Fresnel diffraction is a class of diffraction phenomena occurring when the source of light, the obstacle or aperture, and the screen are at finite distances from one another, such that the incident and diffracted wavefronts are spherical or cylindrical. It utilizes the concept of Fresnel half-period zones to analyze intensity variations.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Qualitative understanding of wavefront division, distinction between Fresnel and Fraunhofer diffraction, diffraction at a straight edge, circular aperture, and narrow wire using zone plates.
PGT extension
Analytical treatment of half-period zones, radius of Fresnel zones, intensity at a point due to a wavefront, zone plate construction, theory, and focal lengths.
Prerequisites
Huygens-Fresnel Principle
Superposition of waves and phase interference
Geometry of spherical and plane wavefronts
Definition and Wavefront Geometry
In Fresnel diffraction, either the source of light or the observation screen or both are at a finite distance from the aperture or obstacle.
The wavefronts arriving at the obstacle and traveling toward the screen are spherical or cylindrical rather than planar.
No lenses are required to focus the rays; observations are made in the near field relative to the aperture.
Fresnel Half-Period Zones (HPZs)
A wavefront can be divided into concentric annular regions called half-period zones such that the path difference from any two corresponding points in consecutive zones to the observation point is lambda/2.
The phase difference between disturbances arriving from consecutive zones at the observation point is pi radians (out of phase).
The resultant amplitude at the observation point is given by the alternating series $R = m1 - m2 + m3 - m4 + $... where amplitudes m decrease slowly as the zone number increases.
Zone Plate
A zone plate is a special screen designed to obstruct alternate Fresnel zones (e.g., even zones), acting somewhat like a convex lens with multiple foci.
The radius of the n-th Fresnel zone is proportional to the square root of n: r_n proportional to sqrt(n).
Unlike a glass lens, a zone plate has multiple focal lengths given by $f_n = r_1^2 / (n \lambda)$, producing numerous chromatic foci.
Comparison with Fraunhofer Diffraction
Fresnel class involves finite source-aperture and aperture-screen distances; Fraunhofer class involves infinite distances.
Fresnel diffraction patterns scale with distance and have curved wavefronts; Fraunhofer patterns are independent of distance in scale factor and use lenses to render wavefronts plane.
Mathematical treatment of Fresnel diffraction involves Fresnel integrals and Cornu's spiral.
Derivations
Radius of n-th Fresnel Half-Period Zone
Conditions: Point source S at distance a from wavefront, observation point P at distance b from wavefront.
Let S be a point source and P be the point of observation at a distance (a + b) from S.
Consider a spherical wavefront with center S and radius a passing through O. Let M_n be the outer boundary of the n-th half-period zone on the wavefront.
The distance from S to the n-th zone point M_n is S$M_n = a$, and the distance from M_n to P is $M_{n_P} = b + n\lambda/2$ by definition of half-period zones.
Using geometry in triangle SM_nP, if r_n is the radius of the n-th zone, expanding using binomial approximation yields r_n^2 \approx \frac{a b}{a + b} n \lambda.
Therefore, the radius of the n-th zone is $r_n = \sqrt{\frac{a b}{a + b} n \lambda}$.
Result: $r_n = \sqrt{\frac{a b}{a + b} n \lambda}$
Focal Length of a Zone Plate
Conditions: Normal incidence of monochromatic light of wavelength lambda on a zone plate of radius r_n for the n-th zone.
The phase condition for constructive interference at the focus F at distance f from the zone plate requires the path difference from the n-th zone to be n\lambda/2.
The distance from the n-th zone boundary to the focus F is \sqrt{f^2 + r_n^2}.
Equating the total path from the source to focus via the n-th zone: \sqrt{f^2 + r_n^2} - $f = n\lambda/2$.
Using the expression for the radius of the n-th zone where a approaches infinity (plane wavefront), $r_n^2 = n \lambda f_1$, substitute into the binomial expansion.
Solving for focal length f yields $f = \frac{r_n^2}{n \lambda} = f_1 / n$, where f_1 is the principal focal length.
