The Twyman-Green Interferometer - adespace · 2013-11-21 · fringes, or points of maxima were the wavefronts intersect, whilst 3b shows 5. When the source is no longer a point, but

Twyman-Green Interferometer Adrian Stannard

The Twyman-Green Interferometer

Adrian Stannard Department of Physics University of Reading

Whiteknights, Reading RG6 6AF UK Abstract This report looks at the theory and application of a Twyman-Green interferometer. Concepts important to interferometery are coherence, alignment and source size, all of which were studied in this experiment. It was found that the bandwidth of the mercury source used in the experiment was Δν=13GHz with a green filter, whilst without a filter the bandwidth was found to be 30GHz. The interferometer was used to observe Fizeau and Haidinger fringes, and then used measure a 136nm depth from a channel etched in a piece of glass; indicating the sensitivity of the device. Analysis was made of the effect of reflections from the front and rear surfaces of the beam-splitter, which gave a strong and weak set of Fizeau fringes with mirror stops 5.3cm apart. The interferometer was finally used to observe the aberrations of two lenses, showing at first hand how a deviation in wave front can help map imperfections on a lens surface.

Introduction The Twyman-Green interferometer, built on the basis of the Michelson interferometer, was designed for the application of optical testing in a production environment. In reference 1 Twyman states “entirely unskilled boys or girls can in a week or so be taught not only to test prisms and lenses and state precisely the nature of their defects, but to correct the optical performance by retouching the surfaces.” This refers to the fact that the instrument he helped develop for the optical workshops of Adam Hilger Ltd gives useful results with great ease. The principle behind the testing is that any aberration of a lens will modify a wavefront such that on interference with a plane wave, the resulting interference gives a contour map of the imperfection, revealing the location and magnitude of any imperfections. It is this contour map that is of use to the boys and girls, as it gives an indication of the amount of polishing required in the workshop.

The main differences between the Twyman-Green and Michelson interferometers is that in the former the light is collimated, and the two interfering beams of light are brought to focus at the eye of the observer. This experiment is initially concerned with alignment of the Twyman-Green apparatus and observing the effect of extending the source. Later, obtrusive effects on the wavefronts in one of the arms of the interferometer will be studied, and used to determine the depth of a channel in a sample of glass. Finally, the interferometer will be used in the manner it was designed for, to examine aberrations in lenses. Theory The Twyman-Green interferometer is one in which a division of amplitude takes place. Interference occurs from the recombination of two parts of a beam of light, which is divided by partial reflection and partial transmission at a beam-splitter. The two beams of light follow different paths in the interferometer, and are

The University of Reading The Department of Physics Page 1


Hg Lamp

Diffuser

Condenser

77A FilterPinhole, S

Collimating

Lens L1

Viewing

Lens L2

Mirror, M1

Mirror, M2

M2’- image of M2

Beam-splitter, B/S

Folding

Mirror

Unskilled Optical

Engineer

E

Hg Lamp

Diffuser

Condenser

77A FilterPinhole, S

Collimating

Lens L1

Viewing

Lens L2

Mirror, M1

Mirror, M2

M2’- image of M2

Beam-splitter, B/S

Folding

Mirror

Unskilled Optical

Engineer

E

Figure 1. Arrangement of the Twyman-Green interferometer for the observation of Fizeau fringes.

reflected back to the beam splitter from mirrors in both interferometer arms. One or both paths may be altered, depending on the requirement of observation, and it is the position of the mirrors, which determine the nature of the interference. In this experiment, mirror M1 (figure 1) can be moved closer to or further from the beam-splitter, and both mirror M1 and M2 can be tilted. M2’ is the image of mirror M2 formed in the beam splitter, and if M2’ is tilted relative to M1, a virtual wedge is formed, i.e. there exists a virtual air gap between M1 and M2’. When the path lengths for the mirrors are nearly equal, the wedge causes the reflected wavefronts to coincide with each other such that Fizeau fringes are formed. The source S is finite in size (in order to let light into the system), localising the fringes in the vicinity of M2’. These will be considered in more detail later.

