Coherence

Physics of Lasers and Modern Optics

Laboratory Manual

EXPERIMENT 13: Coherence DESCRIPTION PURPOSE Learn the principles of coherence: How do lasers differ from light bulbs? INTRODUCTION The description and measurement of Coherence with electromagnetic waves generally assumes that the waves are nearly monochromatic, but not completely synchronized in phase at all times. Purely Coherent light has constant phase while completely incoherent light has randomly varying phase. Real light sources fall somewhere in between. Lasers tend to be highly coherent and incandescent sources tend to be highly incoherent, though, as we shall see, the degree of coherence one measures depends not only on the nature of the source, but also on the construction of the apparatus. Before reading the summary below, read the sections on coherence light in Introduction to Optical Engineering, by Yu and Yang (Y&Y), pp. 222-239. There are more advanced treatments in Hecht Ch. 12, Saleh and Teich Ch. 10, Born and Wolf Ch. X. Now that you have finished reading Y&Y (did you?--really?) Consider following summary (notation and equation numbers as in Y&Y). Superposition Consider the superposition at some fixed point in space of two harmonic waves u1 and u2 with the same basic angular frequency ω, but time-varying phase φ(t). u = u1 (t) + u2 (t) = u01 cos("t # $1 ( t )) + u02 cos("t # $ 2 ( t )) ,

(13-1)

where the time-varying phase difference δ(t) = φ2(t) – φ1(t) tends to smear out the interference pattern. The total intensity at the fixed point is then averaged over the period of measurement. ! 2 I = u1 + u2 = I1 + I2 + 2 I1I2 cos(" ( t )) , (13-2) where I1,2 = <|u1,2|2>. If δ(t) is small or constant over the period of measurement, the third term in Eq. 13-2 is constant and there is good interference. If δ(t) is varies by a large amount over the ! of measurement, the third term in Eq. 13-2 averages to a reduced value or even zero and period there is poor or no interference. Degree of Coherence The Degree of Coherence γ12(τ) is related to the phase fluctuations of a correlation function between two waves combined at a relative time delay τ. I = u1 + u2

! 13 Experiment

2

= I1 + I2 + 2 I1I2 Re[" 12 (# )] ,

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(13-3)

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Physics of Lasers and Modern Optics

" 12 (# ) =

u1 ( t + # ) u2 ( t ) I1I2

Laboratory Manual

.

(13-4)

A practical measurement of the Degree of Coherence amounts to creating an interference pattern between two waves, or of a wave with itself, and measuring the visibility defined as ! I %I (13-5) " = # 12 ($ ) = max min , Imax + Imin where Imax and Imin are, respectively, the maximum and minumum intensity of interference as the phase difference between the two waves is varied over one wavelength. ! The delay τ can be varied using an interferometer. See Example 8.7 for a practical illustration of visibility measurement. Spatial Coherence We normally consider an extended source like a discharge lamp as incoherent because the emissions from the many atoms that make up the source are not synchronized like they are in a laser source. What causes the overall incoherence is that the waves from the individual atoms all have randomly different phases. If we try to do an interference experiment between two coherent (but monochromatic) sources like two Hg lamps, the relative phases on a viewing screen are completely random and the average over phase complete removes the interference term in Eq. 8.31; the superposition of two mutually incoherent sources is just the sum of their independent intensities. But an ‘incoherent’ source will exhibit properties of coherence if we compare different points 1 and 2 on the wave front. This is because the emission from each atom spreads out like a spherical wave with well defined phase and even though the relative phases of the waves at points 1 and 2 are random the relative phase differences depend only on geometry and become quite small far from the source. It is therefore possible to achieve perfect coherence (ν = 1) with an ‘incoherent’ source! The interference on the screen in a Young’s two-slit experiment (See Y&Y Fig. 8.17) with an small extended incoherent source consists of the overlap of the individual interference patterns from the emitting atoms. The visibility (measured spatially across the screen) in this case is

% #Sd ( " = sinc' *, & L$ )

(13-6)

where the S is the source size, d the slit separation, and L the distance from source to the slits. Temporal Coherence ! The self-coherence of wave, as from a laser or localized source, can be represented by the Coherence length Δr, the Coherence Time Δt, or the Coherence Bandwidth Δω = 2π/Δt, which are related by the following equation: (We assume that the bandwidth Δω is small compared to the average angular frequency ω.) "r = c"t = 2#c "$ . Experiment 13

!

