====== Interference and Diffraction (Remote) ====== In this lab we will look at two closely-related phenomena: **wave interference** and **diffraction**. These topics have strong connections to the previous experiment on //standing waves// (which are the result of //interference// of forward and backward moving waves). Unlike previous labs this year where you worked over two sessions towards one large goal, this lab is made up of several smaller investigations on different aspects of interference (in Part 1) and diffraction (in Part 2). ====== Part 1: Interference ====== ---- ===== Lab report template ===== For each project, we will provide a lab template for you to use as your lab notebook. This file includes prompts that you will expand on as you go through the lab. The link below will ask you to sign-in to your UChicago Google Drive and will create a copy of the file for you to edit.
[[https://docs.google.com/document/d/1aJbDXcJ96h2pQCeveTQoIp2uskqweDtEq3HKOxp0f78/copy| Lab Notebook Template]]
===== Wave interference PhET simulation ===== ==== Getting oriented ==== To start, we will use the **Wave Interference** PhET simulation: https://phet.colorado.edu/sims/html/wave-interference/latest/wave-interference_en.html When you open the lab, you have the option to choose a module. * The **Waves** module is a very basic simulation that creates waves from a single source. * The **Interference** module looks at waves created by one or two coherent sources. * The **Slits** module looks at waves created by a single plane-wave source as it encounters a barrier with one or two slits. (//We will not be using this module in this week's lab//.) * The **Diffraction** module looks at light passing through differently-shaped apertures. (//We will not be using this module in this week's lab//.) After selecting a module, the 4 options will now appear along the bottom of the screen so you can flip from one to another. (The previous module will be left undisturbed when you switch from one to the other.) There is also a **home** icon which returns you to the original selection screen. Once again, the orange circle arrow in the lower right corner can be used to return the simulation to its original starting state. ==== Theory of interference ==== We don't have to go over the mathematics of interference before we try to observe it, but we'll include a little bit here if you want to peek. Maybe try the simulation below and come back here if you need help understanding what's going on?
Superposition Interference is simply a consequence of the principle of //superposition//. This means that at any point in space, the total amplitude of the disturbance due to two separate traveling waves is the sum, or superposition, of the amplitudes of the two separate waves. Mathematically, we can write two sinusoidal traveling waves of equal amplitude, $y_1$ and $y_2$, at point $x = 0$ as | $y_1 = A\cos (\omega t + \varphi_1)$\\ $y_2 = A\cos (\omega t + \varphi_2)$, | (1) | where $\varphi_i$  may be a function of position, but not of time. The superposition of these waves yields the total wave $y_T$, | $y_T = y_1 + y_2 = 2A\cos\left(\dfrac{\varphi_2-\varphi_1}{2}\right)\cos \left(\omega t + \dfrac{\varphi_2 + \varphi_1}{2}\right)$. | (2) | This gives a total intensity of | $I_t = \left< y_T^2\right> = 2A^2\cos^2\left(\dfrac{\varphi_2-\varphi_1}{2}\right) = 2A^2\cos^2\left(\dfrac{\Delta\varphi}{2}\right)$. | (3) | Note that the total intensity depends critically on the phase difference $\Delta\varphi$ between the two waves. If $\Delta\varphi$ is zero or an integral multiple of $2\pi$, then the waves are in phase and $I_T$ is twice as large as the sum $I_1+I_2$ (constructive interference). If $\Delta\varphi =(2n+1)\pi$, where $n = 0, 1, 2, \dots$, then the waves are 180 degrees out of phase and $I_T=0$ (destructive interference). A sinusoidal traveling wave, propagating in the $x$-direction and produced at $x_0$ with phase $\varphi_0$, has the form | $Y = A \cos\left(\omega t - k(x-x_0)+\varphi_0 \right) = A \cos \left(2\pi\left(ft - \dfrac{x-x_0}{\lambda}\right) + \varphi_0\right)$, | (4) | where | $\omega = 2\pi f$\\ $k = \dfrac{2\pi}{\lambda}$\\ $v = f\lambda = \dfrac{\omega}{k}$ | (5) | so that the phase at any point $x$ is $\varphi = \varphi_0 - k(x-x_0)$ If two such waves interfere, $\Delta\varphi$ at the point of interference will depend on the distance traveled by each wave. For example, suppose two such waves with identical $\omega$ and $\varphi_0$ are produced on the $x$-axis at $x_1$ and $x_2$. Then, at a distant point $x$, the phase difference will be | $\Delta \varphi = k(x_2-x_1) = 2\pi\left(\dfrac{x_2-x_1}{\lambda}\right)$ | (6) | and the intensity of the total wave at $x$ will be | $I_T = 2A^2\cos^2\left(\pi\left(\dfrac{x_2-x_1}{\lambda}\right)\right)$. | (7) | This analysis applies equally to transverse waves (microwaves and light) and to longitudinal waves (sound). Note that the two waves must have identical frequency and a fixed phase difference at their point of production.