Setting up the Twyman-Green Interferometer Lens L1 is a well-corrected lens in order for plane wave fronts to enter the interferometer, and is fixed to the interferometer bench. To collimate the light from the pinhole, the pinhole itself must be moved. With M1 and M2 blocked off, and beam-splitter removed, auto-collimation is used to position the pinhole. This involves inserting a mirror and reflecting the image of the pinhole back through L1 to the pinhole itself. When the pinhole is at the focal point of the lens, a sharp image of the pinhole is formed, and the light reaching the mirror is therefore collimated (by principle of reversibility). The position of the condenser lens can therefore be optimised to ensure that the light from the mercury source is focused on the pinhole. Before collimating the beam, the components responsible for getting light to L1 (including the Hg source) must be aligned to the optic axis of L1. This is



achieved by using an optical pin and a metal reference plate for L1. The reference plate has a centre marking which coincides with the centre of L1, serving as the datum for the optical pin. The positions of the lamp, condenser assembly and pinhole are then aligned to the optical pin. The diffuser and filter are removed to allow more light through, and assist in aligning of the condenser. The collimation method is carried out first – since the position of the pinhole determines the position of the condenser. The beam splitter is returned to its mounting position, and the offending item blocking off M2 is removed so that M2 is illuminated with the collimated beam, after passing through B/S. Two micrometers allow for adjustment of M2, and it is then positioned so the image of the pinhole coincides with the pinhole itself. The beam splitter is then adjusted so that a M1 is illuminated uniformly. M2 is blocked off whilst M1 is unblocked, and alignment of M1 is performed as it was for M2. In order for fringes to be observed, the path lengths for the two mirrors must be nearly equal, so M1 is moved by its travelling stage until it is at the same distance from the front surface of the beam splitter as M2. This distance is not the optical distance, however, and the consequences of this are discussed later. The differences are within the coherence length of the source for fringes to be observed. The last stage is to place a card at the focal point E of L2 with both mirrors exposed, so that two overlapping spots should be observed. Due to possible irregularities in the motion of the stage for M1, it may be necessary to re-align M1 until the spots coincide. With the filter back in place, an observer placing an eye at E should see some fringes. The position of M1 can then be adjusted a small amount until the greatest contrast is obtained, which is where the path lengths of the two interferometer arms are equal. Adjustments can be made of M2 so that the

fringe spacing is increased or decreased. These fringes are Fizeau fringes.

E

Lens L2S

A

B

C

D

FG

t

i

i

i’i’

n

n’

E

Lens L2S

A

B

C

D

FG

t

i

i

i’i’

n

n’

Figure 2. Construction for observing interference from a point source.

From figure 2, the origin of the Fizeau fringes can be made more quantative. The virtual air gap which is present must have a separation of less than half the coherence length of the source, otherwise the wavetrains will miss each other on reflection and no interference will be observed. The path difference is found from the geometry of figure 2. Path GDE and CFE are equal because they stem from the same wavefront (CG) which was split by amplitude division. Hence the optical path difference for the beams arriving at E is:

nAGBCABn −+=Δ )(' 1 In the figure it is clear that:

'cos itBCAB == 2

iitiACAG sin'tan2sin == 3

and applying Snell’s law:

'sin'sin inin = 4 Substituting 2-4 into equation 1 gives the path difference:



d

xm

Xm+1

θd

xm

Xm+1

θ

Figure 4. Construction to determine fringe spacing for a tilted surface

'cos'2 itn=Δ 5

And so the phase difference is:

'cos'40

itnλπδ = 6

Because both the interfering beams are reflected, it is not necessary to include the π phase changes on reflection. Also, λ0 is the wavelength of the source in vacuum. For constructive interference, the path difference must be an integer number of wavelengths:

0'cos'2 λmitn = 7 where m=0,1… The derivation so far has not considered the effect of tilting one of the reflecting surfaces. It turns out that for small angles, the relationships in equations 5 – 7 are still valid. What does change is how the rays

intersect. Figures 3a and 3b show more clearly what is going on. The incident and reflected rays intersect to form fringes of equal inclination, the spacing of which changes with angle. As the angle increases, the fringe spacing decreases - 3a shows 3 fringes, or points of maxima were the wavefronts intersect, whilst 3b shows 5. When the source is no longer a point, but of finite size (as is the case in a practical situation) then as mentioned, the fringes are localised to the tilted surface. They in fact contour the tilted surface in the same manner as in the contouring of a lens undergoing optical testing.

θ1

1 2 3

θ1

1 2 3

Figure 3a. With an angle between the mirrors, the wavefronts overlap to give fringes of equal thickness.

θ2

12

3 45

θ2

12

3 45

Figure 3b. Increasing the angle between the two mirrors decreases the fringe spacing.

