Understand the relationship
The reasoning behind the result
Numerical aperture represents an acceptance cone
NA = n sin(α); α = asin(NA/n)
n is the positive refractive index on the specimen side and α is the half-angle of the accepted cone, measured from the optical axis. The full cone angle is twice α. The geometric condition 0 < NA ≤ n follows because the sine of a half-angle up to 90° cannot exceed one.
A known NA is checked against the entered medium. Conversely, an angle determines NA before the diffraction model is evaluated. This relationship does not determine the objective's focal length, magnification, aberration correction or working distance.
Rayleigh separation and Abbe period are distinct conventions
rRayleigh = 0.61λ/NAobj; dAbbe = 0.5λ/NAobj
The objective-only Rayleigh scale uses the classical circular-aperture point-image separation convention. The Abbe expression describes a spatial-period limit under its conventional illumination assumptions. Their different coefficients encode different definitions; neither is a universal measurement of what a particular specimen will reveal.
For an appropriate widefield fluorescence application, the detected emission wavelength belongs to the objective-only model. A transmitted-light calculation instead uses the illumination arrangement and wavelength. The tool never infers either wavelength from a specimen label.
The transmitted-light model retains both apertures
r = 1.22λ/(NAobj + NAcond)
This conventional lateral Rayleigh model includes the objective and condenser numerical apertures explicitly. If they are equal, the denominator is twice the objective NA and the expression reduces to 0.61λ/NAobj. An entered smaller condenser aperture changes the result without changing the objective itself.
Condenser adjustment, coherence and specimen contrast affect real images. This formula is not a complete transfer-function calculation and is not extended here to confocal, super-resolution, electron microscopy, axial resolution or optical section thickness.
Inverse solving must retain physical constraints
NAreq = cλ/rtarget − b; (c,b) = (0.61,0), (0.5,0), or (1.22,NAcond)
Rearranging the selected equation gives the exact-target objective NA. A positive requirement above n has no acceptance half-angle in the entered medium. A zero or negative requirement in the transmitted model also has no positive exact solution under its fixed condenser term. These algebraic outcomes remain visible rather than being clipped into a plausible objective. Targets so close to the zero-aperture boundary that subtraction cannot resolve the required NA reliably produce an explicit numerical error.
The curve spans positive objective NAs up to the entered n. It holds wavelength, convention and any condenser term fixed. Logarithmic axes expose the relationship across a broad range; an unavailable exact-target NA is not plotted as an attainable objective point.
Pixel pitch is a separate sampling scale
q = r/p
p is a calibrated specimen-plane pixel pitch in the same length unit as r. The quotient q is the number of pixel intervals across the selected diffraction scale, or across the requested target in inverse mode.
This descriptive ratio is not a complete Nyquist guarantee: spatial-frequency support, pixel response, contrast transfer and the selected resolution convention matter. More pixels do not recover optical detail that was never transmitted. No image-quality threshold is applied.
Follow the numbers
A 550 nm wavelength and NA 0.65
- With n = 1 and NA = 0.65, the acceptance half-angle is asin(0.65) = 40.541602°.
- The objective-only Rayleigh scale is 0.61 × 550 / 0.65 = 516.153846 nm.
- A calibrated 100 nm/pixel pitch gives 516.153846 / 100 = 5.16153846 pixel intervals across that scale.
- The Abbe convention would instead give 550 / (2 × 0.65) = 423.076923 nm. The two values describe different conventions.
The model and pixel ratio are inspectable; actual resolving performance still requires measurement of the configured imaging system.
Quick guide
How to use this calculator
- Select the convention and imaging arrangement before entering a wavelength. The coefficients represent different models and should not be treated as interchangeable measurements.
- Enter objective-side refractive index plus either NA or the acceptance half-angle. For transmitted light, also enter the actual condenser NA and its medium index.
- Alternatively enter an exact target scale and inspect both the algebraic NA requirement and its compatibility with 0 < NA ≤ n.
- An optional calibrated pixel pitch gives samples per chosen model scale. It does not certify sampling adequacy, detection or measured resolution.
Calculation method
Calculation and interpretation
Connect wavelength, medium, acceptance angle and a declared diffraction model without mistaking it for measured resolving performance.
NA = n sin(α); rRayleigh = 0.61λ/NAobj; dAbbe = λ/(2NAobj); rtransmitted = 1.22λ/(NAobj + NAcond); samples per model scale = r/p.
Worked example
A 550 nm wavelength and NA 0.65
The model and pixel ratio are inspectable; actual resolving performance still requires measurement of the configured imaging system.
NA = n sin(α); rRayleigh = 0.61λ/NAobj; dAbbe = λ/(2NAobj); rtransmitted = 1.22λ/(NAobj + NAcond); samples per model scale = r/p.
Supported inputs
Precision and limits
Ideal lateral models only
No measured contrast, signal-to-noise, aberrations, coherence, pinhole, axial response, detector transfer function or specimen preparation is modeled. An algebraically attainable NA is not an equipment or performance approval.
Entered medium and numerical bounds
Positive dimension, scale and magnification inputs support 10⁻¹² through 10¹² in their selected units. Signed displacement components may be exactly zero; nonzero components must have an absolute magnitude from 10⁻¹² through 10¹². These are numerical bounds, not equipment specifications or accuracy claims. Refractive indices and wavelengths are entered, with no material lookup or fixed wavelength default.
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