10 Interference and interferometers

Run this chapter’s code in Google Colab (no installation).

Slides 172–175 (two-beam interference) and 259–266 (Fabry–Perot interferometer at oblique incidence, Mach–Zehnder interferometer).

10.1 Key relations

Two coherent waves $E_1=E_{01}\sin(\omega t-kr_1-\phi_1)$ and $E_2=E_{02}\sin(\omega t-kr_2-\phi_2)$ add up to an intensity (slides 172–173)

\[ I=I_1+I_2+2\sqrt{I_1I_2}\cos\delta,\qquad \delta=k(r_2-r_1)+(\phi_2-\phi_1). \]

Constructive interference: $I_\text{max}=I_1+I_2+2\sqrt{I_1I_2}$, which is $4I_1$ for equal beams.
Destructive interference: $I_\text{min}=I_1+I_2-2\sqrt{I_1I_2}$, which is $0$ for equal beams.
Incoherent waves simply add their intensities, $I=I_1+I_2$ (slide 174).
Fringe visibility $V=(I_\text{max}-I_\text{min})/(I_\text{max}+I_\text{min})=2\sqrt{I_1I_2}/(I_1+I_2)$.

10.2 Mach–Zehnder interferometer, live (slides 264–266)

Light is split, travels two arms with a phase difference $\Phi$, and is recombined. With equal splitting, the average output intensity is $\propto\cos^2(\Phi/2)$ (slide 265). The response to a small phase change is steepest at quadrature, $\Phi=90°$ (slide 266), which is where a modulator is biased.

Show the Python

viewof mz = Inputs.form({
  Phi: Inputs.range([0, 720], {value: 90, step: 1, label: "phase difference Φ (deg)"}),
  k:   Inputs.range([0.05, 0.95], {value: 0.5, step: 0.01, label: "power splitting ratio of the couplers"})
})

Show the Python

mzOut = {
  // two identical couplers with power ratio k; field in bar/cross: √(1−k), j√k
  const a = Math.sqrt(1 - mz.k), b = Math.sqrt(mz.k);
  const f = P => { const p = P*Math.PI/180;
    // E_bar = a·a·e^{-jΦ} − b·b  ;  E_cross = j a b (e^{-jΦ} + 1)
    const br = a*a*Math.cos(p) - b*b, bi = -a*a*Math.sin(p);
    const cr = a*b*Math.sin(p), ci = a*b*(Math.cos(p) + 1);
    return {bar: br*br + bi*bi, cross: cr*cr + ci*ci}; };
  const curve = d3.range(0, 720.1, 1).flatMap(P => { const v = f(P); return [{P, I: v.bar, port: "bar"}, {P, I: v.cross, port: "cross"}]; });
  const now = f(mz.Phi), vals = curve.filter(d => d.port === "bar").map(d => d.I);
  return {curve, now, ER: 10*Math.log10(d3.max(vals) / Math.max(d3.min(vals), 1e-12))};
}
Plot.plot({
  width: Math.min(width, 1050), height: 300, x: {label: "Φ (deg)", ticks: d3.range(0, 721, 90)}, y: {label: "output power / input", domain: [0, 1.02], grid: true},
  color: {legend: true, domain: ["bar", "cross"], range: ["#2a78d6", "#eb6834"]},
  marks: [Plot.line(mzOut.curve, {x: "P", y: "I", stroke: "port", strokeWidth: 2}),
          Plot.ruleX([mz.Phi], {stroke: "#b9b8b2"}),
          Plot.dot([{P: mz.Phi, I: mzOut.now.bar, port: "bar"}, {P: mz.Phi, I: mzOut.now.cross, port: "cross"}], {x: "P", y: "I", fill: "port", r: 6, stroke: "white"})]
})