The angular dependence of fringe spacing on angle is calculated from figure 4. The path difference for two rays leaving the wedge at a point xm away from the origin is 2d (since one ray is travelling back on itself and intersecting with a tilted wavefront at the surface) thus, for a bright fringe:

02 λmd = 8 These fringes are termed “fringes of equal thickness” because the fringe edge is profiled by equation 8. For small angles, the position of the mth fringe is:

θdxm = 9



and for the (m+1) th fringe:

θλ 2/0

1+

=+dxm 10

Thus the fringe separation δx is the difference of these two distances:

θλδ2

01 =−= + mm xxx 11

This confirms figures 3a and 3b, showing quantatively that fringe spacing decreases when the angle is increased. Experiment Having set up the Twyman-Green apparatus, observations of the dependence of fringe spacing on the angle of M2 are clearly observable. As the angle is decreased, the spacing increases until they appear to “fluff out” (M1 and M2’ are then parallel to each other). Conversely, when the angle is increased, the fringes get closer together. This agrees with the account accompanying the derivation of equation 11. In order to observe the fringes, one must focus on the virtual image of M2’ as seen through lens L2. If the lens was not present one would still need to focus near M2’ since the fringes are localised in that

region, but one would only see part of the pattern (which would be quite small). If a point source was used instead of a finite source, the fringes would be non-localised and would be seen anywhere in the intersecting beam region. Hence the purpose of L2 is to magnify the fringes near M2’. The lens is simply behaving like a magnifying glass, and so the position of the fringes (distance of M2’ from L2) must be less than the focal length of L2. The fringes form the an object O, located behind the lens, and thus with an eye at the point E, the focal point of L2, one is observing the magnified virtual image (virtual because it lies on the same side as the object). This virtual image is in fact located behind the object - the position (and hence size) is determined by the entrance aperture of the viewing system (such as the eye). If L2 was removed and the source was extended, then a wider range of incident and reflected angles are introduced. The points of intersection become washed out, causing the fringes to be localised at the surface of M2’. Hence one would thus focus at M2’ to see the fringes. When the mirrors are made parallel to each other a single image of the finite pinhole is obtained from each mirror, superimposed

P1 P2

Focal plane

of L2L2

S

M1M2’

S1’ S2’

P1 P2

Focal plane

of L2L2

S

M1M2’

S1’ S2’

Figure 5. Haidinger fringes are seen at the focal plane of L2 as a result of the finite source S being imaged by two parallel, but not coincident mirrors.



in front of each other (S1’ and S2’ in figure 5). This gives rise to Haidinger fringes when they are imaged to the points P1 and P2. The fringes are seen at the focal plane of L2, and so one needs to stand back to observe them. The fringes are centred on the axis of the lens (since it is responsible for bringing the images into the focal plane), and can be seen to be surrounding the image of the pinhole. On extending the source, it is observed that more light contributes to the fringe system, giving brighter fringes, but it does not affect fringe spacing. This is due to the fact that rays inclined at the same angle are focused to the same point, extending the source allows more light to contribute to those points and brings more fringes into view. From equation 6 one can see that minima occur the phase difference is an odd multiple of π

πλπδ )12('cos'40

+== mitn 12

Rearranging equation 12 we find:

4)12('cos λ

+= mit 13

At the centre of the fringe pattern, θ=0, and so m is found from:

λtm 2

= 14

When t=0.5μm, and using λ0=546nm, m is 1.831. When t is increased to 1mm, m is 3663. Hence as the distance between M1 and M2’ increases, new fringes are generated. Increasing the distance t also has the effect of making the fringes get thicker. This is possibly due to aberrations in the lens diverging the some rays more than others, since as t increases; the angular aperture of the system is changing.

