Calculator Inputs
Example Data Table
This table shows sample PSF decay values from the current calculation, or default values before calculation.
| Radius | Relative Intensity | PSF Value |
|---|---|---|
| 0 µm | 1 | 1 |
| 0.6 µm | 0.882497 | 0.882497 |
| 1.2 µm | 0.606531 | 0.606531 |
| 1.697 µm | 0.367879 | 0.367879 |
| 2.826 µm | 0.0625 | 0.0625 |
Formula Used
Elliptical Gaussian PSF:
PSF(x,y) = A × exp(-0.5 × (((x - x₀)² / σx²) + ((y - y₀)² / σy²)))
Normalized PSF:
Normalized PSF = PSF(x,y) / A
Full width at half maximum:
FWHM = 2 × sqrt(2 × ln(2)) × σ
Total Gaussian energy:
Energy = 2 × π × A × σx × σy
Airy reference estimates:
First zero radius = 0.61 × wavelength / NA
Airy disk diameter = 1.22 × wavelength / NA
How to Use This Calculator
- Enter the peak amplitude for the maximum center intensity.
- Enter sigma X and sigma Y in the same distance unit.
- Add the target point coordinates where the PSF value is needed.
- Set the PSF center coordinates. Use zero for a centered model.
- Choose grid radius and grid points for the heatmap resolution.
- Enter wavelength, numerical aperture, and pixel size for optical checks.
- Press the calculate button. Results will appear above the form.
- Use the CSV or PDF buttons to save the report.
Understanding Point Spread Functions
A point spread function describes how an ideal point appears after passing through an imaging system. A perfect point should stay infinitely small. Real lenses, sensors, motion, and sampling spread it into a pattern. This pattern tells you how sharp the system can be. It also explains blur, contrast loss, and fine detail limits.
Why the Gaussian Model Helps
Many practical studies use a Gaussian model because it is smooth, stable, and easy to compare. The width values, called sigma x and sigma y, control how fast intensity falls away from the center. A small sigma produces a tight spot. A large sigma creates a wider blur. Separate x and y values can describe astigmatism, tracking error, or uneven focus.
What the Results Mean
The calculated PSF value shows intensity at the chosen point. Peak intensity is the value at the center. FWHM converts sigma into a more readable spot width. Total energy estimates the area under the two dimensional curve. The normalized value removes amplitude, so different systems can be compared. The optical estimates use wavelength and numerical aperture to add a diffraction reference.
Using the Chart
The heatmap shows how intensity changes across the selected field. Bright central values mean strong focus. A broad heatmap means more spreading. The radial plot is useful when you want to inspect intensity decay from the center. Use both views before judging performance.
Good Input Practice
Keep all distance inputs in the same unit. Micrometers are common in microscopy and sensor work. Use measured sigma values when available. Use estimated values only for planning. Increase grid range when the spot is clipped at the edge. Increase grid points when the curve looks rough. Compare the FWHM with pixel size to see whether the sensor samples the spot well. Very small pixel coverage can hide detail. Very large coverage can waste resolution.
Practical Limits
This calculator is a mathematical estimator. It does not replace laboratory calibration. Real systems may include coma, spherical aberration, diffraction rings, noise, and detector response. Still, the Gaussian PSF gives a clear view. It helps compare settings, document assumptions, and prepare data for imaging models.
FAQs
1. What is a point spread function?
A point spread function shows how a perfect point source becomes blurred by an imaging system. It describes focus, spread, and resolution behavior.
2. Why does this calculator use a Gaussian model?
A Gaussian model is simple, smooth, and useful for many practical imaging estimates. It is also easy to compare across systems.
3. What does sigma mean?
Sigma controls the width of the spread. A smaller sigma gives a sharper spot. A larger sigma gives a wider blur pattern.
4. What is FWHM?
FWHM means full width at half maximum. It is the spot width where intensity falls to half of the peak value.
5. What is normalized PSF?
Normalized PSF divides the calculated value by peak amplitude. It helps compare spread shape without depending on absolute intensity.
6. What is the Airy disk estimate?
The Airy estimate gives a diffraction reference using wavelength and numerical aperture. It helps compare Gaussian spread with optical resolution limits.
7. Why is pixel size included?
Pixel size helps judge sampling quality. If the FWHM covers too few pixels, the sensor may not capture enough detail.
8. Can this replace measured calibration?
No. It is an estimator. Real imaging systems may include aberrations, noise, diffraction rings, and detector effects that need measurement.