Show the Python

html`<div class="result-box">bar = ${(100*mzOut.now.bar).toFixed(1)} %, cross = ${(100*mzOut.now.cross).toFixed(1)} % ·
extinction ratio of the bar port = <b>${mzOut.ER > 60 ? "> 60" : mzOut.ER.toFixed(1)} dB</b>
${Math.abs(mz.k - 0.5) > 0.01 ? "(couplers not 50/50: the null is no longer perfect)" : "(perfect 50/50 couplers: perfect null)"}</div>`

10.3 Worked example: the Mach–Zehnder as modulator and as wavelength filter

Two uses in photonic integrated circuits:

Modulator or switch: $\Phi$ is set by a phase shifter (thermo-optic as in Chapter 1, or electro-optic, Chapter 6). The steepest slope is at quadrature.
Wavelength filter or (de)multiplexer: an unbalanced MZI with arm-length difference $\Delta L$ has $\Phi=2\pi n_\text{eff}\Delta L/\lambda$, hence a free spectral range $\mathrm{FSR}=\lambda^2/(N_g\Delta L)$.

Show the Python

Phi = np.linspace(0, 2*np.pi, 400)
fig, ax = plt.subplots(1, 2, figsize=(10, 3.8))
ax[0].plot(Phi/np.pi, np.cos(Phi/2)**2); ax[0].plot(0.5, 0.5, "o", color=ol.PALETTE[7], ms=8)
ax[0].set(xlabel="Φ / π", ylabel="I/I₀", title="MZI transfer")
Ng, n_eff, lam0 = 4.2, 2.45, nm(1550)        # typical Si strip waveguide at 1550 nm (assumed values)
dL = c0/(Ng*100e9)
lam = np.linspace(nm(1545), nm(1555), 2000)
phi = 2*np.pi*(n_eff - (Ng - n_eff)*(lam - lam0)/lam0)*dL/lam   # first-order dispersion of n_eff
ax[1].plot(lam*1e9, np.cos(phi/2)**2, label="bar"); ax[1].plot(lam*1e9, np.sin(phi/2)**2, label="cross")
ax[1].set(xlabel="λ (nm)", title=f"unbalanced MZI, ΔL = {dL*1e6:.0f} µm"); ax[1].legend(fontsize=8)
plt.tight_layout(); plt.show()
print(f"ΔL = {dL*1e6:.1f} µm gives FSR = 100 GHz = {lam0**2/(Ng*dL)*1e9:.2f} nm")
print(f"modulator at quadrature: dI/dΦ = I0/2 per radian (slide 266)")

Figure 10.1: Left: MZI transfer function (red dot: quadrature). Right: unbalanced silicon MZI with a 100 GHz free spectral range used as a wavelength interleaver.

ΔL = 713.8 µm gives FSR = 100 GHz = 0.80 nm
modulator at quadrature: dI/dΦ = I0/2 per radian (slide 266)

10.4 Fabry–Perot interferometer at oblique incidence (slides 260–263)

For a plate (etalon) of index $n$ and thickness $L$, a ray at angle $\theta$ inside the plate interferes constructively with its multiply-reflected copies when (slides 261–263)

\[ 2nL\cos\theta=m\lambda . \]

An extended source seen through an etalon therefore shows bright rings, one per order $m$.

Show the Python

lam, L, R, f = nm(633), 5e-3, 0.9, 0.2
x = np.linspace(-8e-3, 8e-3, 700)
X, Y = np.meshgrid(x, x)
theta = np.arctan(np.hypot(X, Y)/f)
F = 4*R/(1 - R)**2
I = 1/(1 + F*np.sin(2*np.pi*L*np.cos(theta)/lam)**2)
fig, ax = plt.subplots(figsize=(4.6, 4.6))
ax.imshow(I, extent=[x[0]*1e3, x[-1]*1e3]*2, cmap="magma"); ax.grid(False)
ax.set(xlabel="x (mm)", ylabel="y (mm)"); plt.show()
m0 = 2*L/lam
print(f"order at the centre m = 2L/λ = {m0:.1f};  first bright ring (m = {np.floor(m0):.0f}) at θ = arccos(m/m0) = "
      f"{np.degrees(np.arccos(np.floor(m0)/m0)):.3f}°")

