1. Theory

Definition: A silicon microring modulator is a ring resonator of a few micrometres to a few tens of micrometres radius, coupled to a bus waveguide, with a pn junction in part of the ring. The voltage shifts the resonance wavelength, and a laser held near the resonance sees the through-port power change (Hu et al.; the IntechOpen chapter). It is a resonant modulator, so a short junction of 40 $μ m$ gives a large effect.
Resonance: The ring resonates when $mλ = n_{e f f} L$ with $L = 2 π R$ , and the resonances repeat every free spectral range $F S R = λ^{2} / (n_{g} L)$ . Hu et al. use an 8 $μ m$ ring on 220 nm SOI with a 420 nm rib and a 70 nm slab, a loaded Q of 1500 and an FSR of 7 nm in the O-band.
Coupling: The through-port dip is deepest near critical coupling, $a = t$ , where the coupling equals the loss (background knowledge); Hu et al. report a dip of about $49 dB$ and the Palermo group $18 dB$ ( $Q = 12000$ ).
Efficiency: The shift is $Δ λ = λ (Δ n_{e f f} / n_{g}) (L_{m o d} / L)$ , 24.5 $pm/V$ for Hu et al. and 22 $pm/V$ for the dual-ring paper, so the effective $V_{π} L$ is 0.57 $V cm$ (computed from 7 nm times 40 $μ m$ divided by twice 24.5 $pm/V$ ), against 1.4 to 2.05 $V cm$ for the Mach-Zehnder of the previous page.
Q and speed: A high Q gives a deep dip and a small voltage per linewidth but a slow photon lifetime $f_{p h} = c / (λ Q)$ ; a low Q (1500) allows 83 to above 110 GHz of electro-optic bandwidth at the price of a large voltage per linewidth.
Temperature: The resonance moves with the temperature of the ring, so each ring has a heater (81.2 $pm/mW$ for Hu et al.) and in practice a feedback lock (the Palermo group’s automatic locking).
Limits: The usable wavelength range is a fraction of the FSR, the response is narrowband, high optical power causes thermal bistability above about 0.3 mW, and the junction must be tuned to the laser.

2. Working Principle

Bias: The junction is reverse-biased (0 to -4 V in Hu et al., 0 to -4 V in the chapter), which widens the depletion region and lowers the capacitance (21 to 16 $fF$ ).
Shift: The change of the carrier density changes $n_{e f f}$ and so the round-trip phase, and the resonance moves to shorter wavelength by about 24.5 pm per volt.
Intensity: The laser sits on the slope of the dip (4 to $6 dB$ of insertion loss in Hu et al.), where the through power follows the Lorentzian line, so a shift of 10 pm changes the power by 6 % (computed for the assumed Lorentzian).
Speed: The bandwidth is limited by the photon lifetime and by the RC time of the diode, combined as $1/ f^{2} = 1/ f_{p h}^{2} + 1/ f_{e}^{2}$ ; Hu et al. measure 83 GHz (0 V) and above 110 GHz (-3 V).
Energy: The energy is $\frac{1}{4} C_{j} V_{pp}^{2}$ per bit: 0.97 fJ at 0.43 Vpp in Hu et al. (computed), which is far below the Mach-Zehnder of the previous page.
Lock: A heater and a monitor hold the resonance on the laser against a temperature drift (about 50 to 70 $pm/K$ , assumed from a thermo-optic coefficient of $1.86 \times 1 0^{- 4}$ per K).

3. Applications

Chip-to-chip optical links: Hu et al. show eyes for PCIe 5.0, 6.0 and 7.0 at 0.43 Vpp, error-free 32 Gbit/s NRZ over 2 km and 80 Gbit/s NRZ with a 3-tap equaliser, and the transmitter energy of 1.96, 1.56 and 0.78 pJ/bit including laser and amplifier.
Scale-out interconnects: At -3 V and 2 Vpp they report 200 Gbaud NRZ, 180 Gbaud PAM4 and 155 Gbaud PAM6 (400 Gbit/s) on one wavelength.
WDM transmitters: The Palermo group’s rings (about 5 $μ m$ , $Q = 12000$ ) carry five channels of 10 Gbit/s at 80 GHz spacing, and 12.5 Gbit/s on one channel, with an automatic wavelength lock.
Early compact modulators: A 12 $μ m$ diameter silicon ring was shown to switch light with less than 0.3 V at 1.5 Gbit/s at 1576 nm (Cornell release, 2005).
Higher extinction ratio: The dual-ring paper uses two rings (R = 15 $μ m$ ) to raise the simulated extinction ratio from 7.9 to $20.1 dB$ , with 20 Gbit/s measured ( $7.13 dB$ ).

