Mach-Zehnder Interferometer for 2-D

GRIN Profile MeasurementInstrument OverviewSample Preparation

Measurement Uncertainty

University of RochesterGradient-Index Research Group

2

2D GRIN Measurement: Mach-Zehnder Interferometer

• Wavelength range covered• 0.355 – 1.064μm, 1.55 – 4.6μm, 8 – 12μm

• Measurement accuracy• δ(Δn) = 1E-5

• Beam size: 65mm x 40mm ellipse• Sample prep requirements

• Thin (~1mm), plane, parallel sample with GRIN “exposed” (index is constant through sample thickness)

tyxnyx ),(2

),(

Measure: , tCalculate: Δn

Laser

Spatial Filter

Beam Splitter

Beam Splitter

Mirror

Mirror

Sample

Lens

Detector

GRIN BlankMeasurement

SliceMeasured

Interferogram

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Mach-Zehnder Sample Prep

GRIN BlankMeasurement

SliceMeasured

Interferogram nt

N

Fringes of #

Radi

alAx

ial

• Sample thickness chosen so that # of fringes can be resolved by imaging optics and detector

• Surface figure error and wedge lead to measurement error

• Thin slice cross-section that “exposes” the gradient for measurement

• Index should be constant through the thickness of the sample (z-axis)

xy

4

Uncertainty due to: Expression

- Figure error measured in air- Wedge error if no known homogeneous region

- Figure error measured in fluid

- Figure error measured in matched fluid(nfluid = nsamp at some location in sample)

- Wedge error if there is a known homogeneous region

Mach-Zehnder Measurement Uncertainty

t

mn

22

2

2

)(

t

mt

t

mm

tn

Best-case uncertainty is determined by a root-sum-square of the terms in the Δn calculation

m = # of fringesλ = wavelengtht = thickness

Uncertainty is increased further by surface figure error and wedge, but it can be minimized by measuring in an index matching fluid

)()( fluidsamp nnt

tn

)1()( sampnt

tn

Use the calculator on the next slide to determine the measurement error for your sample

nt

tn

)(

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Calculation of Measurement UncertaintyRSS uncertainty Sample Prep ErrorsBest-case measurement uncertaintym 35 nsamp 1.78δm 0.02 sample Δn 0.03t (μm) 1500 nfl uid 1.7δt 1 δt (μm) 1 = 1.58 waves (λ = 0.632 μm)λ (μm) 0.6328δλ 0.00005

Figure Error in Fluidδ(Δn) 1.302E-05 δ(Δn) 5.33E-05

Figure Error in Matched FluidUser Input (or wedge if known homog. Region)Calculated Value δ(Δn) 2.00E-05

Figure Error in Air(or wedge if no known homog. Region)δ(Δn) 5.20E-04

• Notes:• Interferometer accuracy is on the order of λ/50 (δm = 0.02)• Wavelength uncertainty is typically small and does not contribute much error• Best uncertainty is achieved by making the sample as thick as possible while

maintaining the ability to resolve fringes (see next slide)

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Calculation of Fringe Density

t

mn

m = # of fringesλ = wavelengtht = thickness

nt

N

Fringes of #

• The fringe density limits the thickness of the sample

• Thicker samples will eventually reach a fringe density that cannot be resolved

• Use calculator below to estimate fringe density in your samples

thickness (μm) 1500wavelength (μm) 0.6328sample Δn 0.03spatial extent ofgradient (mm) 1.2

index slope (mm-1) 0.02500# of fringes 71.1fringe density (mm-1) 59.3

User InputCalculated Value

• Notes:• For wavelengths in the visible

and NIR, the maximum fringe density that can be resolved by the interferometer is ≈100 mm-1

• Samples with large fringe density will require a smaller field of view

• Stitching may be required to measure the full sample