Result: $f_n = \frac{r_1^2}{n \lambda}$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Radius of n-th Fresnel Zone
$$r_n = \sqrt{\frac{a b}{a + b} n \lambda}$$
a = distance of source to wavefront, b = distance of wavefront to screen, n = zone number, lambda = wavelength m
Valid for near-field spherical wavefronts where r_n << a, b
Area of n-th Fresnel Zone
$$A_n \approx \pi \frac{a b}{a + b} \lambda$$
$A_n = area of the n-th zone$, a, b, lambda as defined previously m^2
Approximate equality holds for small zones where n << a/lambda
Focal Length of Zone Plate
$$f_n = \frac{r_n^2}{n \lambda}$$
$r_n = radius of n-th zone$, n = zone index, lambda = wavelength m
Normal incidence of plane waves
Resultant Amplitude at Center of Zone Plate
$$R = m_1 - m_2 + m_3 - m_4 + \dots$$
$m_i = amplitude contribution from the i-th half-period zone$ Arbitrary amplitude units
Assuming uniform illumination and decreasing amplitude per zone
Intensity at Observation Point
$$I \propto R^2$$
I = intensity, R = resultant amplitude W/m^2
Standard wave intensity relation
Phase Difference between Consecutive Zones
$$\Delta \phi = \pi$$
\Delta \phi = phase difference radians
By definition of half-period zones
Path Difference for n-th Zone
$$\Delta_n = n \frac{\lambda}{2}$$
\Delt$a_n = path difference from n-th zone boundary$ m
Exact definition for zone construction
Wavelength relation using Zone Plate measurements
$$\lambda = \frac{r_n^2}{n f_n}$$
r_n, n, f_n as defined previously m
Experimental determination of wavelength
Graphs and experiments
Cornu's Spiral
Draw: A clothoid or Cornu's spiral plotting Fresnel integrals C(v) on the x-axis and S(v) on the y-axis, spiraling symmetrically toward asymptotic points (0.5, 0.5) and (-0.5, -0.5).
Interpret: Used to graphically compute amplitude and phase of diffracted light in Fresnel diffraction problems by measuring vector lengths on the spiral.
Exam use: Tested conceptually for finding intensity distributions near sharp edges and obstacles.
Intensity Distribution at a Straight Edge
Draw: A graph of intensity versus distance on the screen, showing damped oscillations inside the geometrical shadow and a series of maxima and minima outside the geometrical shadow.
Interpret: Demonstrates light bleeding into the shadow region and finite interference fringes near the edge boundary due to Fresnel diffraction.
Exam use: Distinguishing Fresnel diffraction at an edge from rectilinear propagation of light.
Zone Plate Focus Intensity Profile
Draw: Intensity peaks corresponding to primary, secondary, and higher-order foci along the optical axis behind a zone plate.
Interpret: Shows multiple focal points (f, f/3, f/5, etc.) because a zone plate acts simultaneously as a converging and diverging lens element.
Exam use: Comparison questions between glass lenses and zone plates.
Solved numericals
TGT core
Calculate the radius of the first half-period zone of a zone plate for light of wavelength 5000 Å, given that the source is at infinity and the screen is at a distance of 1 m.
Identify given values: lambd$a = 5000 Å = 5 * 10^-7 m$, $a = infinity (so a/(a+b) -> 1)$, b = 1 m, n = 1.
Use the formula for zone radius: $r_n = \sqrt{b n \lambda}$.
A zone plate has a diameter of 1.2 cm for the 10th zone. If light of wavelength 6000 Å falls normally on it, find the principal focal length of the zone plate.
Identify given values: Diameter 2$r_10 = 1.2 cm = 0.012 m$, so $r_10 = 0.6 cm = 0.006 m$.
Number of the zone n = 10, wavelength lambd$a = 6000 Å = 6 * 10^-7 m$.
Use the formula for focal length: $f_n = \frac{r_n^2}{n \lambda}$.
Substitute the values: $f_10 = \frac{(0.006)^2}{10 \times 6 \times 10^{-7}}$.
Find the radius of the third half-period zone of a wavefront of wavelength 589 nm, when the distance from the point source to the wavefront is 2 m and the distance from the wavefront to the observation point is 3 m.
Identify given values: a = 2 m, b = 3 m, n = 3, lambd$a = 589 nm = 589 * 10^-9 m$.
Use the zone radius formula: $r_n = \sqrt{\frac{a b}{a + b} n \lambda}$.
Calculate the distance factor: \frac{a b}{a + b} = \frac{2 \times 3}{2 + 3} = \frac{6}{5} = 1.2 m.
Substitute into the formula: $r_3 = \sqrt{1.2 \times 3 \times 589 \times 10^{-9}}$.
Assumes scalar wave approximation where polarization effects of light are ignored.
Assumes the obliquity factor (1 + cos theta) varies slowly and can be approximated across small zones.
Limited to near-field conditions where source-aperture and aperture-screen distances are finite.
