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I have been attempting to unravel Abbe’s theory, of the role of diffraction in microscopic vision, in a ‘nuts and bolts’ sort of way, meaning in terms that I can understand: diffraction, interference and so on. While Fourier transforms are clearly at the heart of the process the assertion that an object undergoes a Fourier transformation into components (a diffraction pattern) which then undergo a ‘reverse’ transformation into an image is just an assertion, to me. I only feel I understand what is going on when I learn how the processes of diffraction and interference work to produce that pattern. All this is in widely available texts (I use Jenkins & White, Fundamentals of Optics) but I am struggling with the last bit of the process: how the diffraction pattern is re-transformed into the image.

https://lh3.googleusercontent.com/y32sRf0V8G0x19IlSiwuIscME3xdInGRCVbr_CQa9p6U-0h9pwqrG24PnT_ss2KDBY1hIR_9o-5qF652HhnuU7pzbNQbVo_6tIFnxhCxh1ewcwFJIZ1Bq4jZPHGLr2fNHqDPC8Ej6wfverxufuX-cTDLM7Bv-g_fCnRsbhkCtEiDdce896TW24feo1TG6TT2AQyVAozFlrZkaGcFFK8UYEQ52ccwZeAF5zdaxyA4ISEzNUdZp_jkIw4aQ1BqN2s2tfCt0ACeBOHPi6CPqkQewrExkpsgYUUGRdeOPkg9q9gaoN4Qw7r2YXTIbdCrGlCbFE9o6cUIeaJWAwcOtQ1xBVDRnv2VM8oHwuJ9GdXI2bmp8Ot1FqKjqSbwkicUCzVGofzfjYqIyiq0NMe_pNcQhG2pqU4zO04aZRl6yb379F2dGwCR2QBXrnziCFAhb-VAnpXGZv5OfGQmjy79Z5cE8iub2LGKEfIrBuO15JwoLl0ExvoEFxZ4NNPEx_0gZhwlVJcp66SV90vbGC_wAY7pJ_dL9Ixcdn_bMZS7a3itjJd5QRfBKn-JGrkLdf9ftzPTf-QkTNNtH5PEVaOH4dqwhjfUuEglBonMb94wKUDqwHifNZoAo7BV=w1794-h972-no

Figure 1 is my version of the conventional summary of the process. The object is a grating with a 1:1 ‘mark space’ ratio, or equal opaque and transparent elements. This gives rise to zero order light of amplitude >+A and sine elements of frequency F (same as the grating, but magnified x

So how does this happen: how are the elements of the diffraction pattern ‘written’ into the primary image plane as sinusoidal ‘standing waves’ of illumination of the appropriate periodicity? I cannot find anything, for instance in Geoffrey Brooker’s Modern Classical Optics, except the familiar assertion above: a reverse Fourier transform.

https://lh3.googleusercontent.com/e9uyMuCC14-XfNcGXr3VWx4I7nzA54Pi9YU2RKK0cVdDOv9xTT7Mugw3P6gkJMSOYx7_7ddlnPRO0ND8uQd0lSJdbCeu9jsS2jEWmJbO2o8bm9NZ1SLB_os3yo0KNTmz-LO0XN_vAgRriKoYZaobF5i8NBoXQkO_jw_cs0MxFYF6ai7YL3-dFND53CKsNVo2LBmuLbtpWgj1bBtJK2f6LxOOdqphbzUQ6qOwlkSsZJdirSrvIWKGhkcXLyeEsk2ecH2U9_hjT1hGLQEYFQ4A6Q3Z1TUrywPnR96RyHTU58uYfU6OfW8WSEyCQ80KC9iKzNQVlFmRoataO7L8fC8V87KzqTChBeRiva8eiKWvOOWxU5hwsLjjPhE-nHrbU3I3vLyHI0QufZ0_wqRQrvjngTdghdCcFjoXLuEMz9Hf8RNAH5WCohlCUBB09bxMh6CnXGGw7NXOz5_LTiWUgvdmLV-pEnuEaqBhUw838av5C8JnqUJEyfORtkDnpsbHzTZAskwN_4Kzb7HS3bcdQeNO49wKwE_bF-0KefzG0x8ZoneA2JbsHCSGSHzMrykCpqcu2NFqmbdvh1-joWICS58klpC2QpsMDQpWhIcLM86b8c_q01tu=w1233-h996-no

