How To Calculate Primary And Secondary Maxima Slit Intensity

Use this interactive single-slit diffraction calculator to estimate the central maximum intensity, the intensity of a selected secondary maximum, the corresponding angular position, and the screen position. The model uses the standard intensity law for a single slit and the exact secondary-maximum condition tan(beta) = beta.

Expert Guide: How to Calculate Primary and Secondary Maxima Slit Intensity

Calculating primary and secondary maxima slit intensity is a classic optics problem that appears in introductory physics, engineering optics, spectroscopy, laser alignment, and imaging science. The key idea is that a slit does not simply let light pass straight through. Instead, every point across the aperture contributes to a wavefront, and those wavelets interfere with one another. The result is a diffraction pattern made of one large bright central maximum and several weaker side maxima. If you want to know how bright those peaks are, you need the single-slit diffraction intensity formula and the correct condition for locating secondary maxima.

For a uniformly illuminated single slit of width a, illuminated by monochromatic light of wavelength lambda, the far-field or Fraunhofer diffraction intensity at angle theta is

I(theta) = I0 [sin(beta) / beta]^2, where beta = pi a sin(theta) / lambda.

Here, I0 is the central maximum intensity, meaning the peak intensity at theta = 0. This is the primary maximum. The side lobes, called secondary maxima, are much weaker and occur at specific nonzero values of beta. Unlike the dark fringes, which are easy to locate because they satisfy a sin(theta) = m lambda for integer m, the secondary maxima do not occur at such a simple integer condition. Instead, they are found by differentiating the intensity expression and solving the transcendental equation tan(beta) = beta.

100% Relative intensity of the primary central maximum, by definition I0/I0 = 1.

4.72% Relative intensity of the first secondary maximum compared with the central peak.

1.65% Relative intensity of the second secondary maximum compared with the central peak.

What is the primary maximum?

The primary maximum is the broad bright central band at the center of the diffraction pattern. It occurs at theta = 0, so beta also equals zero. If you substitute beta = 0 directly into the formula, you appear to get 0/0, but the correct limiting value is 1. Therefore, I(0) = I0. This central maximum is always the brightest feature of the pattern.

Because the central maximum is so much more intense than the side lobes, it is often used as the reference point. In lab work, you may measure detector counts at the center and normalize all other readings against it. In theoretical calculations, that means the relative central intensity is 1, while all secondary maxima are fractions less than 1.

What are secondary maxima?

Secondary maxima are the smaller bright fringes located on either side of the central maximum. They are not equally bright, and they decrease rapidly in intensity as you move farther from the center. The first secondary maximum is the brightest side lobe, the second is much weaker, and so on.

To find their locations exactly, you solve:

• tan(beta) = beta
• excluding beta = 0, which corresponds to the central maximum

The first few positive solutions are approximately:

• beta1 = 4.493
• beta2 = 7.725
• beta3 = 10.904
• beta4 = 14.066
• beta5 = 17.221

Once a beta value is known, the corresponding relative intensity is computed from [sin(beta)/beta]^2. Multiplying that ratio by I0 gives the absolute intensity of that secondary maximum.

Step by step method to calculate slit intensity

1. Identify the slit width a and wavelength lambda in consistent units.
2. Choose whether you want the central maximum or a secondary maximum.
3. For the central maximum, intensity is simply I0.
4. For a secondary maximum of order n, use the corresponding root beta_n from the equation tan(beta)=beta.
5. Calculate the relative intensity using (sin(beta_n)/beta_n)^2.
6. Find the absolute intensity from I_n = I0 (sin(beta_n)/beta_n)^2.
7. If you also need position, compute sin(theta_n) = beta_n lambda / (pi a).
8. Convert angle to screen position with y_n = L tan(theta_n).

Important distinction: the dark minima satisfy a sin(theta) = m lambda, but the bright side maxima do not. If you use the minima formula to locate maxima, your intensity calculation will be wrong.

Worked example

Suppose a single slit has width a = 20 micrometers and is illuminated by light of wavelength lambda = 550 nm. Let the central intensity be I0 = 1000 arbitrary units, and let the screen be L = 2 m away. We want the first secondary maximum.

1. Use the first exact root: beta1 = 4.493.
2. Compute relative intensity: (sin 4.493 / 4.493)^2 ≈ 0.0472.
3. Compute absolute intensity: I1 = 1000 × 0.0472 ≈ 47.2.
4. Find angle from sin(theta1) = 4.493 × 550e-9 / (pi × 20e-6).
5. This gives theta1 ≈ 2.25 degrees.
6. Screen position is y1 = 2 × tan(2.25 degrees) ≈ 0.0786 m, or about 7.86 cm.