Temporal Coherence and Bandwidth of the Source When the interferometer is set up to observe Fizeau fringes, displacement of the mirror M1 shows that the fringes are only visible over a range, going from a contrast of zero, to a maximum, and then back to zero. This is because of the coherence length of the light source. In order for interference to take place, the coherent wavetrains must cross each other. Once the mirror has moved so that distance t between M1 and M2’ is over half the coherence length of a wavetrain, incident and reflected wavetrains will miss each other, resulting in no interference. The bandwidth of the source is crucial in determining the range over which fringes can be viewed. In other words, if instead of a monochromatic source, a source with a range of wavelengths was used, then equations 8 - 11 show that a range of fringe spacings will result, and as the mirror separation is adjusted, the region will be fluffed out. This was confirmed experimentally by comparing the coherence length with and without the filter in place (see below). The meaning of the term “coherence length” is that it essentially describes the ability of two beams to interfere. When considering a source of light, one needs to consider that the source in this experiment works by the thermal excitation of Mercury atoms. The Mercury atoms, when excited thermally, spontaneously emit light quanta. But as in any thermo-statistical process, the atoms are in constant motion (as defined by a Maxwell-Boltzmann distribution of velocities for that temperature) and collide with each other. This terminates the radiation abruptly, but this is over a short period, so the amplitude remains constant. The time between these collisions is the time for which the phase of the light is predictable, and this is defined as the coherent time (TCOH). The



spatial measure of coherence is given by LCOH = c TCOH where c is the speed of light in vacuo. For a low pressure Hg lamp like the one used in this experiment, the accepted values of TCOH is of the order of 10-10s, and LCOH is 30mm. To measure this on the interferometer, we must note the earlier concerns regarding the passing of coherent wavetrains. The beam-splitter divides the amplitude of a single coherent wavetrain, which recombines to interfere with itself after reflection from M1 and M2. The fringe visibility is a direct indicator of where the wavetrain overlaps. The total path difference between the two positions of zero contrast must be twice the coherence length, since this is where the wavetrains pass each other at its extremes. But because the beam of light is travelling the same distance twice, the positions of the points of zero contrast must be the coherence length. On measuring the positions of M1 with a ruler where these two minimum contrast points occur, it was found that the source had coherence lengths of LCOH =23mm and thus TCOH =7.7x10-11s.

The bandwidth is thus Δν =1/ TCOH giving Δν=1.30x1010s-1. When the filter was removed, it was found that the coherent length was reduced to: LCOH =10mm and thus TCOH =3.33x10-11s. The bandwidth is thus Δν=3x1010s-1. Hence the filter decreases the bandwidth, thus increasing the coherent length of the system. If the mercury source was replaced with a white light bulb, then the fringes will be fluffed out into the constituent colours for successive orders of interference. The other problem in having a polychromatic source is optical dispersion in the beam splitter, which will result in a series of different ray paths due to different wavelengths. A way to overcome it would be to employ a monochromatic filter of suitable narrow bandwidth, but there still remains the problem of temporal coherence of a white source. Accepted values for a tungsten filament lamp are – LCOH =1μm and TCOH ~3x10-15s. Since the definition of TCOH comes the termination time of a wavetrain, it will not make much difference if a filter was used – the fact that the light has a spatial coherence of LCOH =1μm means that the mirror M1 must be positioned within that accuracy, which in practice would be impossible. Some form of increase spatial coherence may be obtained if a very small pinhole (of the order of 10μm) was used to diffract the beam before it was used. Again, this would not be feasible in practice as the light level would then be too low to observe anything.

Figure 6. Observing the reflection of a white surface in the beam-splitter, in order to locate the 50:50 reflection/transmission coating.

Figure 6 shows an important point about the beam-splitter, one can obtain two reflections from it – one from the front 50:50 reflecting surface, and one from the back glass surface. The white card was used to determine the location of the two surfaces – the strongest reflection is clearly the 50:50 coating, which can be seen in the picture to be the side nearest to



M1

M2

45° 45°

a

l

d1

d2

i

M1

M2

45° 45°

a

l

d1

d2

i

Figure 7. Optical Paths for reflection off front surface of the beam splitter.

M1. This means that if M1 is brought a sufficient distance closer to the beam splitter, then the path lengths of M1 and M2’ can be made equal if the back surface of the beam-splitter is used to reflect 4% if the light to M1.

In the apparatus, after moving M1 5.3cm towards the beam splitter, a new weaker set of Fizeau fringes could be observed. Due to the lower visibility, the coherence length LCOH was found to be only 13mm. Figures 7 and 8 show the different optical paths traversed as a consequence of the different reflections. The optical paths can be calculated as follows: In figure 1, to obtain fringes, the paths in the two arms must be equal, i.e.

nldd += 21 14 where n denotes the refractive index of the glass in the beam splitter (and air in this case is taken as unity). From the geometry of figure 7, equation

14 can be written as:

⎟⎠⎞

⎜⎝⎛ −

+=

2211

21

n

nadd 15

For figure 8, the path lengths are now:

M1

M2

45° 45°a

l

l1

l2

i

45°A

B

C

D EF

M1

M2

45° 45°a

l

l1

l2

i

45°A

B

C

D EF

Figure 8. Optical Paths for reflection off back surface of the beam splitter.

nlll −= 21 16

With some geometric substitutions, equation 16 can be expressed as:

⎟⎠⎞

⎜⎝⎛ −

−+=

21

1212

2

n

nadl 17

The difference in optical path is thus:

( )12211 2 −=− nald 18 Substituting in a = 1.95cm, and n=1.5 gives a value of 5.2cm, close to the measured value of displacement.



