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Theoretical Analysis of Temperature Drift of Standard Caliper

Theoretical Analysis of Temperature Drift of Standard Etches

 

Etches based on the F-P multibeam interference principle (including air-gap etalons, solid etalons, and second- or multi-stage solid etalons, etc.) mainly present the following two application challenges:

1. Dispersion of the transmission peak center wavelength caused by the inherent theoretical phase discrepancy of partial or total reflection films.

2. The transmission and reflection peaks shift with temperature, a phenomenon known as temperature drift.

This analysis primarily focuses on the temperature drift effect of the air-gap etalon, the most structurally complex of several commonly used etalons. The calculation and analysis software used is Mathcad.

Air-gap etalons mainly have two structures. The more complex structure is selected for analysis; theoretically, slight modifications can yield the parameters for structure 1.

3. All structures use fused silica as the substrate for both upper and lower reflective surfaces, and zero-expansion glass as the spacer. If other materials are used, the corresponding parameters can be modified in the following analysis.

         

The required parameters are as follows:

Air gap thickness (nm): dair = 1.5 * 10⁵

Refractive indices of each medium:

Incident medium refractive index: ni = 1.44412

Outgoing medium refractive index: nout = 1.44412

Air medium refractive index: nair = 1.00027325

Read in film structure parameters: X:=READEXCEL(".\data.xlsx", "F!A1:H4")

Y:=READEXCEL(".\data.xlsx", "B!A1:H4")

Read in analysis wavelength parameters: Z:=READEXCEL(".\data.xlsx", "Design!A1:A10001")

Front surface parameters: nF:=X(1) dF:=x(2) Define array size: c:=0,1..rows(nF)-1

Rear surface film parameters: nB=Y(1) dB:=Y(2) Define array size: s:=0,1..rows(nB)-1

                                      

Note: Considering the incident and exit directions of the light path, the front surface film system is directly imported according to the TFC simulated film system, and the rear surface film system requires the TFC film system structure to be inverted.

 

Front surface film system characteristic matrix:

 

Post-surface film system characteristic matrix:

The system matrix includes air gaps and entrance/exit medium structures.

Equivalent admittance:        Reflectivity:

 

 

Transmittance:

 

Based on the above calculations, a wavelength-dependent transmittance curve is plotted: λ = 1545, 1545.005, ..., 1555.

      

During theoretical calculations, it was discovered that the thin film characteristic matrix components, calculated using trigonometric functions, suffer from information loss at integer multiples greater than 2n angles. Therefore, accurate thickness values ​​cannot be directly obtained through inverse operations of the thin film's ABCD matrix. Consequently, data analysis was performed on the above calculation results to obtain the required thickness (optical path) components, thereby calculating the necessary temperature drift parameters.

Data Analysis:

Read the analysis wavelength data: L:=Z(0) Define the calculation array size: r:=0..rows(L)-1

(Default 1550±5nm, step size ±0.001nm)

Transmittance corresponding to each analysis wavelength: Pr=T(Lr)

Concatenate the analysis wavelength and corresponding transmittance matrix: A:=augment(L,P)

Obtain the peak wavelength corresponding to the local maximum function (the first and last two points (Peakλ)-1 are removed from the data set due to curve discontinuity):

Peakλ:=localmax(A) n:=rows

The average wavelength free spectral range FsRλ (nm) of the interference curve is obtained by dividing the difference between the wavelengths corresponding to the first and last peaks by the number of intervals.

                         

Note: This step of calculating the free spectral range is somewhat of a shortcut. A more accurate result can be obtained by averaging the wavelength distances between any two adjacent transmission peaks. Considering the anomaly-free characteristic of theoretical calculations compared to actual tests, and setting the analysis wavelength interval to 0.001 nm, the final calculation result can completely ignore the error introduced by this calculation.

Analysis of the average wavelength free harmonic range of the application case: FSRλ = 0.799818182

Reduced frequency free harmonic range FSRU (GHz):

FSRv = (299792458/1550) - [299792458/(1550+FSRλ)]

Analysis of the average frequency free harmonic range of the application case: FSRv = 99.7526716

The above steps complete the principle analysis of the standard etalon's basic structure. Based on this, the parameters of temperature influence are introduced, and the final temperature drift index is analyzed:

Temperature dispersion (°C): Δt: = 70

Coefficient of thermal expansion (CTE) of each medium:

Zero expansion glass CTE: CTEz: = 0.01*10-6

SiO2 film material CTE:CTESiO2：=0.55*10-6

Ta2O5 film material CTE:CTETa2O5：=-44.3*10-6

Temperature coefficient of refractive index of each medium DNT：

SiO2 film material DNT:DNTSiO2：=10*10-6

Ta2O5 film material DNT:DNTTa2O5=121*10-6

Air DNT:DNTair：=0.9454*10-6

Thickness change with temperature: dt=d0+d0*CTE*Δt

Refractive index change with temperature: nt=n0+DNT*Δt

The film thickness and refractive index value after temperature change are calculated and entered into the Excel spreadsheet of the film structure for calculation, and then read to obtain:

Front surface film system parameters: nFt：=X(5) dFt：=X(7) Define array size: c:=0, 1..rows(nF)-1

Back surface film system parameters: nBt:=Y(5) dBl:=Y(7) Define array size: s:=0, 1..rows(nB)-1

The air gap thickness is affected by the expansion of the spacer block and the physical expansion of the film layer, and the final value is:

Air gap etalons are typically used in sealed environments to avoid wavelength drift caused by changes in the refractive index of air. Using the refractive index material equation or the gas equation, we can conclude that if the atomic number density of a gas per unit volume remains constant, the refractive index also remains constant. That is, within a sealed cavity of fixed volume, the gas refractive index does not change with temperature (this is the theoretical basis for zero-temperature-drift sealed cavity etalons).

                                                             

Transmittance:

Based on the above calculations, a wavelength-dependent transmittance curve was plotted: λ = 1549.3, 1549.301, ..., 1549.8.

Transmittance corresponding to each analytical wavelength: Ptr := Tt(Lr)

Concatenate the analytical wavelength and its corresponding transmittance matrix: At := augment(L, Pt)

Obtain the wavelength corresponding to the peak using the local maximum function (the first and last points are removed from the data set due to curve discontinuity):

Peaktλ := localmax(At) n := rows(Peaktλ)-1

The average wavelength free spectral range (FSR) of the interference curve is obtained by dividing the difference between the wavelengths corresponding to the first and last peaks by the number of intervals.

                            

Note: This step of calculating the free spectral range is suspected of being a shortcut. A more accurate result can be obtained by averaging the wavelength distances of any adjacent transmission peaks. Considering the anomaly-free characteristic of theoretical calculations compared to actual tests, and setting the analysis wavelength interval to 0.001 nm, the final calculation result can completely ignore the error introduced by this calculation.

Analysis of the use case: Average wavelength free spectral range: FSRtA = 0.799818182

Average frequency free optical harmonic range FSRu (GHz):

FSRv = (299792458/1550) - [299792458/(1550+FSRλ)]

Analysis of the use case: Average frequency free spectral range: FSRLv = 99.7526716

By comparing the peak transmission wavelength before and after temperature change, the temperature drift of this etalon is: 0.004 nm/70℃.