==== Single source waves ==== To get a feeling for how the simulation works, go into **Waves** mode and create a continuous __light__ wave. Answer the following questions for yourself. //(No need to record them in your report. This is just practice.)// * How can you measure the wavelength? * Do you use the tape measure? Do you use the "two-channel scope"? * How can you estimate the uncertainty? * What are the uncertainties related to //direct// measurements? * What are the uncertainties that get propagated through to calculated values? * How can you lower the uncertainty on measured quantities? * Can you measure one thing multiple ways? (For example, getting wavelength from peak-to-peak and trough-to-trough on the same wave?) * Can you repeat measurements? * Can you measure multiple cycles of a wave and divide by the number of cycles? * Which color has the largest wavelength? Which has the smallest? * //Roughly//, what is the wavelength range (1 m to 10 m? 1 nm to 100 nm? Something else?) ==== Two source interference ==== Now switch to **Interference** mode and again select __light__. If you turn both sources on, you will see that the waves which they produce are **coherent**. That is to say, the two waves are emitted **in-phase** with each other; when the top source is, for example, at a maximum, so is the bottom source. Now click the check boxes to turn on the __screen__ and __intensity__. === Exercises ===
When answering the following questions (and questions in future sections), you do not need to address each bullet point individually. It may be easier – and more concise! – to write a single answer after each group of questions that touches on all the things asked. Keep in mind that the notebook is your record of what you did in the lab and the questions are meant to guide you through your measurements and observations.
You probably want to include some screenshots in your notebook to help illustrate how and where you are making measurements. "Showing" is often more concise than writing out in words.
Choose a particular wavelength of light and a specific source separation (and record them in your notebook). * Qualitatively (that is, //without using numbers//) describe what you see on the screen. * Do you see maxima and minima? Do the positions of these maxima and minima move as you change the wavelength? * If so, how? (Be specific. As you increase wavelength, what happens? As you decrease?) * Do any of the maxima or minima stay put as you change wavelength? * What happens as you increase or decrease the source separation? (Again, be specific.) Now let's get quantitative. For one wavelength and separation, find a **maximum** (other than the central maximum) on the screen... * Measure the light's wavelength and estimate the uncertainty. * Measure the distance from each light source to the maximum and estimate the uncertainty in this distance. * What is the difference between the distances from each source? * How does this path length difference compare to the wavelength of the light? Now, repeat for one **minimum** on the screen... * Measure the distance from each light source to the minimum and estimate the uncertainty. * What is the difference between the distances from each source? * How does this path length difference compare to the wavelength of the light? Putting all this investigation together... What mathematical relationship between the two path lengths do we need to see constructive interference (a maximum) and what relationship do we need for destructive interference (a minimum)?
Why do we ask about uncertainties so much? There have been a lot of student questions recently about why we ask you to estimate uncertainties __so much__. And again here, we've just asked you to estimate the uncertainty in your wavelength and in the measured distances between sources and maxima/minima. So why do we do this? Whenever we compare values that have an inherent fuzziness – like the lengths that you measure here – the uncertainty helps us quantify how much difference is normal and how much should cause us to worry. At the simplest level, we can say that two numbers are "consistent with each other within uncertainties" (e.g. if you measure a value for the speed of light of $(3.05 \pm 0.10) x 10^8$ m/s and the literature value is $3.00 x 10^8$ m/s, you would say your measurement is "consistent with the literature value within uncertainties"), but there are more sophisticated methods we can use too (like the $t^{\prime}$ test we used in PHYS 131 when looking at pendulum periods.) Here, we are comparing a path length difference and a wavelength and looking for some relationship. The uncertainties on these two values help us to have confidence in whatever relationship we think we see. (Repeated measurements under different conditions can help build that confidence, but we're stopping after measuring for one maximum and one minimum today.) So, if your wavelength is $\lambda = 101 \pm 5$ nm and your path length difference is $\Delta= 24 \pm 2$ nm, you would say your measurements are consistent with $\Delta = \lambda/4$ (and not consistent with, for example, $\Delta = \lambda/3$ or $\Delta = \lambda/2$).