Optical Testing When a warm hand is placed in one of the arms of the interferometer, it causes fluctuations in the air surrounding the hand, in turn giving fluctuations in refractive index. This results in distortions of the wavefronts, showing up as shifts in the interference fringes. Another way to change the distribution of a gas such as air is by pressure fluctuations. A portable hi-fi was placed in front of one of the arms and positioned such that the speaker diaphragm was aligned to the optical path. With a suitable choice of music and volume level, the fringes could be observed to jump about in time to the music. It was observed that heavy metal music gave the most disturbing movements, whilst Tchaikovsky’s 1st piano concerto led to gentle swaying during the adagio. Sadly this could not be recorded photographically due to the length of exposure times required. A piece of glass was placed in front of M1 in the interferometer, and was covered in 10 fringes. Since the width of the glass block was found to be 35.5mm, the fringe spacing must be 3.55mm. Using equation 11, we can determine the angle between M1 and M2’:

rad

mm

x

5

3

90

10x69.7

10x55.3x210x546

2

−

−

−

=

==δλθ

A channel was then introduced into the glass block by polishing with a rubbing compound. After sufficient pressure and time, the channel was deep enough to cause a noticeable fringe shift. Figure 9 is a photograph taken with the channel rubbed further to show the effect. Before the second dose of polishing, the fringe shift was observed to be one quarter of a

fringe. By creating the channel, one has taken out a piece of glass of thickness 2n2t and replaced it with the same thickness of 2n1t. The factor of 2 comes from the wave passing through the glass block twice, n1 and n2 denote refractive indices of air and glass respectively, and t is the depth of the channel. For m multiples of wavelengths shifted, we can write:

λmnnt =− )(2 12 19a and hence the depth of the channel is

)(2 12 nnmt−

=λ 19b

Substituting into 19b m=1/4, n1=1, n2=1.5 and 546nm for wavelength, the depth of the channel is found to be:

t=136.5nm This value is quoted without error simply because it is being used to illustrate that the technique can measure to fractions of a wavelength of light.

Figure 9. Fringe shift due to a channel created in a piece of glass.



Figure 10. Setting up of a Twyman-Green interferometer with a lens-testing rig.

Lens Testing For optical lens testing, M2 was removed, and the optical test rig inserted. The rig has changed little from Twyman’s original design (reference 2). Twyman actually worked in the research department of the Hilger lens making company, and it was for improving the production and consistency of lenses and prisms that prompted the development. During the course of World War One, supplies of good quality abrasives for polishing lenses became quite scarce (reference 1) and so a reliable means of testing became all the more urgent. Figure 10 shows a photograph of such a test rig. It is first aligned by placing a circular mirror in the lens support jaws, and with M1 blocked off, the rig can be adjusted about a front pivot until the image of the pinhole is returned back to the pinhole. The back plate of the unit supports the convex mirror, which is chosen to suit the lens being tested.

In this experiment, a lens was randomly selected from a collection of ‘faulty’ lenses, and found to have a focal length of 20cm. It was not known to what degree of aberration it had; the aim was to simulate an inspection engineer picking a random sample from his unskilled lens makers. The requirement in the rig is that the convex mirror must be placed so that its radius of curvature coincides with the focal length of the lens (figure 11). For the 20cm focal length lens, there are two choices of radii for the spherical mirror: 1.91cm and 12.96cm. The former,

r

Lens L3 under test

Spherical Mirror M3

Focal length of L3

r

Lens L3 under test

Spherical Mirror M3

Focal length of L3

Figure 11. The optical set-up of the lens-testing rig. If the system was aberration free, then it would be equivalent to having M2 in the interferometer arm.