Resonance, Coupling and Quality Factor

Name	Formula	Related Formulae	Variables/Constants
Resonance condition and free spectral range	$m λ_{m} = n_{e f f} L, L = 2 π R$ $F S R = \frac{λ ^{2}}{n _{g} L}$	$n_{g} = n_{e f f} - λ \frac{d n _{e f f}}{d λ}$ $Δ f_{F S R} = \frac{c}{n _{g} L}$	$m : in t e g er or d er o f t h e r eso nan ce$ $n_{e f f} : e f f ec t i v e in d e x$ $n_{g} : g r o u p in d e x$ $R : r in g r a d i u s$ $L : c i r c u m f er e n ce$ $λ : v a c uu m w a v e l e n g t h$ $c : s p ee d o f l i g h t in v a c uu m, H u e t a l . : R = 8 μ m, F S R 7 nm, O - ban d (1313.56 nm), t h e I n t ec h O p e n c ha pt er : R = 7.5 μ m, λ = 1552.3 nm, n_{e f f} = 2.5694 (f i tt e d), n_{g} ab o u t 3.89 (s im u l a t e d), an d a p r in t e d or d er m = 28, P a l er m o g r o u p : R ab o u t 5 μ m, 1.3 μ m ban d$ $(co m p u t e d : H u e t a l . L = 50.27 μ m, so F S R = 7 nm n ee d s n_{g} = 4.90 an d n_{g} = 4.2 or 4.4 w o u l d g i v e 8.2 or 7.8 nm; t h e c ha pt er : m = n_{e f f} L / λ = 78.00 (t h e p r in t e d m = 28 w o u l d n ee d n_{e f f} = 0.92, w hi c h i s im p oss ib l e f or s i l i co n), an d i t s F S R i s 13.1 nm f or n_{g} = 3.89)$
All-pass ring transmission and critical coupling	$T = \frac{a ^{2} + t ^{2} - 2 a t c o s φ}{1 + a ^{2} t ^{2} - 2 a t c o s φ}, φ = \frac{2 π n _{e f f} L}{λ}$	$T_{0} = \frac{( a - t ) ^{2}}{( 1 - a t ) ^{2}}$ $a = t \Rightarrow T_{0} = 0 (critical coupling)$	$a : am pl i t u d e t r an s mi ss i o n o f o n e r o u n d t r i p (l oss)$ $t : se l f - co u pl in g am pl i t u d e o f t h e b u s co u pl er (c a l l e d τ b y H u e t a l . an d σ in t h e c ha pt er)$ $φ : r o u n d - t r i p p ha se, κ^{2} + t^{2} = 1 f or a l oss l ess co u pl er (t h e c ha pt er)$ $T_{0} : t r an s mi ss i o n a t r eso nan ce, t h e f or m o f T i s a s p r in t e d b y H u e t a l . an d t h e c ha pt er, t h e co u pl in g s t a t e m e n t i s ba c k g r o u n d k n o w l e d g e, f i tt e d v a l u es o f t h e c ha pt er f or t h e 5 μ m c a r r i er - inj ec t i o n r in g : t = 0.9944, a = 0.9931, an d f or t h e 7.5 μ m d e pl e t i o n r in g : κ^{2} = 0.0354$ $(co m p u t e d w i t h t h e f i tt e d v a l u es : T_{0} = 0.0109, t ha t i s - 19.6 d B; κ^{2} = 1 - t^{2} = 0.0112 i s b e l o w t h e r o u n d - t r i p l oss 1 - a^{2} = 0.0138, so t h e r in g i s s l i g h tl y u n d er - co u pl e d)$
Quality factor, photon lifetime and finesse	$Q = ω_{0} τ_{p h} = \frac{λ _{0}}{Δ λ _{F W H M}}$ $\frac{1}{Q _{L}} = \frac{1}{Q _{i}} + \frac{1}{Q _{c}}$	$\frac{1}{τ _{p h}} = \frac{1}{τ _{c}} + \frac{1}{τ _{in}}$ $F = \frac{F S R}{Δ λ _{F W H M}}$	$ω_{0} : an g u l a r f r e q u e n cy o f t h e r eso nan ce$ $τ_{p h} : e n er g y (p h o t o n) l i f e t im e$ $Δ λ_{F W H M} : f u l l w i d t h a t ha l f ma x im u m o f t h e d i p$ $Q_{L}, Q_{i}, Q_{c} : l o a d e d, in t r in s i c an d co u pl in g Q$ $τ_{c}, τ_{in} : co u pl in g an d in t r in s i c l i f e t im es$ $F : f in esse, H u e t a l . : Q = 1500 l o a d e d, F W H M 0.88 nm, a s im u l a t e d Q o f 974 an d a p r in t e d^{'} in t r in s i c^{'} r an g e o f 1315 t o 1600, d u a l - r in g p a p er : Q_{c} = 20000 an d Q_{i} = 80000 t o 100000, P a l er m o g r o u p : Q = 12000, t h e I n t ec h O p e n c ha pt er : a f i tt e d l i f e t im e o f 9.07 p s$ $(co m p u t e d : H u e t a l . Q = 1313.56/0.88 = 1493 an d f in esse 7.95; d u a l - r in g p a p er w i t h Q_{c} = 20000 an d Q_{i} = 80000 : Q_{L} = 16000 an d τ_{p h} = Q / ω_{0} = 13.3 p s a t 1563 nm, w h er e a s i t s p u l se - d ec a y s im u l a t i o n g i v es 28 p s, w hi c h i s t h e 1/ e t im e o f t h e f i e l d, 2 τ_{p h} = 26.6 p s)$
Coupled-mode equation of the ring	$\frac{d a}{d t} = (i (ω_{0} - ω) - \frac{1}{2 τ _{p h}}) a + \frac{1}{2 τ _{c}} s_{+}$	$\frac{d A}{d t} = (2 π c j (\frac{1}{λ} - \frac{1}{λ _{0}}) - \frac{1}{τ}) A + j μ S_{i}$	$a, A : m o d e am pl i t u d e$ $s_{+}, S_{i} : in p u t w a v e$ $μ : co u pl in g r a t e w i t h μ^{2} = 2/ τ_{c} in t h e c ha pt er, t h e f i r s t f or m i s t ha t o f t h e d u a l - r in g p a p er an d u ses t h e e n er g y l i f e t im e τ_{p h} (t h e f i e l d d ec a y s a s 1/ (2 τ_{p h})), t h e seco n d i s t ha t o f t h e I n t ec h O p e n c ha pt er an d u ses τ a s t h e d ec a y t im e o f t h e am pl i t u d e (m y r e a d in g o f i t s^{'} 1/ τ^{'} an d i t s t e x t), t h e tw o p a p er s t h er e f or e d e f in e t h e l i f e t im e d i f f er e n tl y b y a f a c t or o f tw o$ $(co m p u t e d f or t h e c ha pt e r^{'} s τ = 9.07 p s a t 1552.3 nm : ω τ = 11006 i f τ w er e t h e e n er g y l i f e t im e, or Q = ω τ /2 = 5503 i f i t i s t h e am pl i t u d e d ec a y t im e)$