Common misconceptions
Common error: Fresnel diffraction requires lenses to observe patterns.
Correct: Fresnel diffraction occurs at finite distances without lenses; lenses are used in Fraunhofer diffraction.
Lenses are employed to render wavefronts planar in Fraunhofer setups, whereas Fresnel deals directly with spherical/cylindrical wavefronts.
Common error: A zone plate has only one focal point like a convex lens.
Correct: A zone plate has multiple focal points corresponding to different odd integers (f, f/3, f/5, etc.).
Higher order diffraction orders constructively interfere at multiple axial positions due to zone boundary phase conditions.
Quick revision
Fresnel diffraction involves finite source-to-screen distances with spherical or cylindrical wavefronts.
Half-period zones divide the wavefront with consecutive path differences of lambda/2.
Radius of n-th Fresnel zone is proportional to the square root of n.
A zone plate acts as a converging lens with multiple foci and chromatic aberration.
Must remember
In Fresnel diffraction, either the source, the screen, or both are at a finite distance from the aperture.
The wavefronts arriving at the obstacle and traveling toward the screen are spherical or cylindrical.
No lenses are required to focus the rays in Fresnel diffraction; observations are made in the near field.
The path difference from consecutive half-period zones to the observation point is $\lambda/2$, giving a phase difference of $\pi$ radians.
The resultant amplitude at an observation point is given by the alternating series $R = m_1 - m_2 + m_3 - m_4 + \dots$.
The radius of the $n$-th Fresnel zone is proportional to the square root of $n$: $r_n \propto \sqrt{n}$.
A zone plate acts like a convex lens with multiple focal lengths given by $f_n = r_1^2 / (n \lambda)$.
Glossary
Fresnel DiffractionDiffraction phenomenon where the source or screen or both are at finite distances from the aperture, involving spherical or cylindrical wavefronts.
Half-Period Zone (HPZ)Concentric annular regions into which a wavefront is divided such that the path difference between consecutive zones at the observation point is $\lambda/2$.
Zone PlateA device constructed by alternately blocking even or odd Fresnel zones to act as a multi-focus lens.
Resultant Amplitude ($R$)The net optical disturbance at an observation point formed by summing the contributions from all half-period zones as $R = m_1 - m_2 + m_3 - \dots$.
Fresnel vs Fraunhofer Diffraction
Parameter
Fresnel Diffraction
Fraunhofer Diffraction
Distance of Source/Screen
Finite distance from aperture
Infinite (very large) distance
Nature of Wavefront
Spherical or cylindrical
Plane wavefront
Use of Lenses
Not required
Required to focus rays
Mathematical Treatment
Fresnel integrals, Cornu's spiral
Fourier transforms, simple geometry
Practice Quiz
Choose an option to reveal the explanation.
1. In Fresnel diffraction, what is the nature of the wavefronts arriving at the obstacle?
Explanation: In Fresnel diffraction, the source and/or screen are at finite distances, producing spherical or cylindrical wavefronts.
2. What is the path difference between disturbances arriving at the observation point from two consecutive half-period zones?
Explanation: By definition, the path difference from consecutive half-period zones to the observation point is $\lambda / 2$.
3. What is the phase difference between disturbances arriving from consecutive half-period zones?
Explanation: A path difference of $\lambda / 2$ corresponds to a phase difference of $\pi$ radians (out of phase).
4. The resultant amplitude $R$ at the observation point due to half-period zones is given by which series?
Explanation: Consecutive zones send disturbances that are out of phase by $\pi$, leading to the alternating series $R = m_1 - m_2 + m_3 - m_4 + \dots$.
5. How does the radius $r_n$ of the $n$-th Fresnel zone vary with the zone number $n$?
Explanation: The radius of the $n$-th Fresnel zone is proportional to the square root of $n$: $r_n \propto \sqrt{n}$.
6. Are lenses required to observe Fresnel diffraction patterns?
Explanation: Fresnel diffraction takes place at finite distances without the use of lenses, unlike Fraunhofer diffraction.
7. What is the formula for the focal length $f_n$ of a zone plate in terms of its $n$-th radius $r_n$ and wavelength $\lambda$?
Explanation: The focal length of a zone plate is given by $f_n = \frac{r_n^2}{n \lambda}$.
8. What is a zone plate designed to do?
Explanation: A zone plate obstructs alternate Fresnel zones (e.g., even zones) to enhance constructive interference, functioning similarly to a convex lens.
