasgow simulation study of an AlGaInAs/InP electroabsorption modulated laser with quantum-well intermixing (eprints.gla.ac.uk/274428; title not recorded; read through a fetch-tool summary); Chiu et al., InGaAsP multiquantum-well traveling-wave electroabsorption modulator, IEEE Photonics Technology Letters (2002) (UCSB file chiu02ptl.pdf; title not recorded; read through a fetch-tool summary).

Ques: Does the infinite-well Stark formula overestimate or underestimate the shift?

The formula $Δ E = - C_{s} m^{*} e^{2} F^{2} L_{w}^{4} / ℏ^{2}$ with $C_{s} = 2.19 \times 1 0^{- 3}$ is the second-order shift of an infinite well. A finite-difference solution reproduces it (computed: $- 1.51$ meV against $- 1.52$ meV at 100 $kV/cm$ for 10.4 nm and $0.045 m_{0}$ ). With the assumed masses of 0.045 and 0.40 $m_{0}$ the transition shifts by 15.0 meV, or 29 nm at 1550 nm (computed). A finite barrier lets the wave function leak, so the effective well is wider and, since the shift grows as $L_{w}^{4}$ , larger. For assumed offsets of 150 and 100 meV the second-order shifts are 6.9 and 2.0 times the infinite-well values (computed), so the formula underestimates the confinement part. The exciton adds a Coulomb term that the formula omits, with measured red shifts of up to 2.5 times the binding energy (Miller et al.) (my reading). Reference: Miller et al., Phys. Rev. B 32, 1043-1060 (1985) (abstract page).

Quartz 5

Explorer

Indium Phosphide Quantum-Well Electroabsorption Modulator

1. Theory

2. Working Principle

3. Applications

Quantum-Confined Stark Effect and Absorption

Extinction, Efficiency and Loss

Chirp and Dispersion

Speed, Drive and Integration

Important Questions

Ques: Is the extinction ratio of an EAM $27 dB$ , $40 dB$ , 20 to $35 dB$ or $5 dB$ ?

Ques: Is the efficiency 20 or 30 $dB/V$ , and is it constant?

Ques: Is the zero-bias loss of the 300 $μ m$ modulator 29 cm $^{- 1}$ or 84 cm $^{- 1}$ ?

Ques: Why does the Glasgow insertion loss of $7.6 dB$ not follow from 400 cm $^{- 1}$ ?

Ques: Which bandwidth is realistic: 20, 22, 41, 53 or more than 100 GHz?

Ques: Is the chirp parameter 0, 2 or above 4?

Ques: How much does the temperature change the detuning, and why must the bias follow?

Ques: Do the 802.3df dispersion numbers follow from the fibre formula?

Ques: Why do the drive swings differ from 0.5 V to 2 V?

Ques: Is the polarization weighting 3/4 and 1/4 or 1.5 and 0.5?

Ques: Does the infinite-well Stark formula overestimate or underestimate the shift?

Graph View

Table of Contents

Backlinks

Quartz 5

Explorer

Indium Phosphide Quantum-Well Electroabsorption Modulator

1. Theory

2. Working Principle

3. Applications

Quantum-Confined Stark Effect and Absorption

Extinction, Efficiency and Loss

Chirp and Dispersion

Speed, Drive and Integration

Important Questions

Ques: Is the extinction ratio of an EAM 27 dB, 40 dB, 20 to 35 dB or 5 dB?

Ques: Is the efficiency 20 or 30 dB/V, and is it constant?

Ques: Is the zero-bias loss of the 300 μm modulator 29 cm−1 or 84 cm−1?

Ques: Why does the Glasgow insertion loss of 7.6 dB not follow from 400 cm−1?

Ques: Which bandwidth is realistic: 20, 22, 41, 53 or more than 100 GHz?

Ques: Is the chirp parameter 0, 2 or above 4?

Ques: How much does the temperature change the detuning, and why must the bias follow?

Ques: Do the 802.3df dispersion numbers follow from the fibre formula?

Ques: Why do the drive swings differ from 0.5 V to 2 V?

Ques: Is the polarization weighting 3/4 and 1/4 or 1.5 and 0.5?

Ques: Does the infinite-well Stark formula overestimate or underestimate the shift?

Graph View

Table of Contents

Backlinks

Ques: Is the extinction ratio of an EAM $27 dB$ , $40 dB$ , 20 to $35 dB$ or $5 dB$ ?

Ques: Is the efficiency 20 or 30 $dB/V$ , and is it constant?

Ques: Is the zero-bias loss of the 300 $μ m$ modulator 29 cm $^{- 1}$ or 84 cm $^{- 1}$ ?

Ques: Why does the Glasgow insertion loss of $7.6 dB$ not follow from 400 cm $^{- 1}$ ?