Electro-Optic Tuning and Modulation Depth

Name	Formula	Related Formulae	Variables/Constants
Resonance shift and voltage-length product of a ring	$Δ λ_{r es} = λ \frac{Δ n _{e f f}}{n _{g}} \frac{L _{m o d}}{L}$ $V_{π} L = \frac{F S R \cdot L _{m o d}}{2 ( d λ _{r es} / d V )}$	$Δ n_{e f f} = \frac{n _{g}}{λ} \frac{L}{L _{m o d}} \frac{d λ _{r es}}{d V} V$	$L_{m o d} : l e n g t h o f t h e d o p e d j u n c t i o n in t h e r in g$ $d λ_{r es} / d V : s hi f t p er v o l t, t h e f i r s t f or m i s t h e s t an d a r d p er t u r ba t i o n r es u l t (ba c k g r o u n d k n o w l e d g e), t h e seco n d i s t h e d e f ini t i o n o f H u e t a l ., H u e t a l . : 24.5 pm/V (0 t o - 4 V a v er a g e, w a f er r an g e 22.8 t o 25.5), L_{m o d} = 40 μ m, V_{π} L ab o u t 0.57 V cm, d u a l - r in g p a p er : 22 pm/V an d a f i t (s hi f t in p m) - 4.5 V^{2} + 31 V + 0.13 f or t h e r e v er se bia s V in v o l t s, w i t h 0 t o 2 V s ai d t o s hi f t t h e r eso nan ce ab o u t 30 p m, t h e c ha pt er g i v es p o l y n o mia l f i t s f or Δ n_{e f f} (V) (n e x t r o w)$ $(co m p u t e d : V_{π} L = 7 x 0.004/ (2 x 0.0245) = 0.571 V cm, e q u a l t o H u e t a l ., w hi c h m e an s V_{π} = 143 V f or o n e p a ss; w i t h n_{g} = 4.90 an d L_{m o d} / L = 0.796 t h e in d e x c han g e i s 1.15 \times 1 0^{- 4} p er v o l t, 2.1 t im es t h e 5.5 \times 1 0^{- 5} p er v o l t o f t h e M a c h - Z e hn d er o f t h e p r e v i o u s p a g e; t h e f i t o f t h e d u a l - r in g p a p er g i v es 26.6, 44.1 an d 52.6 p m a t 1, 2 an d 3 V, a s l o p e o f 31 pm/V a t 0 V an d 13 pm/V a t 2 V, an d a m e an o f 22.1 pm/V o v er 0 t o 2 V)$
Voltage per linewidth and intensity modulation	$T (λ) = 1 - \frac{1 - T _{0}}{1 + ( 2 Δ λ /Δ λ _{F W H M} ) ^{2}}$ $V_{F W H M} = \frac{Δ λ _{F W H M}}{d λ _{r es} / d V}$	$\frac{d T}{d λ} = (1 - T_{0}) \frac{2 x}{( 1 + x ^{2} ) ^{2}} \frac{2}{Δ λ _{F W H M}}, x = \frac{2Δ λ}{Δ λ _{F W H M}}$	$T_{0} : t r an s mi ss i o n a t t h e ce n t r e o f t h e d i p$ $Δ λ : d e t u nin g o f t h e l a ser f r o m t h e ce n t r e$ $x : d e t u nin g in ha l f - w i d t h s$ $V_{F W H M} : v o l t a g e t ha t m o v es t h e r eso nan ce b y o n e l in e w i d t h, t h e L or e n t z ian f or m i s ba c k g r o u n d k n o w l e d g e (a s in g l e - m o d e r in g), H u e t a l . : E R o f t h e d i p ab o u t 49 d B (D C), o p er a t in g p o in t s se t a t 4, 5 an d 6 d B o f in ser t i o n l oss, 0.43 V pp f or 32 G bi t / s N R Z an d 0.56 V pp f or 80 G bi t / s N R Z w i t h a 3 - t a p e q u a l i ser, t h e d u a l - r in g p a p er : s in g l e - r in g e x t in c t i o n r a t i o 7.9 d B (s im u l a t e d, d r i v e ab o u t ha l f t h e F W H M), m e a s u r e d 5.9 an d 5.8 d B$ $(co m p u t e d w i t h T_{0} = - 49 d B, an a ss u m e d L or e n t z ian an d 24.5 pm/V : V_{F W H M} = 35.9 V f or H u e t a l . (0.88 nm), b u t 4.4 V f or a Q o f 12000 a t 1.3 μ m (0.108 nm); a t 6 d B o f l oss t h e l a ser s i t s 0.255 nm f r o m t h e ce n t r e, t h e s l o p e i s 1.48 \times 1 0^{- 3} p er p m, an d 0.43 V pp (10.5 p m) c han g es t h e t h r o ug h p o w er b y 6.2 %, an e x t in c t i o n r a t i o o f 0.27 d B)$
Depletion-model fits of a 7.5 micrometre ring	$f (V) = a_{0} + a_{1} V + a_{2} V^{2} + a_{3} V^{3} + a_{4} V^{4}$	$C_{p n} = C^{'} (V) \cdot 0.75 L$	$t h e p r in t e d coe f f i c i e n t s (V in v o l t s, v a l i d 0 t o 5 V) : Δ n_{e f f} : - 4.3 \times 1 0^{- 7}, 7.3 \times 1 0^{- 5}, 8.0 \times 1 0^{- 6}, 1.1 \times 1 0^{- 6}, 5.2 \times 1 0^{- 8}, Δ α in dB/cm : 0.01, 1.5, 0.17, - 2.3 \times 1 0^{- 2}, 1.0 \times 1 0^{- 3}, C^{'} in fF/ μ m : 0.71, - 0.14, 5.5 \times 1 0^{- 2}, - 1.2 \times 1 0^{- 2}, 1.0 \times 1 0^{- 3}, t h e I n t ec h O p e n c ha pt er : r ib 500 nm w i d e, 220 nm hi g h, 90 nm s l ab, d o p in g ab o u t 2 \times 1 0^{18} c m^{- 3} o n ab o u t 75 % o f t h e r in g, R_{p} = R_{n} = 30 Ω, C_{p n} (0 V) = 25 fF, 25 G bi t / s m e a s u r e d w i t h a 4.4 V pp d r i v e$ $(co m p u t e d : C^{'} (0 V) = 0.71 fF/ μ m o v er 0.75 x 47.1 μ m = 25.1 fF, w hi c h a g r ees w i t h t h e p r in t e d 25 fF, an d C (4 V) = 18.3 fF; Δ n_{e f f} (4 V) = 5.03 \times 1 0^{- 4} an d Δ α (4 V) = 7.5 dB/cm; w i t h n_{g} = 3.89 an d t h e 75 % f r a c t i o n (a ss u m e d) t h e 0 t o 4 V s hi f t i s 151 p m, 37.7 pm/V)$