==== Beats ==== In the simulation above, we considered what happened when we had two sources which produced in-phase waves of identical frequency. You observed interference in that the wave intensity changed depending on the position; in some locations the two waves constructively added to form an intensity maximum, while in others they destructively cancelled each other out. Importantly, in this case the intensity at a given position was //constant in time//. A spot was, for example, //always// an intensity maximum or //always// an intensity minimum. This is, however, not the only way we can get a phase difference between two waves and interference. We can use two wave sources which have //different frequencies//. Now, at a given position, the wave amplitude will increase when the phase difference between the two waves is small (that is, when the two waves nearly overlap) and will decrease when the phase difference is close to 180° (that is, when the crest of one wave is nearly in line with the trough of the other). Whether the waves add constructively or destructively will change continuously; a given point will cycle from fully constructive to fully destructive and back. This slow variation in the amplitude of the sum of the two frequencies is called //beating//. The frequency of the beat is given by the difference between the two frequencies from each source. Figure 1 shows this amplitude variation created when two waves of similar frequency interfere at a point.
| {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:ophysics_beats.gif?direct&400 |}} | | **Figure 1**: Two waves of identical amplitude, but slightly different frequency add to produce a wave whose amplitude varies with time with a frequency equal to the frequency difference between the two waves.(Image borrowed from [[https://ophysics.com/waves10.html|oPhysics]].) Click image to see animation. |
=== Hearing beats (mixed tones) === In order to //experience// beats, we have produced a few sound files that mix pure tones of equal volume but different frequency. Listen to these files and see if you can hear the beat frequencies. ^ Sound File ^ Frequency 1 ^ Frequency 2 ^ Beat Frequency ^ | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:800hz-700hz.mp3 |}} | 800 Hz | 700 Hz | 100 Hz | | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:800hz-750hz.mp3 |}} | 800 Hz | 750 Hz | 50 Hz | | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:800hz-790hz.mp3 |}} | 800 Hz | 790 Hz | 10 Hz | | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:800hz-799hz.mp3 |}} | 800 Hz | 799 Hz | 1 Hz | | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:800hz.mp3 |}} | 800 Hz | 800 Hz | 0 Hz | == Exercises == Now, take a listen to the two //mystery// tones below. ^ Mystery Tone 1 ^ Mystery Tone 2 ^ | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:mystery_frequency_1.mp3 |}} | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:mystery_frequency_2.mp3 |}} | Using this [[https://www.szynalski.com/tone-generator/|online tone generator]], you can play the mystery tone and a tone of known frequency at the same time. * By listening for when the beat frequency goes to zero, can you identify the frequencies of the two mystery tones? * Are either (or both) the tones an integer frequency, or does your mystery tone seem to fall in between the values available in the tone generator? Can you estimate the mystery frequency value anyway? === Hearing beats (separate channels) === We can now use an adjustable tone generator that will produce different frequencies in each channel of your speakers (or headphones) – one tone in the left channel and one in the right channel: * **oPhysics Interactive Beats Simulation**: [[https://ophysics.com/waves10.html]]
What if I don't have headphones or separable stereo speakers? This part only works if you can listen to each channel separately. It's easiest to do with headphones, but it can also be done if you have separate (and small) left and right speakers. If you don't have access to speakers or headphones like this, then please read through the following section anyway, but write "I do not have access to the appropriate equipment for this part of the experiment and cannot complete this section" on your lab report template.
== Exercises == Choose two frequencies that are different (but close) at about 200 Hz. Place **both** the left and right headphones close to one of your ears. Adjust one of the frequencies until beats are clearly audible. * Describe what you hear. * Based on the frequencies you chose, what beat frequency should you hear? Is that what you detect?
