DWDM Engineering Formulas — Quick Reference
This chapter is the applied formula card for the Optical-Transport series. Where Ch01 derives the physics from first principles, Ch03 covers amplifier noise theory, Ch04 explains FEC and framing overhead, and Ch05 tackles power optimisation — this chapter collects the 18 most-used engineering formulas into a single quick-reference with worked examples for each. Keep this open during link-budget spreadsheets, OSNR validation checks, and capacity planning exercises.
Key Concepts at a Glance
| Formula Area | Core Question It Answers | Key Formula |
|---|---|---|
| dBm/mW Conversion | What is this power reading in linear units? | P(mW) = 10^(dBm/10) |
| Wavelength/Frequency | Which ITU channel does this wavelength map to? | v(THz) = 299792.458 / lambda(nm) |
| Span Loss | How much power does this fibre span consume? | alpha*L + splices + connectors + margin |
| Link Budget | Does the signal arrive above receiver sensitivity? | P_rx = P_tx - total_losses |
| OSNR Estimation | Will this modulation format survive N amplified spans? | 58 + P_ch - NF - 10*log10(N) - L_span |
| Shannon Limit | What is the theoretical max bit-rate for this channel? | C = B * log2(1 + SNR) |
| Channel Bit Rate | What net throughput does this transponder deliver? | Baud * bits/sym * 2(DP) / (1+FEC) |
| System Capacity | How much total throughput fits on this fibre? | N_channels * rate_per_channel |
| Maximum Reach | How many spans before OSNR falls below threshold? | N_max = 10^((budget)/10) |
flowchart TD START([Which formula do I need?]) --> Q1{What am I calculating?} Q1 -->|Power units| A1[dBm/mW Conversion<br>Section 1] Q1 -->|ITU channel| A2[Wavelength/Frequency<br>Section 2] Q1 -->|Fibre loss| Q2{Single span or<br>amplified chain?} Q2 -->|Single span| A3[Span Loss + Link Budget<br>Sections 3-4] Q2 -->|Amplified| Q3{Noise or reach?} Q3 -->|OSNR check| A4[OSNR Estimation<br>Section 5] Q3 -->|Max distance| A5[Maximum Spans<br>Section 9] Q1 -->|Capacity| Q4{Per-channel or system?} Q4 -->|Per-channel| A6[Channel Bit Rate<br>Section 7] Q4 -->|System total| A7[System Capacity<br>Section 8] Q1 -->|Amplifier design| Q5{Single EDFA or cascade?} Q5 -->|Single| A8[ASE Noise + EDFA Power<br>Sections 5b-6] Q5 -->|Cascade| A9[Friis Formula + Raman NF<br>Sections 10-11] Q1 -->|Latency or sites| A10[Infrastructure<br>Sections 12-13]
1. Conversions
These two conversions underpin every other formula in optical engineering. Memorise the three anchor points (0 dBm = 1 mW, +3 dB = double, -3 dB = half) and derive the rest.
1.1 dBm to mW and Back
dBm to mW: P(mW) = 10^(dBm / 10)
mW to dBm: P(dBm) = 10 × log₁₀(P in mW)
Worked example: an OSA reads a channel at -8.5 dBm.
P = 10^(−8.5 / 10) = 10^(−0.85) = 0.141 mW = 141 µW
Highlight: Key Insight Memorise three values and derive all others: 0 dBm = 1 mW, +3 dBm = 2 mW (doubling), -3 dBm = 0.5 mW (halving).
Quick-reference anchors:
0 dBm = 1 mW (the reference point)
+3 dBm = 2 mW (doubling = +3 dB)
−3 dBm = 0.5 mW (halving = −3 dB)
From these: +10 dBm = 10 mW (because +10 = +3 +3 +3 +1 ≈ 2×2×2×1.25)
−10 dBm = 0.1 mW (divide by 10)
+20 dBm = 100 mW (divide by 100)
−20 dBm = 0.01 mW (= 10 µW)
−30 dBm = 0.001 mW (= 1 µW)
1.2 Wavelength to Frequency (ITU Grid)
The speed of light in vacuum links wavelength and frequency. This is essential for mapping transponder readings to ITU channel numbers.
ν (THz) = 299,792.458 / λ (nm)
λ (nm) = 299,792.458 / ν (THz)
Worked example 1: transponder displays lambda = 1548.52 nm.