Speed: Photon Lifetime and RC

Name	Formula	Related Formulae	Variables/Constants
Photon-lifetime-limited bandwidth	$f_{p h} = \frac{1}{2 π τ _{p h}} = \frac{c}{λ Q}$	$f_{p h} = \frac{c α}{π n _{g}}$	$α : p o w er l oss coe f f i c i e n t o f t h e c a v i t y (t h e f or m p r in t e d b y H u e t a l .), τ_{p h} an d Q a s in t h e e a r l i er r o w, H u e t a l . : Q = 1500, t h e P a l er m o g r o u p : Q = 12000, t h e I n t ec h O p e n c ha pt er : Q = 5503 or 11006 d e p e n d in g o n t h e l i f e t im e co n v e n t i o n, t h e d u a l - r in g p a p er : Q_{L} ab o u t 16190 (m e a s u r e d, r in g 1)$ $(co m p u t e d : 152 G H z f or Q = 1500 a t 1313.56 nm; 19.2 G H z f or Q = 12000 a t 1.3 μ m; 35.1 G H z or 17.5 G H z f or t h e 7.5 μ m r in g; 11.8 G H z f or t h e d u a l - r in g p a p e r^{'} s Q_{L} o f 16190)$
Combined electro-optic bandwidth	$\frac{1}{f _{E O}^{2}} = \frac{1}{f _{p h}^{2}} + \frac{1}{f _{e}^{2}}, f_{e} = \frac{1}{2 π ( R _{s} + R _{d r} ) C _{j}}$	$f_{e} (at 0 V) and at - 3 V$	$R_{s} : ser i es r es i s t an ce o f t h e d i o d e$ $R_{d r} : o u tp u t r es i s t an ce o f t h e d r i v er$ $C_{j} : j u n c t i o n c a p a c i t an ce, H u e t a l . : C_{j} = 21 fF (0 V) an d 16 fF (- 3 V), R_{s} = 31 an d 32 Ω, m e a s u r e d 83 G H z (6 d B l oss), 97 G H z (5 d B) an d ab o v e 110 G H z (4 d B) a t 0 V, an d ab o v e 110 G H z a t - 3 V, s im u l a t e d 85 G H z (0 V) an d 100 G H z (- 3 V), t h e d r i v er r es i s t an ce i s n o t s t a t e d (t ak e n a s 50 Ω, a ss u m e d), bi s t abi l i t y p a p er : 25 G H z s t a t e d f or a 7.5 μ m r in g$ $(co m p u t e d w i t h R_{d r} = 50 Ω : f_{e} = 93.6 G H z a t 0 V an d co mbin e d 79.7 G H z (m e a s u r e d 83); a t - 3 V f_{e} = 121.3 G H z an d co mbin e d 94.9 G H z (m e a s u r e d ab o v e 110); f or t h e 7.5 μ m r in g w i t h 60 + 50 Ω an d 25 fF f_{e} = 57.9 G H z an d t h e co mbin e d v a l u e i s 30.0 G H z f or Q = 5503 or 16.8 G H z f or Q = 11006)$