This result immediately shows the most important physical trend: the first side lobe is visible, but it is much weaker than the central maximum. If your detector saturates at the center, you may barely detect the higher-order maxima unless your dynamic range is good.

Comparison table: exact side maxima data

Secondary maximum order n	Exact beta root	Relative intensity I_n / I0	Percent of central maximum	Interpretation
1	4.493	0.0472	4.72%	Brightest side lobe and usually the easiest secondary peak to observe
2	7.725	0.0165	1.65%	Already much weaker than the first side lobe
3	10.904	0.00834	0.834%	Often difficult to measure accurately without low-noise detection
4	14.066	0.00503	0.503%	Typically a faint outer lobe
5	17.221	0.00337	0.337%	Very weak and sensitive to alignment and background light

Why slit width matters

The slit width appears in beta through the ratio a/lambda. If the slit becomes narrower while wavelength stays fixed, beta changes more slowly with angle, so the diffraction pattern spreads out. That means the central maximum becomes wider and all side maxima move farther apart in angle. If the slit becomes wider, the pattern contracts and the fringes crowd closer to the axis.

However, the relative intensities of the maxima do not depend on slit width alone once you are evaluating at the exact maxima. The percentages in the table above come from the shape of the function (sin beta / beta)^2. What slit width changes is where those maxima occur in angle, not the normalized brightness ratios themselves.

Comparison table: exact maxima positions versus a common approximation

Order n	Exact beta from tan(beta)=beta	Approximation (n + 1/2)pi	Absolute error	Relative error
1	4.493	4.712	0.219	4.87%
2	7.725	7.854	0.129	1.67%
3	10.904	10.996	0.092	0.84%
4	14.066	14.137	0.071	0.50%

This comparison shows why students are sometimes taught that side maxima occur roughly halfway between minima. That approximation gets better for higher order maxima, but for precise calculations, especially for the first side lobe, the exact root should be used.

Common mistakes when calculating slit maxima

• Confusing minima and maxima: minima use integer multiples of lambda, but secondary maxima require solving tan(beta)=beta.
• Forgetting unit conversion: wavelength is often given in nanometers and slit width in micrometers. Convert both to meters before calculating.
• Using small-angle formulas too early: for modest angles, the approximation sin(theta) ≈ tan(theta) ≈ theta can help, but exact trigonometric evaluation is safer.
• Ignoring detector scaling: measured intensity may be in counts, volts, or arbitrary units. The formula still works as long as you use a consistent central intensity reference.
• Not handling beta = 0 correctly: the central maximum is not undefined. Its limit is exactly I0.

Physical interpretation of the intensity drop

The reason side maxima are weak is that the wave contributions from different parts of the slit increasingly cancel one another as angle grows. At the center, all contributions arrive almost perfectly in phase, so the field adds strongly and the intensity peaks. Away from the center, some portions of the slit contribute with phase delays that create partial cancellation. At the minima, cancellation is strong enough to reduce intensity to zero. Between minima, incomplete cancellation produces a smaller local maximum.

When this model is valid

This calculator assumes a single slit in the Fraunhofer diffraction regime with uniform illumination and monochromatic light. That means:

• The screen is far enough away, or a lens is used to observe the far-field pattern.
• The slit is narrow enough for diffraction to be prominent.
• The incoming light has a reasonably well-defined wavelength.
• The slit transmission is approximately uniform across its width.

If your setup involves a double slit, a diffraction grating, partial coherence, nonuniform beam profile, or a finite detector aperture, the pattern changes. In those cases, the single-slit envelope may still matter, but it will not be the whole story.

Authoritative references for deeper study

If you want to validate the underlying physics or go deeper into diffraction theory, these sources are reliable starting points:

• National Institute of Standards and Technology (NIST): Basic Optical Radiometry
• Georgia State University: Single-Slit Diffraction
• University-level optics reference: Single-Slit Diffraction

Bottom line

To calculate primary and secondary maxima slit intensity correctly, start from the single-slit intensity law I = I0 (sin beta / beta)^2. The primary maximum occurs at theta = 0 with intensity I0. The secondary maxima are found at the nonzero solutions of tan(beta)=beta, and their intensities are fixed fractions of the central maximum. The first side lobe is about 4.72% of the central peak, the second is about 1.65%, and each additional side lobe becomes progressively weaker. Once beta is known, you can also calculate the angular location and the screen position. That complete workflow is exactly what the calculator above automates.

Educational note: this page focuses on single-slit diffraction maxima. If you are analyzing multi-slit interference, the single-slit pattern typically acts as an envelope over the finer interference fringes.