Lenses Experiment: PO2.1

Lenses Experiment (PO2.1)

• The aim is to verify the lenses formula and study lens magnification alongside chromatic aberration

• Needed apparatus includes: optical pins (object), a screen, convex lens, red and blue filters, wire meshes (object), optical bench, ruler, and a light source

• Parallel light rays bend through a convex lens to meet at the real principal focus

• Use this formula for lenses, whether diverging or converging, when applying a 'real is positive' sign convention:

  1/u + 1/v = 1/f

  Where: - f is focal length - u = object distance - v = image distance

Verifying the Lens Formula:

• Get an approximate focal length by holding the convex lens to a wall near a window
• Focus the transmitting light rays
• Configure an illuminated pin as the object along an optical bench, with the object, lens, and screen all in alignment
• Shift the screen to focus light and subsequently document both the object distance (u) and image distance (v) for five object distances
• Each object distance has to be between f and 2f

Studying Lens Magnification:

• Repeat the process from verifying lenses but for a variety of sizes in objects
• Find the image's position and final determined size

Studying Chromatic Aberration:

• Use a wire mesh as the object and add a blue filter, making sure its by the object and the source of illumination
• Correct the lens to ensure the image is focused on the screen at a decided object distance
• Measure and record u and v, making sure u stays between f and 2f
• Pick four more object distances, reiterating the first step and compiling the results in a table
• Redo the prior steps and tables using a red filter

Results:

• You can verify the lens formula from plotting a graph of 1/u against 1/v, relating the focal length of the lens determined from measuring/ initial approximate value in a percentage error calculation
• With a lens with a red or blue filter, measure the focal length

Conclusions:

• Link the magnification m = s’/s to the data of v/u to check if results show if an ideal lens is being used
• List the cause and state other forms of lens aberrations.

Laboratory Data Sheet

• Tables will need to be completed for different filter and no filter scenarios
• The tables require object distance, image distance, size, and magnification, as well as the result of v/u

Measuring Wavelength of Laser Light

• The determination of laser lights wavelength is the whole objective of this experiment
• The correlation nλ = d sin θ shows how a grating causes maxima of strength and the angles at maxima help determine the laser light's wavelength
• Equipment: laser, Convex/Concave lens, Optical Bench, lens holders, support for slits, set of coarse gratings projector screen
• Shine light through a grating (array of slits), broadening the laser beam until it goes through the array
• Focus the pattern the screen after shining diverged light
• A different slit space requires the procedure to be repeated
• Select one patch and find n by counting from the center and measure the distance to get angle theta
• Finish answering with nλ = d sin θ

Fraunhofer Diffraction

• Necessary gear: green laser (563.5 nm) on 2-axis translation stage, 1m optical bench, 1 slide holder, a slide with four single slits/ 3 diffraction gratings/ four double slits/ multiple slits, pinhole, a slide containing an etched Fourier transformed function, along with a screen and tape measure
• The purpose aims to understand/test Fraunhofer diffraction alongside its Fourier analysis
• The theory explains, shining a laser through a small slit, where if light travels in lines, the screen will show the slit's image in shadow
• For small slits, a diffraction pattern appears with a central bright fringe along the slit-screen axis with dark/bright fringes on both sides
• Light bends to diffract

Diffraction Qualities

• Diffraction assumes light acts like a wave
• A ray is made perpendicular to wave fronts, which tells the direction of light
• Use Huygens' principle to make any point on a wave front act as the point source emitting circular wave fronts
• The diffraction pattern comes from the interference of waves made by the point sources
• Fraunhofer and Fresnel diffraction separate the diffraction pattern,
• Fresnel diffraction describes a near-view where wave effects have little control on the aperture's shadow,
• Fraunhofer diffraction is based on the view far from the aperture where geometric optic is completely worthless

Math Regarding the Phenomenon

• Assume a general aperture illuminated reflects how its geometry with f(x,y) can be given with the electric field:

  E(x, y) = E0f(x, y)e-iωt,

• The oscillator can be re-expressed as linear combinations with infinite plane antennas given by:

E(x, y) = Eo ∫∫ f(kx,ky)e(kxx+kyy)-iωtdkdky,

• A characterized formula wave vectors show as inverse Fourier transform is:

f(kx,ky) = ℑ-1{f(x, y)} = 1/2π ∫∫ e-ikxx-ikyy f(x, y)dxdy

• Each antenna creates a plane wave:

ei(kxx+kyy)-iωt → ei(kxx+kyy+ikzz)-iωt with kx2 + ky2 + kz2 = (2π/λ)2 = ω2 / c2

• The field will use angles (θx , θy) with respect to the z axis, such that:

  sin θx = kx/k and sin θy = ky/k

• The treatment stays within paraxial approximation, making k stay with k ≈ k= 2π/λ, with:

theta x = x’/L and theta y = y’/L

• The field amplitude for combining both equations is:

E(x’, y’) = f(kx,ky)(x’/L, y’/L)

• In paraxial approximation, the far field diffraction pattern is just a scaled inverse Fourier transform of the aperture
• If a single slit is considered, and the vertical dimension made one-dimensional, then:

f(x) = E0, x ∈ [-a/2, a/2]

• Also, f(x) = 0, x ∉ [-a/2, a/2]
• The field at a point on the screen converts to:

E(x’) = 1/2π ∫ f(kx)eikxdx = 1/2π ∫ E0eikxdx = E0/2πik eika/2_e-ika/2 = 2aE0/2πka sin(kxa/2)

• Using kx and defining akx / 2 = ak sin θ / 2 ≡ β, the result is: E(β ) = aE0/2π sin β β

Grating Width

• When treatment is valid, there is infinitely sharp diffraction maxima With a small width the given peak shows with this equation,

wkd sin θ / 2 = π δ sin θ = λ/w

W is grating width

• To properly distinct peaks, widening the grating has to be done
• Gratings often separate different wavelengths, therefore each angle depends on the incident light's wavelength
• If peaks overlap the outcome seems like a brighter peak
• The lowest possible difference of wavelength is

lambda/ delta lambda min to shows how the chromatic resolution is identified: ℜ ≡ λ/ ∆λ min If ℜ ≡ (w m)/d = mN with N equaling lines, that means resolution in first order is N

Experimental Procedure

• Put laser equipment to a point where its aiming at the wall and paper taped to it, be aware its very dangerous to look directly at the laser's light
• Tape measure records distance to wall
• Equation (10) and (12) calculate the values and of maxima and minima, putting one side of 4 slits with different opening values
• Measure/Sketch and compare theoretical results with the 3 equations
• Continue data collection, measuring and taking data from slits

Spectrometer Experiment (PO2.3)

• Spectrometer is used as necessary equipment to produce spectra
• Collimator, table, and telescope are its parts
• To find glass prism refractive index, be prepared to note readings between different reading arrangements
• Given conditions show that A = r + rt D = (i + i') - (r + r') A+D/2 helps discover u via snells law

Traveling Microscope Experiment (PO2.2)

• Microscope can determine refractive indices of glass and water
• traveling microscope with vernier scale, glass block, beaker, lycopodium powder or chalk dust, and fine sand is requires
• The change between direction/ medium is refraction
• Given diagrams, we can see the relationship with snells law:

aµ b = (sin APQ) / (sin AP' Q) and aµ b = I/bµ a = (PQ)/(P'Q)

Newton's Rings Experiment

• Determines sodium light wavelength and plots graph between n and d
• Plano-convex lens, optical arrangement, glass plate must be there
• Formula: lambda = D²n+m - D²n/ 4mR
• An air filmed from differing surface has an interference which creates ring, using the formula DN2 = 4nr(lambda), helps relate everything