With both phones at one ear, the mechanism for hearing beats is easily understood: while the two waves are in phase, the air pressure driving the ear drum is larger and the sound is perceived as louder. While out of phase, the pressure is less and is perceived as quieter.
Next, put on the headphones so that one sound plays in one ear and one sound plays in the your other. In this mode, each ear drum is presented with a single frequency and should, therefore, not experience beats. Without changing the frequencies used earlier, place one phone on each ear and listen for beats. * What do you hear? Repeat the above exercise using frequencies of about 500 Hz and 1000 Hz. * Can you still hear beats at this higher frequency?
The physiology involved here may be summarized as follows: Each eardrum is forced to vibrate by the small pressure changes caused by the vibration of its headphone. The eardrum is mechanically coupled through small bones to another membrane at the entrance to the cochlea, a fluid-filled spiral tube. Inside the cochlea is a membrane containing hair cells which are set in motion by the vibration of the fluid. The hair cells closest to the cochlear entrance are sensitive to high frequencies, while those farther from the entrance are sensitive to lower frequencies. The hair cells convert the mechanical vibrations to electrical signals. The signals from the left and right ears are //mixed// in each of two sets of neurons in the brain stem. One set of neurons is sensitive to high frequencies (typically kHz.) and detects //intensity//. At high frequencies, the sound shadow cast by the head gives rise to differences in intensity from left to right and is used to judge direction of the sound source. The other set of neurons, sensitive to lower frequencies (typically 200 Hz.), detects //phase// or time differences to judge the direction of the sound source. The detection of beats with different frequencies sent to the left and right ears depends on the functioning of the lower frequency, phase-sensing neurons.
It has been observed that some people can hear beats in the two-ear mode while others cannot. * According to the above model the ability to hear beats in this mode should be better at low frequencies. Is this the case for you? = ====== Part 2: Diffraction ====== ---- ===== Diffraction patterns ===== To help us start to understand diffraction, let's look at this video from //Veritasium// about a very curious phenomenon called "Arago's spot". Observation of this non-intuitive phenomenon helped build the evidence for light's behavior as a wave. | | The unusual patterns predicted by diffraction theory – like Arago's spot – were difficult to see using the technology of the 1800s (i.e., candles, slits, and screens.) Today, though, with the use of coherent laser beams and CCD camera detectors, it has become much easier to see these very curious – and pretty! – diffraction patterns, and to see the wave nature of light with our own eyes. A simple setup is shown in Fig. 3. A helium-neon laser produces red light which passes through a diverging lens to spread the beam over a wide area (to cover the CCD of the TV camera and to reduce the beam intensity). There are several objects provided for the beam to pass through (e.g., a sewing needle, a razor blade edge, and some ink spots on a glass slide) which are shown in more detail in Fig. 1(b). | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:patterns_setup_small.png?direct&500 |}} | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:patterns_objects_small.png?direct&250 |}} | | (a): The apparatus in action | (b): Diffracting objects | | **Figure 3**: Diffraction pattern apparatus || As we move the different objects into the beam to cast a shadow onto the TV camera, different diffraction patterns appear on the monitor. These images are shown in Fig. 4. | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:razor_notch_small.jpg?direct&200 |}} | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:razor_edge_small.jpg?direct&200 |}} | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:pin_head_small.jpg?direct&200 |}} | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:irregular_dot_small.jpg?direct&200 |}} | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:round_dot_small.jpg?direct&200 |}} | | **Figure 4**: Diffraction images of (from left to right) the notch in the razor blade, the razor blade edge, the pin, an irregular dot, and a symmetric round dot. (**Click on each image to expand! The small thumbnail images will appear distorted.