Figure 10.2: Fabry–Perot rings seen through a 5 mm air-spaced etalon (R = 0.9) at 633 nm, observed in the focal plane of a lens (f = 200 mm).

order at the centre m = 2L/λ = 15797.8;  first bright ring (m = 15797) at θ = arccos(m/m0) = 0.572°

Going deeper

Why does the FSR of an unbalanced MZI depend on $N_g$ and not on $n_\text{eff}$? Derive it from $\Phi(\lambda)$ with $n_\text{eff}(\lambda)$.
Cascading unbalanced MZIs (lattice filters) and the arrayed-waveguide grating (AWG), a “many-arm” interferometer that behaves like a diffraction grating (Chapter 13).
Extinction ratio limited by non-ideal couplers: derive the extinction ratio as a function of the splitting ratio, and compare with the live panel.
Interferometric sensors: the fiber gyroscope (Sagnac), gravitational-wave detectors (Michelson with Fabry–Perot arms). Estimate the phase sensitivity limited by shot noise.

Complements

P. Hariharan, Basics of Interferometry: compact and practical.
L. Chrostowski, M. Hochberg, Silicon Photonics Design, Cambridge: MZI, ring and AWG design in silicon photonics.

10.5 Try it yourself

Two beams with $I_2=I_1/10$ interfere. What is the fringe visibility? Is it worth equalising them in an experiment?
A thermo-optic MZI switch needs $\Phi=\pi$. Using the numbers of Chapter 1, how much heater power is roughly needed if the heater raises the waveguide temperature by 10 K per mW?
Tilt the etalon of the ring figure by 1°. What happens to the rings?