ν = 299,792.458 / 1548.52 = 193.600 THz → ITU channel C21 (100 GHz grid)
Worked example 2: provision a channel at 193.4375 THz.
λ = 299,792.458 / 193.4375 = 1549.72 nm
2. Span Design and Link Budget
These formulas determine whether a signal survives the physical fibre path between two sites — the foundation of every optical network design.
2.1 Span Loss
Total optical attenuation across a fibre span including all impairments.
Span_loss (dB) = (α × L) + (N_s × L_s) + (N_c × L_c) + margin
| Parameter | Meaning | Typical Value |
|---|---|---|
| alpha | Fibre attenuation (dB/km) | 0.20 for C-band G.652D |
| L | Span length (km) | Design-specific |
| N_s, L_s | Fusion splices, loss per splice | 0.05 dB each |
| N_c, L_c | Connector pairs, loss per pair | 0.3 dB each |
| margin | Ageing/repair allowance | 0.5-1.0 dB for new builds |
Worked example: 72 km span, 4 fusion splices, 2 connector pairs, 0.5 dB ageing margin.
Span loss = (0.20 × 72) + (4 × 0.05) + (2 × 0.3) + 0.5
= 14.4 + 0.2 + 0.6 + 0.5
= 15.7 dB
2.2 Link Budget (Received Power)
Determines whether the received signal exceeds the minimum detectable level.
P_received (dBm) = P_launch (dBm) − total_losses (dB)
Link works if: P_received > receiver_sensitivity + system_margin
Worked example: can a 400ZR+ pluggable span 35 km without amplification?
Given: P_launch = 0 dBm, receiver sensitivity = −22 dBm, system margin = 3 dB
Span loss = (0.20 × 35) + (2 × 0.05) + (2 × 0.3) = 7.0 + 0.1 + 0.6 = 7.7 dB
P_received = 0 − 7.7 = −7.7 dBm
Minimum needed = −22 + 3 = −19 dBm
−7.7 > −19 → YES, link works with 11.3 dB of headroom.
No amplifier needed.
2.3 Maximum Unamplified Reach
The longest span achievable before an amplifier becomes mandatory. See Ch01 for the physics behind attenuation limits.
L_max (km) = (P_launch − P_sensitivity − margin) / α
Worked example: coherent 100G transponder, P_launch = +1 dBm, sensitivity = -24 dBm, margin = 4 dB.
L_max = (1 − (−24) − 4) / 0.20 = 21 / 0.20 = 105 km
Add connectors and splices (~1 dB) and the effective reach drops to approximately 100 km.
3. OSNR and Amplifier Noise
The critical formulas for amplified multi-span link design. OSNR is the single metric that determines whether a given modulation format survives the noise accumulated across a chain of EDFAs. See Ch03 for amplifier theory and Ch05 for nonlinear impairment interactions.
3.1 OSNR Estimation (Cascaded EDFAs)
The most-used formula in amplified network design.
OSNR (dB) ≈ 58 + P_ch − NF − 10·log₁₀(N) − L_span
| Parameter | Meaning |
|---|---|
| 58 | Physical constant from hvdelta_v (Planck, C-band frequency, 0.1 nm ref BW) |
| P_ch | Per-channel launch power (dBm) |
| NF | Amplifier noise figure (dB) |
| N | Number of spans |
| L_span | Loss per span (dB), assuming gain = loss at each EDFA |
Highlight: Key Insight Every doubling of span count costs exactly 3 dB of OSNR (because 10*log10(2) = 3.01). This is the fundamental reach-cost relationship in amplified optical networks.
Worked example: design check for a 640 km link with 80 km spans.
Given: P_ch = −1 dBm, NF = 5.5 dB, N = 8 spans, L_span = 16.8 dB
OSNR = 58 + (−1) − 5.5 − 10·log₁₀(8) − 16.8
= 58 − 1 − 5.5 − 9.03 − 16.8
= 25.67 dB
Check: DP-16QAM needs ~22 dB. Margin = 25.67 − 22 = 3.67 dB. PASS.
DP-QPSK needs ~12 dB. Margin = 25.67 − 12 = 13.67 dB. Very comfortable.
3.2 ASE Noise Power per Amplifier
The exact noise contribution of each individual EDFA.
P_ASE = 2 × NF_lin × h × ν × Δν × (G − 1)
| Constant | Value |
|---|---|
| NF_lin | 10^(NF_dB / 10) |
| h | 6.626 x 10^-34 J*s |
| v | 193.4 x 10^12 Hz |
| delta_v | 12.5 x 10^9 Hz (0.1 nm reference bandwidth) |
| G | 10^(G_dB / 10) (linear gain) |
| Factor of 2 | ASE in both polarisations |
Worked example: EDFA with NF = 5.5 dB and gain = 18 dB.
NF_lin = 10^(5.5/10) = 3.55
G_lin = 10^(18/10) = 63.1
h × ν × Δν = 6.626e-34 × 193.4e12 × 12.5e9 = 1.602 × 10⁻⁹
P_ASE = 2 × 3.55 × 1.602e-9 × (63.1 − 1) = 0.707 µW = −31.5 dBm
Highlight: Tip Quick shortcut: for NF = 5.5 dB, G = 20 dB the ASE noise is approximately 1.1 uW (-29.5 dBm). Scale linearly with (G-1) for other gains.
3.3 Cascaded Noise Figure (Friis Formula)
System noise figure when multiple amplifiers are in series. The first-stage NF dominates — always place the lowest-NF amplifier at the front.
NF_total = NF₁ + (NF₂ − 1)/G₁ + (NF₃ − 1)/(G₁ × G₂) + ...
All values in LINEAR (not dB). Convert: NF_lin = 10^(NF_dB/10), G_lin = 10^(G_dB/10).
Worked example: pre-amp (NF = 4.5 dB, G = 20 dB) followed by booster (NF = 6.0 dB, G = 15 dB).
NF₁ = 10^(4.5/10) = 2.82, G₁ = 10^(20/10) = 100
NF₂ = 10^(6.0/10) = 3.98
NF_total = 2.82 + (3.98 − 1)/100 = 2.82 + 0.030 = 2.85
NF_total (dB) = 10·log₁₀(2.85) = 4.55 dB
The 6 dB second stage only adds 0.05 dB to the system NF.
The FIRST amplifier dominates — put the best NF there.
Highlight: Warning A common design mistake is placing a high-gain booster before the low-NF pre-amplifier. The Friis formula shows why this destroys noise performance — the first stage’s NF appears without division, while subsequent stages are divided by all preceding gains.
3.4 Raman Effective Noise Figure
The combined NF of a hybrid Raman + EDFA amplification scheme. Raman pre-amplification in the transmission fibre improves the effective NF because the signal arrives at the EDFA already boosted. See Ch03 for Raman gain physics.
NF_effective (dB) ≈ NF_EDFA (dB) − G_Raman (dB)
Worked example: EDFA NF = 5.5 dB, Raman provides 12 dB of distributed gain.
NF_effective = 5.5 − 12 = −6.5 dB (effective)
Highlight: Note A negative NF is physically impossible for a standalone amplifier but valid for the combined system — the distributed Raman gain makes the signal arrive at the EDFA strong enough that the EDFA’s noise has less relative impact.
4. EDFA Power Budget
4.1 Per-Channel Power from Total Output
Determine individual channel power when the EDFA total output is known — essential for setting launch power in loaded systems.
P_ch (dBm) = P_total (dBm) − 10·log₁₀(N_channels)
P_total (dBm) = P_ch (dBm) + 10·log₁₀(N_channels)
Worked example 1: EDFA total output = +19 dBm, 80 channels loaded.
P_ch = 19 − 10·log₁₀(80) = 19 − 19.03 = −0.03 dBm ≈ 0 dBm per channel
Worked example 2: need +2 dBm per channel for 40 channels.
P_total = 2 + 10·log₁₀(40) = 2 + 16.02 = +18 dBm (63 mW total)
4.2 Number of Amplifier Sites
How many powered shelters are required along a route.
N_amps = ceil(Route_distance / Span_length) − 1
N_sites = N_amps + 2 (include both terminal sites)
Worked example: Riyadh to Dammam, 420 km, 80 km span design.
N_spans = ceil(420 / 80) = 6 spans
N_amp_sites = 6 − 1 = 5 inline amplifier sites
N_total_sites = 5 + 2 = 7 (including Riyadh and Dammam terminals)
5. Capacity and Spectral Efficiency
These formulas translate physical-layer parameters into the throughput numbers that network planners and capacity engineers care about. See Ch02 for modulation format theory and Ch04 for FEC overhead impact.
5.1 Shannon Capacity Limit
The theoretical ceiling — no real system exceeds this regardless of modulation or coding.
C = B × log₂(1 + SNR_linear)
SNR_linear = 10^(OSNR_dB / 10)
Worked example: 75 GHz channel with OSNR = 26 dB.
SNR_linear = 10^(26/10) = 398.1
C = 75 × 10⁹ × log₂(1 + 398.1)
= 75 × 10⁹ × 8.64
= 648 Gbps theoretical maximum
A real 400G channel in this slot uses 400/648 = 62% of Shannon.
A real 600G channel would use 600/648 = 93% of Shannon — close to the limit.
5.2 Channel Bit Rate
Calculate the net data rate from transponder parameters.
Bit_rate_raw = Baud_rate × bits_per_symbol × N_polarisations
Bit_rate_net = Bit_rate_raw / (1 + FEC_overhead)
Worked example 1: 64 GBd DP-16QAM transponder.
Raw = 64 × 4 (16QAM) × 2 (DP) = 512 Gbps
Net = 512 / 1.25 (25% SD-FEC) = 409.6 Gbps ≈ 400G
Worked example 2: 800G transponder at 96 GBd — what modulation?
800G net → 800 × 1.25 = 1000 Gbps raw
1000 / 96 / 2 = 5.2 bits per symbol per polarisation
→ need at least 64QAM (6 bits) but that's too sensitive to noise
→ industry uses DP-16QAM (4 bits) + probabilistic constellation shaping (PCS)
to reach ~5.2 effective bits without full 64QAM penalty
5.3 System Capacity
Total throughput of a DWDM system across the full spectrum.
Total_capacity = N_channels × rate_per_channel
N_channels = floor(Band_width / Channel_spacing)
Worked example: C-band system with 75 GHz spacing at 400G per channel.
C-band usable: ~4.8 THz (1530-1565 nm ≈ 4800 GHz)
N_channels = floor(4800 / 75) = 64 channels
(practical: 88 channels with flex-grid optimisation)
Total = 88 × 400G = 35.2 Tbps per fibre pair
C+L: ~176 channels × 400G = 70.4 Tbps per fibre pair
5.4 Spectral Efficiency
How efficiently the available optical spectrum is utilised.
SE (b/s/Hz) = Net_bit_rate / Channel_spacing
SE_system (b/s/Hz) = Total_capacity / Total_bandwidth
Worked example: 400G DP-16QAM in a 75 GHz slot.
SE = 400 Gbps / 75 GHz = 5.33 b/s/Hz
| Configuration | SE (b/s/Hz) | Typical Use |
|---|---|---|
| 100G DP-QPSK in 50 GHz | 2.0 | Long-haul, subsea |
| 200G DP-QPSK in 50 GHz | 4.0 | Regional backbone |
| 400G DP-16QAM in 75 GHz | 5.3 | Metro, DCI |
| 800G DP-16QAM in 100 GHz | 8.0 | Next-gen metro |
6. Reach and Dispersion
6.1 Maximum Spans for a Given Modulation
Rearranging the OSNR formula to solve for N gives the maximum number of amplified spans before OSNR drops below the modulation threshold.
N_max = 10^((58 + P_ch − NF − L_span − OSNR_required) / 10)
Worked example: 400G DP-16QAM reach. Given P_ch = 0 dBm, NF = 5.5 dB, L_span = 17 dB, OSNR_required = 22 dB, margin = 3 dB.
N_max = 10^((58 + 0 − 5.5 − 17 − 22 − 3) / 10)
= 10^(10.5 / 10)
= 10^1.05
= 11.2 → 11 spans maximum
At 80 km/span: max reach = 11 × 80 = 880 km for DP-16QAM.