**) ||||| === Exercises === Look over the five images in Fig. 4 above. * For **one** of the images, describe the pattern you see and (in one or two sentences) explain how the symmetry (or asymmetry) of the object leads to the pattern. ===== Diffraction through a single slit ===== Now let's make some quantitative measurements by looking at diffraction through a single slit. ==== Theory ==== === Interference from two point sources === In the interference portion of the lab, we saw that light from two point sources would interfere at different positions in space due to the superposition of the waves from each source. | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:interference_two_point_sources.png?direct&300 |}} | | **Figure 5**: Interference from two point sources | Figure 5 represents the wave disturbance originating from two coherent point sources, S1 and S2, separated by a distance $d$. If we place a detector at some point, the intensity at the point is given by the superposition principle. The intensity will be a __maximum__ (i.e., the waves will be in phase) if the path difference, $\Delta r = r_2−r_1$, is an integral number of wavelengths, i.e. $\Delta r = n\lambda$. Since the path length difference for these two sources is $\Delta r = d\sin\theta$ (where $\theta$ is defined in the figure), the angles at which the intensity will be a maximum, denoted by $\theta_{\textrm{max}}$, obey the following relation: | $n\lambda = d\sin\theta_{\textrm{max}}$ (for $n=0,1,2,\dots$). | (1) | The intensity observed at the detector will be a __minimum__ if the two sources are out of phase, i.e., $\Delta r = (n+1/2)\lambda$. Therefore, denoting angles where we expect an intensity minimum by $\theta_{\textrm{min}}$, we have | $(n+1/2)\lambda = d\sin \theta_{\textrm{min}}$ (for $n=0,1,2,\dots$). | (2) | === Single slit diffraction === Now, consider an extended source, e.g., a slit, having width a as shown in Fig. 6. We can consider the pattern of diffracted light coming from such an extended source as that resulting from the interference pattern of an infinite number of point sources within the slit, all oscillating in phase. | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:single_slit_diffraction.png?direct&300 |}} | | **Figure 6**: Single slit diffraction | To find the angles where the intensity will be a //minimum//, let us divide the extended source of width a into two equal regions: the top and bottom halves of the slit. Now suppose that we are at such an angle $\theta$ that the path difference between a source at the top of the slit and one at the //middle// of the slit is exactly one-half wavelength; then it is clear that these two will //cancel// and give no contribution at the detector. Also, it should be clear that the contribution from the //entire// slit will as well be zero at this angle. This is because for any source in the top half of the slit, we can find one in the bottom half (namely one a distance a/2 from the upper source) which will, when added, give zero intensity. Thus, our condition for a minimum intensity becomes | $\Delta r = \dfrac{a}{2}\sin\theta_{\textrm{min}} = \dfrac{n\lambda}{2}$ (for $n=1,2,3,\dots$). | (3) | (Note that $n = 0$ is not included.) Thus, | $a\sin \theta_{\textrm{min}} = nλ$ (for $n=1,2,3,\dots$). | (4) | The maxima will occur approximately half way between the minima. A single slit diffraction pattern is shown in Fig. 7. | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:single_slit_pattern.png?direct&400 |}} | | **Figure 7**: Single slit diffraction pattern | ==== Apparatus and data ==== To create diffraction patterns, we will direct a laser through a specially-designed slit holder and onto a white screen. (See Fig. 8) In front of the screen, we mount a photodiode (light sensor) attached to a slide potentiometer (a variable resistor) so that we can measure the light intensity as a function of position along the screen. Both the intensity and detector position are read out and recorded on a computer. In addition, we have a gain knob which allows us to amplify the signal from the photodiode before sending it to the computer. | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:diffraction_scanner_schematic.png?direct&400 |}} | | **Figure 8**: Laser with diffraction scanner | We set up our photodetector a distance $L = 71.0 \pm 0.2$ cm from the diffraction slit holder and project the output onto our photodetector. The diffraction pattern we get when we pass the light through a single slit of width $a = 0.08$ mm is shown in Figure 9. In Fig. 9(a), we turn the voltage gain down so that we can see (almost) the entire central maximum peak. In Fig. 9(b), we increase the voltage gain so that we can better see the smaller amplitude maxima on either side. (The flat top to the central maximum is because the detector can put out only a maximum of 5 V. We cannot amplify the signal beyond that point.) | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:single_0.08mm.png?direct&400 |}} | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:single_0.08mm_3.png?direct&400 |}} | | (a) Single slit, zoomed-out | (b) Single slit, zoomed-in | | **Figure 9**: Light intensity (arbitrary units) of the diffraction pattern produced by single slit. (**Click to enlarge.**) || === Exercises === Let us try to determine the wavelength of the laser from the diffraction pattern. From the data provided in Fig. 7 for the single slit diffraction, fill out the following table for the first 4 minima on either side of the central maximum. In order to calculate the minimum angle, use the fact that $\theta_{\textrm{min}}$ is one of the angles in the right triangle with sides $L = 71.0 \pm 0.2$ cm and $x_{\textrm{min}}$.