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In a typical self-coherence experiment, such as using a single source with a Michelson interferometer, the visibility will be

% #l ( " = sinc' * , & $r )

(13-8)

Where l is the path difference between the two arms of the interferometer. We can measure self-coherence by interfering a wave with itself using a Michelson Interfer! (Y&F Fig. 8.18). The wave is split into two beams sent along two different paths and ometer then recombined to form interference fringes. The path difference l amounts to a time delay τ = c/l between the waves. The dependence of the fringe visibility ν on the path difference l allows calculation of the coherence length Δr. The coherence time Δt follows from Eq. 13-7. ADDITIONAL READING M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980). Optics, 4th ed., by Eugene Hecht (Addison-Wesley, Reading MA, 2002). B. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York 1991). F. T. S. Wu and X. Yang, Introduction to Optical Engineering (Cambridge, 1997).

Experiment 13

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PROCEDURE ACTIVITY 1: TEMPORAL COHERENCE OF A LASER Equipment: Michelson Interferometer, He-Ne laser, Semiconductor Laser Similar to experiments described in Y&Y sections 8.6.2 and 8.7 to measure the coherence length/time for several sources. Relevant Equations 13-7&8. The laser coherence length will be measured with a long-baseline Michelson Interferometer (Y&Y Fig. 8.18). You will measure the fringe visibility (Eq. 8.27) as a function of path difference l from ~0 to 2 m. First, make a careful sketch of the apparatus, noting the type and location of all elements. Then align the interferometer with l ≈ 0 and the beams exiting parallel and expanded on the viewing screen with uniform intensity that can be controlled with the variable retarder. Then aim the combined beam at the detector through the small iris. Block the reference (shortest) beam and measure the signal beam power. Block the signal beam and adjust the reference beam power to the same value. Be sure to minimize and account for the background signal from the detector. With the beam powers balanced, vary the relative phase using the variable retarder and determine the max and min intensities. Repeat for a range of path differences l (0 m to 2 m for the He-Ne laser and 0 to 20 cm for the semiconductor laser). Plot the visibility vs. path difference l. Questions: 1) Why must the individual beam intensities be equal at the detector? 2) Calculate the coherence length defined as the path difference at which the visibility drops to 50%. 3) Calculate the coherence time and bandwidth. ACTIVITY 2: TEMPORAL COHERENCE OF EMISSION SOURCES Equipment: Michelson interferometer, Hg lamp with filter, Na lamp, LED, incandescent lamp Place the Gaertner Michelson interferometer 35 cm from the Hg lamp and green filter with the lens facing you. Half the light passes straight to mirror 2, the movable mirror, and half is reflected to mirror 1, the tiltable mirror. Adjust the position of mirror 2 to the 0 position as indicated on the sliding scale. View the lamp light from the output port of the interferometer and adjust the position of mirror 2 until clear fringes are visible. Tilt mirror 1 until 4-5 vertical fringes appear across the field of view. Adjust the path length until the fringes have maximum contrast and record the position of mirror 2—this is the reference position. Now move mirror 2 until the fringes are no longer visible, so that the intensity is uniform independent of mirror 1 tilt or mirror 2 position. Define the coherence length as the path difference l where the visibility first drops to zero. Now do the same thing with the Na lamp, but without a filter. Record again the reference position of mirror 2 (at highest contrast). Now move mirror 2 until you obtain minimum contrast and record this position and calculate its distance l1 from the reference position. Repeat to meas-

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ure the position of minimum contrast at 5 successive locations. Plots the distances ln vs. n and calculate the slope. The distance between these two positions is related to the separation Δλ of the Na D lines by the relation 2ΔL = λ2/Δλ where λ is the wavelength you measured by counting fringes and ΔL is the distance mirror 2 moves in going from highest contrast to lowest contrast. Calculate Δλ. Now replace the Hg lamp and return the Michelson to its reference position. Then place the incandescent (‘white light’) source at the interferometer input and locate its fringes. Note, this an extremely delicate task because the coherence length is so short. You can make it easier by using the green filter to narrow the bandwidth. Measure the coherence length of the incandescent source with the green filter and without any filter. If time permits, measure the coherence length of one of an LED source. Questions: 1) Calculate the coherence time and bandwidth for all sources. 2) Explain in 2-3 clear sentences how an ‘incoherent’ source like an incandescent lamp can exhibit interference fringes, especially as it emits such a large range (bandwidth) of frequencies. ACTIVITY 3: SPATIAL COHERENCE Equipment: TBD To Be Determined…

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From: Intro. to Optical Engineering, by F.T.S. Yu and X. Yang, pp 222-239. © 1997 Cambridge U. Press. For use by UNL PHYS 343 students only.