Figure 12a. Optical testing of lens of focal length 20cm

M3 at Paraxial focus M3 at Marginal focus M3 Tilted Figure 12b. Optical testing of lens of focal length 17cm

M3 at Paraxial focus M3 at Marginal focus M3 Tilted

although it could be used, is not desirable, because it would mean that the wavefront is focused to a much smaller area, imposing the risk of adding further distortions to it. It would be desirable to have a mirror of curvature just less than the length under test so that it can be situated directly behind the lens. This is not practical, and so the mirror that can be positioned closest to the lens is chosen from the range on offer (in this case 12.96cm). The lens L3 to be tested is aligned by an optical pin attached to the rig, ensuring concentricity. M3’s position on the rig can be checked by observing the image of the pinhole that it generates, with M1 blocked off. When it is set up so that its radius of curvature is at the focal point of the lens, the image is sharp, and when it is not in the focal region the image becomes blurred.

Because of the change of optical paths, the position of M1 need to be adjusted to take into account the test rig. Because M3 on the test rig is further away from the beam-splitter than M2 was, M1 need to be moved further away. Alignment of M1 will then need to be checked (block off M3/L3). Once aligned, put the rig back in the system - the tilt of M3 needs to be adjusted until the pinhole image coincides with that from M1, in a similar manner to setting up before. The system then provides fringes detailing the change in wavefront on passing through the lens. The position of M3 is the fine tuned to observe the change in wavefront at the paraxial and marginal regions. Photography For the high-resolution pan-F films, it was found that 10 seconds with aperture f4.5 was a suitable exposure time with the camera. The aperture was not open fully due to the zoom mechanism of the lens



employed (this was with a processor driven SLR camera which determined the best aperture stop for a given zoom magnification). Film developing times were performed according to manufacturers recommendations. Results Two different lenses were tested and photographed, the results shown in figures 12a and 12b. In both cases the lenses suffer from primary spherical aberration. Referring to an optical testing chart (reference 3) it is apparent that the lenses suffer no other aberration (such as coma or astigmatism). The way to interpret the pictures is to consider the fringes as contouring the lens surface. One can see that the first lens at paraxial (or Gaussian) focus has a much larger central field than the second lens, and there are very few outer fringes. The second lens has many fringes (not so visible in photograph), which suggests that it has more spherical aberration. To confirm this figure 13 illustrated what is

taking place. The wavefront from M1 serves as the reference plane Σ’’, whilst the wavefronts passing through the test rig undergo distortions, represented by the shape of planes Σ’. For the second lens, it is clear that in order to get a smaller central field, the wavefront has to be distorted more. Conclusions The experiment has thus shown that the Twyman-Green interferometer can be used to test optical components to a high degree of accuracy. The principle operation is that any optical component placed on one of the interferometer arms will modify the wavefront in that arm. The effect is made observable by interfering the modified wavefront with its planar counterpart, giving rise to a change in fringe pattern which contours the optical topology of the component. It was also observed that the device could only function within the coherence length of the illuminating

Σ’ Σ’Σ’’ Σ’’

Lens 1 (f=20cm) Lens 2 (f=17cm)

Σ’ Σ’Σ’’ Σ’’

Lens 1 (f=20cm) Lens 2 (f=17cm) Figure 13. A contour profile of the fringes observed at paraxial focus for the two tested lenses. The wavefronts denoted Σ’ represent the approximate distortion of the wavefronts after passing through the test lens, whilst Σ’’ denotes the reference plane wave from M1.



source. For the Hg lamp, the coherence length LCOH was found to be approximately 23mm, 7mm less than accepted values. The discrepancy was due in part to the difficulty of observing the fringes at low contrast by the experimenter, and also since it was for purpose of demonstrating the reason for using a filter, it is not important if the results differ by a small amount. Hence the experiment showed that with a filter in place, narrowing the bandwidth helped to improve the coherence of the source. References 1. F.Twyman, "Prism and Lens Making", Adam Hilger LTD 1942, pp 119-124. 2. F.Twyman, "Interferometers for the Experimental Study of Optical Systems from the point of view of the Wave Theory", Phil.Mag. Vol35, No.205. Jan 1918, pp 49-58. 3.Cagnet, Françon, and Thrier "Atlas of Optical Phenomena" MSC Optics Laboratory pp 12-15.

On extending the source too far, Haidinger’s fringes can seem to take on a world of their own.


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The Twyman-Green Interferometer - adespace · 2013-11-21 · fringes, or points of maxima were the wavefronts intersect, whilst 3b shows 5. When the source is no longer a point, but