Temperature, Power, Energy and Channels

Name	Formula	Related Formulae	Variables/Constants
Thermal drift and heater tuning	$\frac{d λ _{r es}}{d T} = \frac{λ}{n _{g}} \frac{d n _{e f f}}{d T}$ $P_{F S R} = \frac{F S R}{d λ _{r es} / d P}$	$Δ T_{w in} = \frac{Δ λ _{w in}}{d λ _{r es} / d T}$	$d n_{e f f} / d T : t h er m o - o pt i c coe f f i c i e n t o f t h e g u i d e d m o d e (t ak e n a s 1.86 \times 1 0^{- 4} K^{- 1}, a ss u m e d f r o m ba c k g r o u n d k n o w l e d g e, n o so u r ce I r e a d g i v es a d r i f t in pm/K)$ $d λ_{r es} / d P : h e a t er e f f i c i e n cy$ $P_{F S R} : h e a t er p o w er f or o n e f r ee s p ec t r a l r an g e, H u e t a l . : h e a t er 81.2 pm/mW (77.6 t o 84.5), 54 Ω, an d a 1.5 K o p er a t in g r an g e f or 64 G ba u d N R Z, t h e d u a l - r in g p a p er : 80.5 pm/mW, t h e P a l er m o g r o u p : 23 mi cr o w a tt p er G H z, 44 mW/FSR$ $(co m p u t e d w i t h t h e a ss u m e d coe f f i c i e n t : 49.9 pm/K f or n_{g} = 4.9 a t 1313.56 nm, 58.2 pm/K f or n_{g} = 4.2, an d 68.6 pm/K f or n_{g} = 4.2 a t 1550 nm; H u e t a l . 7/0.0812 = 86 mW/FSR, t h e d u a l - r in g p a p er 77 mW/FSR f or an a ss u m e d n_{g} = 4.2 an d F S R = 6.17 nm, t h e P a l er m o g r o u p 44; a 1.5 K w in d o w i s 75 p m, 0.085 o f t h e l in e w i d t h, an d t r a c k in g 1 K t ak es 0.61 mW)$
Optical power and thermal bistability	$P_{d B m} = 10 lo g_{10} \frac{P}{1 mW}$	$Δ λ_{t h} = R_{t h} η_{ab s} P \frac{d λ _{r es}}{d T}$	$P : p o w er in t h e b u s$ $R_{t h} : t h er ma l r es i s t an ce o f t h e r in g$ $η_{ab s} : f r a c t i o n o f t h e p o w er ab sor b e d, t h e seco n d f or m i s ba c k g r o u n d k n o w l e d g e, t h e bi s t abi l i t y p a p er (7.5 μ m r in g, 25 G H z) : a sy mm e t r i c l in e f r o m ab o u t 0.3 mW (- 5.23 dBm), t h er ma l r e d s hi f t d o minan t ab o v e ab o u t 1 mW, s w ee p 0.063 t o 15.85 mW (- 12 t o + 12 dBm), r e v er se b r e ak d o w n a t - 8 V, o p er a t in g a t + 0.6 V w i t h 0.5 an d 1 V pp, t h e t h er ma l t im e co n s t an t i s in mi cr oseco n d s an d t h e c a r r i er t im e co n s t an t in nan oseco n d s, H u e t a l . : 500 mi cr o w a tt s in t h e b u s$ $(co m p u t e d : 10 l o g 10 0.3 = - 5.23 dBm, 10 l o g 10 0.063 = - 12.0 dBm, 10 l o g 10 15.85 = 12.0 dBm; 500 mi cr o w a tt s i s - 3.0 dBm, ab o v e t h e o n se t o f 0.3 mW)$
Dynamic energy per bit of a depletion ring	$E_{bi t} = \frac{1}{4} C_{j} V_{pp}^{2} (NRZ)$	$E_{p} = C_{j} V_{pp}^{2} \sum_{i = 1}^{N - 1} (N - i) (\frac{i}{N - 1})^{2}, E_{b} = \frac{E _{p}}{N ^{2} l o g _{2} N}$	$N : n u mb er o f P A M l e v e l s$ $E_{p} : e n er g y p er sy mb o l$ $E_{b} : e n er g y p er bi t, t h e f a c t or 1/4 f o l l o w s t h e G er mani u m an d S i l i co n - G er mani u m E l ec t r o ab sor pt i o n M o d u l a t or p a g e, t h e s u m i s t h e f or m o f H u e t a l ., H u e t a l . : 0.97 f J / bi t f or N R Z a t 0.43 V pp an d z er o bia s, 0.78 f J / bi t f or P A M 4 a t 0.73 V pp, an d a t - 3 V an d 2 V pp 16 f J / bi t (N R Z), 4.44 f J / bi t (P A M 4) an d 2.89 f J / bi t (P A M 6), t h e main t e x t p r in t s p J / bi t f or t h e f i r s t v a l u es, t h e P a l er m o g r o u p : 473 f J / bi t a t 9 G bi t / s w i t h a 2 V pp C M O S d r i v er$ $(co m p u t e d : 21 fF x 0.4 3^{2} /4 = 0.971 f J; t h e P A M 4 coe f f i c i e n t i s 2.222/32 = 0.0694, so 0.777 f J a t 0.73 V; 16 fF a t 2 V g i v es 16.0 f J (N R Z), 4.44 f J (P A M 4) an d 2.89 f J (P A M 6, coe f f i c i e n t 0.0451); t h e 473 f J / bi t a t 2 V pp n ee d s an e f f ec t i v e c a p a c i t an ce o f 473 fF, so i t co n t ain s m or e t han t h e j u n c t i o n c ha r g e)$
Channel spacing and number of channels	$N_{c h} = \frac{Δ f _{F S R}}{Δ f _{c h}}$ $Δ λ_{c h} = \frac{λ ^{2}}{c} Δ f_{c h}$	$Δ f_{F S R} = \frac{c F S R}{λ ^{2}}$	$Δ f_{c h} : c hann e l s p a c in g in f r e q u e n cy$ $Δ f_{F S R} : f r ee s p ec t r a l r an g e in f r e q u e n cy, P a l er m o g r o u p : 80 G H z s p a c in g an d m or e t han 20 c hann e l s a t 1.3 μ m, f i v e c hann e l s o f 10 G bi t / s, H u e t a l . : F S R 7 nm an d l in e w i d t h 0.88 nm, t h e F S R o f t h e P a l er m o r in g i s n o t p r in t e d (i t i s d er i v e d h er e f r o m 44 mW/FSR d i v i d e d b y 23 mi cr o w a tt p er G H z, m y r e a d in g)$ $(co m p u t e d : 80 G H z i s 0.451 nm a t 1300 nm; t h e d er i v e d F S R i s 1913 G H z (10.8 nm), w hi c h h o l d s 23.9 c hann e l s, in l in e w i t h^{'} m or e t han 2 0^{'}; H u e t a l . 7 nm i s 1216 G H z, w hi c h h o l d s 15.2 c hann e l s o f 80 G H z, b u t t h e i r l in e w i d t h i s 153 G H z, so 80 G H z s p a c in g w o u l d o v er l a p)$