# Interference and interferometers {#sec-interference} [![Open in Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/{{< var github_user >}}/{{< var repo >}}/blob/main/{{< var subdir >}}/notebooks/interference.ipynb){.colab-link}  Run this chapter's code in Google Colab (no installation). ```{python} #| include: false import numpy as np import matplotlib.pyplot as plt import optolib as ol ol.use_style() c0, nm, um = ol.c0, ol.nm, ol.um ``` *Slides 172–175 (two-beam interference) and 259–266 (Fabry–Perot interferometer at oblique incidence, Mach–Zehnder interferometer).* ## Key relations Two coherent waves $E_1=E_{01}\sin(\omega t-kr_1-\phi_1)$ and $E_2=E_{02}\sin(\omega t-kr_2-\phi_2)$ add up to an intensity (slides 172–173) $$ I=I_1+I_2+2\sqrt{I_1I_2}\cos\delta,\qquad \delta=k(r_2-r_1)+(\phi_2-\phi_1). $$ - Constructive interference: $I_\text{max}=I_1+I_2+2\sqrt{I_1I_2}$, which is $4I_1$ for equal beams. - Destructive interference: $I_\text{min}=I_1+I_2-2\sqrt{I_1I_2}$, which is $0$ for equal beams. - **Incoherent** waves simply add their intensities, $I=I_1+I_2$ (slide 174). - Fringe visibility $V=(I_\text{max}-I_\text{min})/(I_\text{max}+I_\text{min})=2\sqrt{I_1I_2}/(I_1+I_2)$. ## Mach–Zehnder interferometer, live (slides 264–266) Light is split, travels two arms with a phase difference $\Phi$, and is recombined. With equal splitting, the average output intensity is $\propto\cos^2(\Phi/2)$ (slide 265). The response to a small phase change is steepest at **quadrature**, $\Phi=90°$ (slide 266), which is where a modulator is biased. ```{ojs} //| echo: false viewof mz = Inputs.form({ Phi: Inputs.range([0, 720], {value: 90, step: 1, label: "phase difference Φ (deg)"}), k: Inputs.range([0.05, 0.95], {value: 0.5, step: 0.01, label: "power splitting ratio of the couplers"}) }) ``` ```{ojs} //| echo: false mzOut = { // two identical couplers with power ratio k; field in bar/cross: √(1−k), j√k const a = Math.sqrt(1 - mz.k), b = Math.sqrt(mz.k); const f = P => { const p = P*Math.PI/180; // E_bar = a·a·e^{-jΦ} − b·b ; E_cross = j a b (e^{-jΦ} + 1) const br = a*a*Math.cos(p) - b*b, bi = -a*a*Math.sin(p); const cr = a*b*Math.sin(p), ci = a*b*(Math.cos(p) + 1); return {bar: br*br + bi*bi, cross: cr*cr + ci*ci}; }; const curve = d3.range(0, 720.1, 1).flatMap(P => { const v = f(P); return [{P, I: v.bar, port: "bar"}, {P, I: v.cross, port: "cross"}]; }); const now = f(mz.Phi), vals = curve.filter(d => d.port === "bar").map(d => d.I); return {curve, now, ER: 10*Math.log10(d3.max(vals) / Math.max(d3.min(vals), 1e-12))}; } Plot.plot({ width: Math.min(width, 1050), height: 300, x: {label: "Φ (deg)", ticks: d3.range(0, 721, 90)}, y: {label: "output power / input", domain: [0, 1.02], grid: true}, color: {legend: true, domain: ["bar", "cross"], range: ["#2a78d6", "#eb6834"]}, marks: [Plot.line(mzOut.curve, {x: "P", y: "I", stroke: "port", strokeWidth: 2}), Plot.ruleX([mz.Phi], {stroke: "#b9b8b2"}), Plot.dot([{P: mz.Phi, I: mzOut.now.bar, port: "bar"}, {P: mz.Phi, I: mzOut.now.cross, port: "cross"}], {x: "P", y: "I", fill: "port", r: 6, stroke: "white"})] }) ``` ```{ojs} //| echo: false html`<div class="result-box">bar = ${(100*mzOut.now.bar).toFixed(1)} %, cross = ${(100*mzOut.now.cross).toFixed(1)} % · extinction ratio of the bar port = <b>${mzOut.ER > 60 ? "> 60" : mzOut.ER.toFixed(1)} dB</b> ${Math.abs(mz.k - 0.5) > 0.01 ? "(couplers not 50/50: the null is no longer perfect)" : "(perfect 50/50 couplers: perfect null)"}</div>` ``` ## Worked example: the Mach–Zehnder as modulator and as wavelength filter Two uses in photonic integrated circuits: 1. **Modulator or switch**: $\Phi$ is set by a phase shifter (thermo-optic as in @sec-em, or electro-optic, Chapter 6). The steepest slope is at quadrature. 