Comparison — DP-QPSK vs DP-16QAM (OSNR_required = 12 dB):
N_max = 10^((58 + 0 − 5.5 − 17 − 12 − 3) / 10) = 10^(20.5/10) = 112 spans
Max reach = 112 × 80 = 8960 km → effectively unlimited for terrestrial.
| Modulation | OSNR Required (dB) | N_max (spans) | Reach at 80 km/span |
|---|---|---|---|
| DP-QPSK | 12 | 112 | 8,960 km (terrestrial-unlimited) |
| DP-16QAM | 22 | 11 | 880 km |
Highlight: Key Insight DP-QPSK’s 10 dB OSNR advantage over DP-16QAM translates to a 10x reach advantage. This is why subsea and ultra-long-haul systems still use lower-order modulation — the reach penalty of higher spectral efficiency is severe.
6.2 Dispersion-Limited Reach (Legacy 10G OOK)
Only relevant for legacy direct-detect systems without coherent DSP. Modern coherent transponders (100G+) compensate chromatic dispersion entirely in DSP — this formula applies only to 10G NRZ planning and access network design.
L_disp (km) = 10⁵ / (D × Δλ × B²)
| Parameter | Meaning | Typical Value |
|---|---|---|
| D | Dispersion coefficient | 17 ps/nm/km (G.652D at 1550 nm) |
| delta_lambda | Source spectral width | 0.1 nm (DFB laser) |
| B | Bit rate (Gbps) | 10 for legacy OOK |
Worked example: 10G NRZ OOK.
L_disp = 100,000 / (17 × 0.1 × 100) = 100,000 / 170 = 588 km
Highlight: Note Modern coherent systems make this formula obsolete for new deployments. The DSP in a coherent receiver can compensate tens of thousands of ps/nm of accumulated dispersion, making chromatic dispersion a non-issue for reach planning on any link that uses 100G+ coherent optics.
7. Infrastructure
7.1 Fibre Latency
One-way propagation delay through optical fibre. Light travels at approximately c/n = 200,000 km/s in glass (refractive index n approximately 1.5), yielding 5 microseconds per kilometre.
Latency = L × 5 µs/km (≈ speed of light in glass: c/n ≈ 200,000 km/s)
RTT = 2 × Latency + processing_delay
Worked example 1: Riyadh to Jeddah, 950 km fibre route.
One-way = 950 × 5 = 4750 µs = 4.75 ms
RTT = 2 × 4.75 + 0.02 (DSP) ≈ 9.52 ms
Worked example 2: data centre interconnect, 15 km.
One-way = 15 × 5 = 75 µs = 0.075 ms
RTT = 0.15 + 0.02 = 0.17 ms
8. Consolidated Formula Card
The complete formula set in a single reference block for quick lookup during design work.
┌─────────────────────────────────────────────────────────────────┐
│ CONVERSIONS │
│ dBm → mW: P = 10^(dBm/10) │
│ mW → dBm: P = 10·log₁₀(mW) │
│ λ → ν: ν(THz) = 299792.458 / λ(nm) │
│ │
│ SPAN DESIGN │
│ Span loss = α·L + N_s·0.05 + N_c·0.3 + margin │
│ P_received = P_launch − total_losses │
│ L_max = (P_launch − sensitivity − margin) / α │
│ │
│ OSNR & NOISE │
│ OSNR ≈ 58 + P_ch − NF − 10·log₁₀(N) − L_span │
│ P_ASE = 2 × NF_lin × h·ν·Δν × (G − 1) │
│ NF_cascade = NF₁ + (NF₂−1)/G₁ + ... │
│ NF_raman_eff = NF_EDFA − G_Raman │
│ │
│ CAPACITY │
│ Bit rate = Baud × bits/sym × 2(DP) / (1 + FEC) │
│ System = N_channels × rate_per_channel │
│ Shannon max = B × log₂(1 + SNR_linear) │
│ SE = bit_rate / channel_spacing │
│ │
│ INFRASTRUCTURE │
│ N_amps = ceil(distance / span_length) − 1 │
│ P_ch = P_total − 10·log₁₀(N_channels) │
│ Latency = distance_km × 5 µs/km │
│ N_max_spans = 10^((58+P−NF−L−OSNR_req−margin)/10) │
└─────────────────────────────────────────────────────────────────┘
See Also
- Optical Physics and Link Engineering — first-principles derivation of attenuation, dB math, and OSNR physics underlying these formulas
- Optical Amplifiers — EDFA, Raman, and Wideband C+L — amplifier noise theory behind the OSNR and ASE formulas in §3
- Fibre Nonlinearities and Power Optimisation — nonlinear impairment interactions that set the practical ceiling on per-channel launch power