Important Questions

Ques: Is the photon lifetime of the 7.5 micrometre depletion ring 9.07 ps or half of that?

The IntechOpen chapter fits a lifetime of 9.07 ps with an amplitude equation that decays as $1/ τ$ , whereas the dual-ring paper writes $1/ (2 τ)$ and calls $τ$ the energy lifetime. If $τ = 9.07$ ps is the amplitude decay time, the loaded Q is $ω τ /2 = 5503$ and $f_{p h} = 35.1$ GHz; if it is an energy lifetime, Q is 11006 and $f_{p h} = 17.5$ GHz (computed). With 25 $fF$ and 60 + 50 $Ω$ the combined bandwidth would be 30.0 or 16.8 GHz (computed, driver resistance assumed). Because 25 GHz and 25 Gbit/s are reported for this ring, I judge the amplitude reading, Q about 5500, more likely (my reading). The dual-ring 28 ps is the same factor of two (13.3 ps against 26.6 ps, computed). Reference: IntechOpen chapter 54847 on models of carrier-injection and carrier-depletion ring modulators (title and authors not recorded; read through a fetch-tool summary); ‘Cascaded uncoupled dual-ring modulator’, arXiv 1407.5704 (read through a fetch-tool summary; authors not recorded); arXiv 1911.01135 on the nonlinear and thermal behaviour of a silicon ring modulator (title and authors not recorded; read through a fetch-tool summary).

Ques: Which group index belongs to a silicon ring, 3.89, 4.9 or 5.0, and what is the mode order of the 7.5 micrometre ring?