9. Unlike a standard glass convex lens, a zone plate possesses:
Explanation: A zone plate produces numerous foci corresponding to different odd integers, leading to multiple focal lengths.
10. What is the approximate area $A_n$ of the $n$-th Fresnel zone for small zone numbers?
Explanation: The area of the $n$-th zone is approximately constant and given by $A_n \approx \pi \frac{a b}{a + b} \lambda$.
11. In Fraunhofer diffraction, the source and screen are considered to be at:
Explanation: Fraunhofer class involves infinite distances, meaning incoming and outgoing wavefronts are planar.
12. Intensity $I$ at an observation point is related to the resultant amplitude $R$ by:
Explanation: Intensity is proportional to the square of the amplitude of the wave: $I \propto R^2$.
13. Which mathematical tools are typically used in the rigorous treatment of Fresnel diffraction?
Explanation: Fresnel diffraction treatment involves Fresnel integrals and Cornu's spiral to account for path variations.
14. If the zone number $n$ increases, how do the amplitude contributions $m_i$ from successive zones generally behave?
Explanation: The amplitudes $m_i$ decrease slowly as the zone number increases due to increasing obliquity and distance.
15. Which parameter can be determined experimentally if $r_n$, $n$, and $f_n$ of a zone plate are known?
Explanation: Using $\lambda = \frac{r_n^2}{n f_n}$, the wavelength of light can be measured.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The radius of the first Fresnel zone of a zone plate is $1\text{ mm}$ for a wavelength of $5000\text{ \AA}$. What is its primary focal length?
2. In a Fresnel diffraction setup, if the distance between the source and screen is doubled while keeping other parameters constant, how does the radius of the first half-period zone change?
Explanation: The radius formula contains the term $a b / (a + b)$. Scaling distances by a factor of 2 scales the radius by $\sqrt{2}$.
3. The phase difference between the contributions from the first and third half-period zones at the point of observation is:
Explanation: Consecutive zones differ by $\pi$. Therefore, the 1st and 3rd zones differ by $2\pi$, meaning they are in phase.
4. Why does a zone plate produce multiple foci instead of a single focal point like a glass lens?
Explanation: Higher-order diffraction zones constructively interfere at multiple axial points, giving secondary foci at $f/3, f/5$, etc.
5. A zone plate behaves optically most similarly to which of the following optical components?
Explanation: A zone plate focuses light by diffraction rather than refraction, but its focal properties resemble a convex lens.
6. If every even half-period zone of a zone plate is completely opaque, what happens to the intensity of light at the focal point compared to an unobstructed wavefront?
Explanation: Blocking even zones removes negative terms from the series, causing all remaining zones to add constructively and increasing intensity.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the essential condition that differentiates Fresnel diffraction from Fraunhofer diffraction regarding distance parameters?
2. The area of the $n$-th Fresnel zone is approximately independent of:
Explanation: The formula $A_n \approx \pi \frac{a b}{a + b} \lambda$ shows that zone area is approximately independent of $n$ for small values.
3. What is the path difference corresponding to the boundary of the $n$-th Fresnel zone?
Explanation: The path difference from the $n$-th zone boundary to the observation point compared to the wavefront pole is $n \lambda / 2$.
4. In the series $R = m_1 - m_2 + m_3 - m_4 + \dots$, why do successive terms have alternating signs?
Explanation: A path difference of $\lambda/2$ introduces a phase difference of $\pi$, reversing the sign of successive amplitude contributions.
5. If a zone plate has a focal length of $1\text{ m}$ for yellow light, what will be its focal length for light of a shorter wavelength?
Explanation: Since $f \propto 1/\lambda$, a shorter wavelength yields a larger focal length ($f = r_1^2 / \lambda$). Note: careful with $r_1$ definition; since $r_1 = \sqrt{ab\lambda/(a+b)}$, substituting shows $f$ depends on $\lambda$ inversely if $r_1$ is fixed, but if zones are drawn for a specific $\lambda_0$, $f$ varies accordingly.
6. Which of the following describes the wavefront geometry in Fresnel diffraction?
Explanation: Finite distances imply that wavefronts arriving at and leaving obstacles are curved (spherical or cylindrical).
7. What happens to the resultant amplitude $R$ if the entire wavefront is unobstructed?
Explanation: For an unobstructed wavefront, $R = \frac{m_1}{2}$ due to the rapid decrease in zone amplitudes.
8. The radius of the 4th Fresnel zone, compared to the 1st Fresnel zone, is larger by a factor of: