Interferometry — Abridged Guide

Quick-reference guide to interferometric measurement. For full derivations and worked examples, see the Comprehensive Guide.

Comprehensive Interferometry Guide →

1.Introduction to Interferometry

Interferometry measures physical quantities by superposing two coherent beams and analyzing the resulting intensity pattern. The fringe pattern encodes optical path differences with sub-wavelength sensitivity — down to sub-nanometer precision with phase-shifting techniques.

For quick estimates, remember that one fringe = λ/2 of surface height change in reflection. At HeNe (632.8 nm), one fringe ≈ 316 nm.

2.Fundamentals of Interference

Two-beam interference

I = 2I_0(1 + \cos\delta), \quad \delta = \frac{2\pi}{\lambda}\,\text{OPD}

Constructive interference (bright fringes) occurs when OPD = mλ; destructive (dark fringes) when OPD = (m + ½)λ. Fringe visibility V = (I_max − I_min)/(I_max + I_min) must exceed ~0.2 for reliable detection.

The coherence length sets the maximum usable OPD. For a HeNe laser, l_c ≈ 20 cm. For white light, l_c ≈ 1 μm — which is exploited for absolute position sensing in WLI and OCT.

Coherence length

l_c = \frac{\lambda^2}{\Delta\lambda} \approx \frac{c}{\Delta\nu}

3.Interferometer Architectures

Six major architectures cover most applications. Michelson and Fizeau dominate surface testing; Mach–Zehnder for refractive index measurement; Fabry–Pérot for spectroscopy; Sagnac for rotation sensing.

Architecture	Passes	Best For
Michelson	Double	Displacement, FTIR, surface testing
Mach–Zehnder	Single	Refractive index, fiber sensors
Fabry–Pérot	Multiple	Spectroscopy, wavelength filtering
Fizeau	Common path	Surface figure, optical shop testing
Sagnac	Common path	Rotation sensing (FOGs)
Twyman–Green	Double	Optical component testing

Fizeau interferometers are preferred in optical shops because their common-path design inherently rejects vibration and source noise — both beams share the same optical path until the last surface.

Fabry–Pérot transmission

\frac{I_t}{I_i} = \frac{1}{1 + F\sin^2(\delta/2)}, \quad F = \frac{4R}{(1-R)^2}

Free spectral range

\text{FSR} = \frac{c}{2nd}

4.Fringe Analysis and Interpretation

Fringe spacing from tilt

\Lambda = \frac{\lambda}{\alpha}

Fringes are contour lines of OPD. Straight fringes indicate tilt; rings indicate curvature mismatch; deviations from the ideal pattern reveal surface figure error. One fringe of deviation = λ/2 surface error in reflection.

In a Fizeau test, count fringes across the aperture for total tilt. Then assess fringe straightness — deviations of 0.1 fringe correspond to ~32 nm of surface figure error at HeNe.

5.Phase Measurement Techniques

Hariharan 5-step PSI

\delta = \arctan\!\left(\frac{2(I_2 - I_4)}{2I_3 - I_5 - I_1}\right)

Phase-shifting interferometry captures multiple intensity frames at known phase offsets (typically 90° steps) to extract the phase at every pixel. This transforms fringe-counting (λ/20 resolution) into quantitative surface mapping (λ/1000 resolution).

Use the Hariharan 5-step algorithm as the default for PSI — it is insensitive to PZT step-size calibration errors to second order. For vibrating environments, switch to simultaneous PSI (pixelated-mask camera) or single-shot spatial-carrier methods.

Algorithm	Frames	Step Size	Calibration Robustness
3-step	3	90°	Low — sensitive to step error
4-step	4	90°	Medium — cancels first-order error
Hariharan 5-step	5	90°	High — insensitive to linear + quadratic error
Carré	4	Unknown (self-calibrating)	High — no step calibration needed

6.Coherence and Source Requirements

Source coherence length must exceed the maximum OPD in the interferometer. HeNe lasers (l_c > 20 cm) suit most equal-path instruments. Frequency-stabilized HeNe (l_c > 300 m) extends to long-path applications. Broadband sources (l_c ~ 1–30 μm) enable absolute distance sensing via coherence gating.

For Fizeau surface testing, a standard HeNe is almost always sufficient — the OPD is set by the cavity gap (typically < 100 mm). For scanning Michelson or Mach–Zehnder with large OPD, upgrade to a frequency-stabilized source.

Source	Coherence Length	Best For
HeNe (multimode)	~20 cm	Fizeau, short-path Michelson
HeNe (stabilized)	> 300 m	Long-path, large-aperture testing
Diode laser (external cavity)	1–10 m	Compact systems, wavelength tuning
SLD (broadband)	10–30 μm	OCT, white-light interferometry
Halogen lamp	~1 μm	VSI, thin-film metrology

7.Error Sources and Limitations

OPD error from air temperature

\Delta\text{OPD} = d \times \frac{dn}{dT} \times \Delta T

dn/dT ≈ −0.93 × 10⁻⁶ /°C for air at 632.8 nm.

Environmental errors (vibration, air turbulence, thermal drift) dominate most interferometric measurements. A 1°C temperature fluctuation over a 100 mm air path introduces ~93 nm of OPD error — nearly λ/7.

Three practical rules: (1) isolate vibration (pneumatic table), (2) enclose the beam path (reduce turbulence), (3) let the system thermally equilibrate before measuring. For λ/100 work, all three are mandatory.

8.Components and Hardware

Critical components include the beam splitter (50:50 for maximum visibility), reference optic (surface figure sets accuracy floor), PZT actuator (sub-nm resolution, < 2% hysteresis), and camera (≥ 10-bit, global shutter, ≥ 2 pixels per fringe).

When specifying a reference flat, remember that its figure error is your accuracy floor — not your repeatability floor. A λ/20 reference limits absolute accuracy to λ/20 even if your PSI repeatability is λ/1000. For better absolute accuracy, invest in a calibrated λ/50+ reference or perform three-flat calibration.

9.Metrology Applications

Interferometry spans a vast application range: surface figure testing (Fizeau + PSI), displacement measurement (heterodyne Michelson), thin-film thickness (spectrophotometric), rotation sensing (fiber Sagnac), tissue imaging (OCT), and gravitational wave detection (LIGO).

Application	Architecture	Typical Precision
Surface figure	Fizeau / Twyman–Green	λ/100 RMS
Displacement	Heterodyne Michelson	0.3–1 nm
Thin-film thickness	Spectrophotometric / WLI	±1–5 nm
Rotation rate	Fiber Sagnac	0.001°/hr
Tissue imaging	SD-OCT / SS-OCT	1–15 μm axial

If the test surface is rough or has step features, skip PSI and go directly to vertical scanning interferometry (WLI) — PSI requires smooth, continuous surfaces with < λ/2 slope between adjacent pixels.

10.Interferometer Selection and Specification

Start with four questions: (1) What are you measuring? (2) What precision do you need? (3) What is the sample geometry? (4) What are the environmental constraints? The answers map directly to architecture and component choices.

For general-purpose optical shop testing, a Fizeau interferometer with HeNe source, λ/20 reference flat, megapixel camera, and Hariharan PSI is the default starting point. Deviate from this only when the application demands it (vibration → simultaneous PSI; rough surfaces → WLI; large OPD → stabilized laser).

Continue Learning

Comprehensive Guide →Fringe Analysis Calculator →Coherence Length Calculator →

The Comprehensive Guide includes 6 worked examples, 6 SVG diagrams, and 10 references.