The sources do not agree. Hu et al. state an FSR of 7 nm for R = 8 $μ m$ , which needs $n_{g} = 4.90$ at 1313.56 nm (computed). The Palermo group gives 23 microwatt per GHz and 44 $mW/FSR$ , from which I derive an FSR of 10.8 nm and $n_{g} = 4.99$ for R = 5 $μ m$ (computed, my reading). The chapter simulates $n_{g} = 3.89$ for a 500 nm rib near 1552 nm. I cannot tell which is right without a mode simulation; I give more weight to the stated FSR of Hu et al. The chapter’s order m = 28 is wrong: $n_{e f f} L / λ = 78.00$ (computed), so I use 78. Reference: Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above); Texas A&M WDM ring-resonator transmitter paper (people.engr.tamu.edu/spalermo/docs/2014_wdm_ring_resonator_transmitter_oic.pdf; title and authors not recorded; read through a fetch-tool summary, which found no photon lifetime, bandwidth or capacitance); IntechOpen chapter 54847 on models of carrier-injection and carrier-depletion ring modulators (title and authors not recorded; read through a fetch-tool summary).

Ques: Is the Q of Hu et al. loaded or intrinsic, and can the extinction ratio exceed the wafer range?

Hu et al. give a loaded Q of 1500 and a wafer range of 1315 to 1600 that the summary calls intrinsic. A loaded Q cannot exceed the intrinsic Q, because $1/ Q_{L} = 1/ Q_{i} + 1/ Q_{c}$ , so a loaded value of 1500 cannot sit inside a range whose lower end is 1315 if that range is intrinsic; I read it as the range of the loaded Q (my reading). Likewise the extinction ratio of the dip is given as $49 dB$ while the wafer range is 26.7 to $48.4 dB$ ; $49 dB$ is above the upper end, so I treat it as a best-case value near $48 dB$ (my reading). The measured Q of 1500 agrees with $λ /Δ λ = 1493$ (computed). Reference: Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above).

Ques: What is the shift per volt: 22, 24.5, 31 or 37.7 pm?

Hu et al. measure 24.5 $pm/V$ on average between 0 and -4 V (wafer range 22.8 to 25.5). The dual-ring paper gives 22 $pm/V$ for the 15 $μ m$ ring, and its fit $- 4.5 V^{2} + 31 V + 0.13$ pm gives a slope of 31 $pm/V$ at 0 V and a mean of 22.1 $pm/V$ over 0 to 2 V (computed); the text says 0 to 2 V shifts the resonance about 30 pm, which fits the initial slope rather than the 44 pm at 2 V, so I read it as 31 $pm/V$ (my reading). My own estimate from the chapter’s polynomial, with 75 % of the ring doped and $n_{g} = 3.89$ (assumed), is 37.7 $pm/V$ over 0 to 4 V (computed). I would quote 22 to 25 $pm/V$ as the measured range. Reference: Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above); ‘Cascaded uncoupled dual-ring modulator’, arXiv 1407.5704 (read through a fetch-tool summary; authors not recorded); IntechOpen chapter 54847 on models of carrier-injection and carrier-depletion ring modulators (title and authors not recorded; read through a fetch-tool summary).

Ques: Can 0.43 Vpp really carry 32 Gbit/s, and is the energy 0.97 fJ or 0.97 pJ per bit?

The main text of Hu et al. prints pJ/bit, the methods fJ/bit. The formula settles it: $\frac{1}{4} \times 21 fF \times 0.4 3^{2} = 0.971$ fJ (computed), and the PAM4 value 0.78 fJ, 16 fJ, 4.44 fJ and 2.89 fJ also follow from their sum (computed), so fJ is right. The optical swing is small: for a Lorentzian dip of $49 dB$ at $6 dB$ of insertion loss, 0.43 Vpp (10.5 pm) changes the through power by 6.2 % and the extinction ratio is $0.27 dB$ (computed, assumed Lorentzian). I could not check how the eye closes; the source mentions an amplifier before the receiver, which I take as the reason (my reading). Reference: Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above).

Ques: Are 83, 97 and above 110 GHz consistent with the model of Hu et al.?

At 0 V the model agrees: with 21 $fF$ , 31 + 50 $Ω$ (the 50 $Ω$ driver resistance is assumed) and $f_{p h} = 152$ GHz the combined bandwidth is 79.7 GHz (computed), against 83 GHz measured at $6 dB$ of insertion loss. At -3 V the model gives 94.9 GHz (computed), but the measured value is above 110 GHz at 4, 5 and $6 dB$ . The measured bandwidth also rises as the laser moves further from resonance (83, 97, above 110 GHz at 6, 5, $4 dB$ ), which cannot come from the RC and lifetime terms alone; I read it as peaking of the response with detuning (my reading, background knowledge). So the formula is a floor, not a prediction. Reference: Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above).

Ques: Why does a low-Q ring need so many volts per linewidth?