2. **Wavelength filter or (de)multiplexer**: an unbalanced MZI with arm-length difference $\Delta L$ has $\Phi=2\pi n_\text{eff}\Delta L/\lambda$, hence a free spectral range $\mathrm{FSR}=\lambda^2/(N_g\Delta L)$. ```{python} #| label: fig-mzi #| fig-cap: "Left: MZI transfer function (red dot: quadrature). Right: unbalanced silicon MZI with a 100 GHz free spectral range used as a wavelength interleaver." Phi = np.linspace(0, 2*np.pi, 400) fig, ax = plt.subplots(1, 2, figsize=(10, 3.8)) ax[0].plot(Phi/np.pi, np.cos(Phi/2)**2); ax[0].plot(0.5, 0.5, "o", color=ol.PALETTE[7], ms=8) ax[0].set(xlabel="Φ / π", ylabel="I/I₀", title="MZI transfer") Ng, n_eff, lam0 = 4.2, 2.45, nm(1550) # typical Si strip waveguide at 1550 nm (assumed values) dL = c0/(Ng*100e9) lam = np.linspace(nm(1545), nm(1555), 2000) phi = 2*np.pi*(n_eff - (Ng - n_eff)*(lam - lam0)/lam0)*dL/lam # first-order dispersion of n_eff ax[1].plot(lam*1e9, np.cos(phi/2)**2, label="bar"); ax[1].plot(lam*1e9, np.sin(phi/2)**2, label="cross") ax[1].set(xlabel="λ (nm)", title=f"unbalanced MZI, ΔL = {dL*1e6:.0f} µm"); ax[1].legend(fontsize=8) plt.tight_layout(); plt.show() print(f"ΔL = {dL*1e6:.1f} µm gives FSR = 100 GHz = {lam0**2/(Ng*dL)*1e9:.2f} nm") print(f"modulator at quadrature: dI/dΦ = I0/2 per radian (slide 266)") ``` ## Fabry–Perot interferometer at oblique incidence (slides 260–263) For a plate (etalon) of index $n$ and thickness $L$, a ray at angle $\theta$ inside the plate interferes constructively with its multiply-reflected copies when (slides 261–263) $$ 2nL\cos\theta=m\lambda . $$ An extended source seen through an etalon therefore shows **bright rings**, one per order $m$. ```{python} #| label: fig-rings #| fig-cap: "Fabry–Perot rings seen through a 5 mm air-spaced etalon (R = 0.9) at 633 nm, observed in the focal plane of a lens (f = 200 mm)." lam, L, R, f = nm(633), 5e-3, 0.9, 0.2 x = np.linspace(-8e-3, 8e-3, 700) X, Y = np.meshgrid(x, x) theta = np.arctan(np.hypot(X, Y)/f) F = 4*R/(1 - R)**2 I = 1/(1 + F*np.sin(2*np.pi*L*np.cos(theta)/lam)**2) fig, ax = plt.subplots(figsize=(4.6, 4.6)) ax.imshow(I, extent=[x[0]*1e3, x[-1]*1e3]*2, cmap="magma"); ax.grid(False) ax.set(xlabel="x (mm)", ylabel="y (mm)"); plt.show() m0 = 2*L/lam print(f"order at the centre m = 2L/λ = {m0:.1f}; first bright ring (m = {np.floor(m0):.0f}) at θ = arccos(m/m0) = " f"{np.degrees(np.arccos(np.floor(m0)/m0)):.3f}°") ``` ::: {.callout-tip collapse="true"} ## Going deeper - Why does the FSR of an unbalanced MZI depend on $N_g$ and not on $n_\text{eff}$? Derive it from $\Phi(\lambda)$ with $n_\text{eff}(\lambda)$. - Cascading unbalanced MZIs (lattice filters) and the arrayed-waveguide grating (AWG), a "many-arm" interferometer that behaves like a diffraction grating (@sec-diffraction). - Extinction ratio limited by non-ideal couplers: derive the extinction ratio as a function of the splitting ratio, and compare with the live panel. - Interferometric sensors: the fiber gyroscope (Sagnac), gravitational-wave detectors (Michelson with Fabry–Perot arms). Estimate the phase sensitivity limited by shot noise. ::: ::: {.callout-note collapse="true"} ## Complements - P. Hariharan, *Basics of Interferometry*: compact and practical. - L. Chrostowski, M. Hochberg, *Silicon Photonics Design*, Cambridge: MZI, ring and AWG design in silicon photonics. ::: ## Try it yourself 1. Two beams with $I_2=I_1/10$ interfere. What is the fringe visibility? Is it worth equalising them in an experiment? 2. A thermo-optic MZI switch needs $\Phi=\pi$. Using the numbers of @sec-em, how much heater power is roughly needed if the heater raises the waveguide temperature by 10 K per mW? 3. Tilt the etalon of the ring figure by 1°. What happens to the rings?