A note on uncertainties. It is possible to estimate uncertainties on all the measured quantities and to propagate them through to a final result. It will be a bit of a complicated formula to derive, but once you have it, it should not be too hard to use. (You may want to look over this page for a reminder of the general propagation formula using partial derivatives.) If you are unable to use the propagation formula, you may use the following approximation: $\dfrac{\delta \lambda}{\lambda} = \dfrac{\delta x_{\textrm{min,left}}}{x_{min}}$. For example, if you think you can estimate the position of the first minimum with $\delta x_{\textrm{min,left}} = 0.20$ cm and the distance from the center is $x_{\textrm{min}} = 4.00$ cm, then the fractional uncertainty in your wavelength will be $\delta \lambda/\lambda = \delta x_{\textrm{min,left}}/x_{\textrm{min}} = (0.20)/4.00 = 0.05 = 5\%$.
^ diffraction order, $n$ ^ location of the left minimum, $x_{\textrm{min,left}}$ (cm) ^ location of the right minimum, $x_{\textrm{min,right}}$ (cm) ^ average distance from the center, $x_{\textrm{min}} = (x_{\textrm{min,right}} - x_{\textrm{min, left}})/2$ (cm) ^ $\sin\theta_{\textrm{min}}$ ^ $a\sin\theta_{\textrm{min}}$ (cm) ^ | 1 | | | | | | | 2 | | | | | | | 3 | | | | | | | 4 | | | | | | * Now, use this data to determine the laser wavelength, $\lambda$. You can use whatever method you like (and explain), but it should incorporate all four measured distances. * How does your value compare to the literature value of $\lambda$ = 632.8 nm? ===== Diffraction grating ===== ==== Theory ==== We saw above that interference from two point sources separated by distance $d$ produced maxima on a detector at angles given by | $n\lambda = d\sin \theta_{\textrm{max}}$ (for $n=0,1,2,\dots$). | (5) | When we change from two point sources to two narrow slits, we now have a combination of the interference due to two sources and the diffraction due to the non-zero slit width; the result is an intensity pattern similar to that shown in Fig. 10. Note that the two slit pattern of Fig. 10 would lie entirely beneath the single slit pattern of Fig. 7, showing that the larger overall shape (i.e. the envelope) of the two slit pattern is still dictated by the widths of the individual slits. | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:two_slit_pattern.png?direct&400 |}} | | **Figure 10**: Two-slit diffraction | We can, in fact, show this experimentally as illustrated in Fig. 11. The single slit diffraction pattern for slit width a = 0.08 mm is shown again in Fig. 11(a). The diffraction pattern for light through two slits of width $a$ = 0.08 mm and separation $d$ = 0.25 mm is shown in Fig. 11(b). The overlap of the two is shown in Fig. 11(c), with the double-slit pattern clearly falling completely under the envelope formed by the single-slit pattern. | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:single_0.08mm.png?direct&400 |}} | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:double_0.08mm_0.25mm.png?direct&400 |}} | | (a) Single slit diffraction | (b) Double-slit diffraction (with the same $\lambda$ and $L$) | | {{ phylabs:lab_courses:remote_courses:phys-120_130_140-remote-wiki-home:interference-and-diffraction-remote:comparison_0.08mm.png?direct&400 |}} || | (c) An overlay of both patterns showing that the double slit patterns falls completely underneath the single-slit pattern || | **Figure 11**: Comparison of single and double-slit diffraction patterns. (**Click to expand.**) || As we continue to add equally-spaced slits – 3 slits, 4 slits, etc. – the position of these maxima do not change, but they become progressively narrower. In the limit of many slits, we effectively find destructive interference everywhere except right at the maximum angle. Such a device is called a **diffraction grating**, and it might have hundreds or thousands of slits (or more commonly, shallow grooves which act the same way) and therefore can achieve diffraction spectra with incredibly sharp peaks.