The resonance must move by a fraction of the linewidth to modulate, and the linewidth shrinks with Q. For Hu et al., 0.88 nm divided by 24.5 $pm/V$ is 35.9 V per linewidth (computed), so a swing of 0.5 V moves the dip by only 1.4 % of its width, whereas a ring of $Q = 12000$ at 1.3 $μ m$ (0.108 nm) needs 4.4 V per linewidth (computed, same shift per volt assumed). The low-Q ring pays with modulation depth but gains bandwidth (152 GHz against 19 GHz, computed) and a lower sensitivity to temperature relative to its width. The choice is a trade between depth, speed and the lock, and the sources use both ends. Reference: Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above); Texas A&M WDM ring-resonator transmitter paper (people.engr.tamu.edu/spalermo/docs/2014_wdm_ring_resonator_transmitter_oic.pdf; title and authors not recorded; read through a fetch-tool summary, which found no photon lifetime, bandwidth or capacitance).

Ques: How much temperature drift can a ring tolerate?

No source I read gives the drift in pm/K. With an assumed thermo-optic coefficient of $1.86 \times 1 0^{- 4}$ per K (background knowledge) the drift is 49.9 $pm/K$ for $n_{g} = 4.9$ at 1313.56 nm, or 68.6 $pm/K$ at 1550 nm for $n_{g} = 4.2$ (computed). Hu et al. state a 1.5 K operating range for a 64 Gbaud link, which is 75 pm (computed, assumed coefficient), only 0.085 of the 0.88 nm linewidth. Their heater gives 81.2 $pm/mW$ , so holding 1 K costs 0.61 mW (computed), and one free spectral range costs 86 $mW/FSR$ (computed), against 77 (dual-ring paper, with an assumed $n_{g}$ of 4.2) and 44 $mW/FSR$ for the Palermo group. These heater figures differ by a factor of two, and I cannot say why. Reference: Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above); ‘Cascaded uncoupled dual-ring modulator’, arXiv 1407.5704 (read through a fetch-tool summary; authors not recorded); Texas A&M WDM ring-resonator transmitter paper (people.engr.tamu.edu/spalermo/docs/2014_wdm_ring_resonator_transmitter_oic.pdf; title and authors not recorded; read through a fetch-tool summary, which found no photon lifetime, bandwidth or capacitance).

Ques: At what optical power does a ring modulator start to behave nonlinearly?

The bistability study of a 7.5 $μ m$ ring finds an asymmetric line from about 0.3 mW (-5.23 $dBm$ , computed from 10 log10 0.3) in the bus, with the thermal red shift dominating above about 1 mW and the sweep running from 0.063 to 15.85 mW (-12 to +12 $dBm$ , computed). Carriers act in nanoseconds and the heating in microseconds, so the two effects can be separated by speed. Hu et al. used 500 microwatts (-3.0 $dBm$ , computed), which is above the 0.3 mW onset of the other ring; the onset depends on the Q and the absorption, so the numbers cannot be compared directly (my reading). I would keep the bus power below about 0.3 mW unless the lock is active. Reference: arXiv 1911.01135 on the nonlinear and thermal behaviour of a silicon ring modulator (title and authors not recorded; read through a fetch-tool summary); Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above).

Ques: How does the ring compare with the Mach-Zehnder of the previous page?

The ring is far smaller and cheaper in energy: Hu et al. use a 40 $μ m$ junction and 0.97 fJ/bit at 0.43 Vpp, whereas the Xiao junction (750 $μ m$ , 102 $fF$ ) takes 1.08 pJ/bit at 6.5 V (computed on the previous page). The ratio is 1108 (computed), made of a voltage factor $(6.5/0.43)^{2} = 229$ and a capacitance factor 4.9. The ring gives 0.57 $V cm$ against 1.4 to 2.05 $V cm$ . The price is a narrow optical band (a fraction of a 7 nm FSR), the thermal lock, a small extinction ratio at low swing ( $0.27 dB$ computed against 3.3 to $3.9 dB$ for the Mach-Zehnder) and the sensitivity to optical power. Reference: Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above); Silicon Carrier-Depletion Mach-Zehnder Modulator page of this vault.

Ques: How many wavelength channels can one ring cover?

The number is FSR divided by the channel spacing. The Palermo group states 80 GHz spacing and more than 20 channels at 1.3 $μ m$ ; the derived FSR of 1913 GHz (from 44 $mW/FSR$ divided by 23 microwatt per GHz, my reading) holds 23.9 channels (computed), which agrees. Hu et al. have an FSR of 7 nm, that is 1216 GHz, so 15.2 channels of 80 GHz (computed), but their linewidth is 0.88 nm or 153 GHz, so channels 80 GHz apart would overlap and the spacing must be wider than the 0.88 nm linewidth (my reading). A low Q therefore lowers the channel count, a high Q (12000, linewidth 0.108 nm) allows the dense grid. The sources do not describe crosstalk or the demultiplexing. Reference: Texas A&M WDM ring-resonator transmitter paper (people.engr.tamu.edu/spalermo/docs/2014_wdm_ring_resonator_transmitter_oic.pdf; title and authors not recorded; read through a fetch-tool summary, which found no photon lifetime, bandwidth or capacitance); Hu et al., ‘400-Gbps/lambda Ultrafast Silicon Microring Modulator for Scalable Optical Compute Interconnects’, arXiv 2509.01555 (read through a fetch-tool summary; only ‘Hu et al.’ is recorded as the author; the summary listed internal inconsistencies, some of which are discussed in the questions above).

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Silicon Microring Modulator