Figure 2 is my guess. 2B follows rays from both edges of a slit object to their positions in the image plane, via a rather dumpy objective represented by its two equivalent planes. 2A notes that there is a path difference

All of which is great, but they are the wrong way up. See fig. 1. As will be the 1st order rays, here in antiphase to the zero order. If they interfere constructively on axis then there would be a dark spot in the brightfield image where a bright spot should be. What please is wrong with my picture? (fig. 2) What have I ignored / misunderstood / got mixed up? Or, is fig. 2 right and there is something wrong with the account in Fig. 1? I don’t think there is: I am a long time microscope user and it seems to ‘add up’.

Any help

https://lh3.googleusercontent.com/y32sRf0V8G0x19IlSiwuIscME3xdInGRCVbr_CQa9p6U-0h9pwqrG24PnT_ss2KDBY1hIR_9o-5qF652HhnuU7pzbNQbVo_6tIFnxhCxh1ewcwFJIZ1Bq4jZPHGLr2fNHqDPC8Ej6wfverxufuX-cTDLM7Bv-g_fCnRsbhkCtEiDdce896TW24feo1TG6TT2AQyVAozFlrZkaGcFFK8UYEQ52ccwZeAF5zdaxyA4ISEzNUdZp_jkIw4aQ1BqN2s2tfCt0ACeBOHPi6CPqkQewrExkpsgYUUGRdeOPkg9q9gaoN4Qw7r2YXTIbdCrGlCbFE9o6cUIeaJWAwcOtQ1xBVDRnv2VM8oHwuJ9GdXI2bmp8Ot1FqKjqSbwkicUCzVGofzfjYqIyiq0NMe_pNcQhG2pqU4zO04aZRl6yb379F2dGwCR2QBXrnziCFAhb-VAnpXGZv5OfGQmjy79Z5cE8iub2LGKEfIrBuO15JwoLl0ExvoEFxZ4NNPEx_0gZhwlVJcp66SV90vbGC_wAY7pJ_dL9Ixcdn_bMZS7a3itjJd5QRfBKn-JGrkLdf9ftzPTf-QkTNNtH5PEVaOH4dqwhjfUuEglBonMb94wKUDqwHifNZoAo7BV=w1794-h972-no

Figure 1 is my version of the conventional summary of the process. The object is a grating with a 1:1 ‘mark space’ ratio, or equal opaque and transparent elements. This gives rise to zero order light of amplitude >+A and sine elements of frequency F (same as the grating, but magnified x

**M**) and amplitude -A (from the 1st maximum), also of 3F with amplitude +A/3 (from the 3rd maximum), 5F at –A/5 (from the 5th maximum) and so on: all the ‘odd’ orders alternating in phase and antiphase and diminishing in amplitude. These are represented in the figure as**b, c**,**d**and**e**. The brightfield image is all these combined in**g**, and the darkfield image is all except the zero order, in**f**.So how does this happen: how are the elements of the diffraction pattern ‘written’ into the primary image plane as sinusoidal ‘standing waves’ of illumination of the appropriate periodicity? I cannot find anything, for instance in Geoffrey Brooker’s Modern Classical Optics, except the familiar assertion above: a reverse Fourier transform.

https://lh3.googleusercontent.com/e9uyMuCC14-XfNcGXr3VWx4I7nzA54Pi9YU2RKK0cVdDOv9xTT7Mugw3P6gkJMSOYx7_7ddlnPRO0ND8uQd0lSJdbCeu9jsS2jEWmJbO2o8bm9NZ1SLB_os3yo0KNTmz-LO0XN_vAgRriKoYZaobF5i8NBoXQkO_jw_cs0MxFYF6ai7YL3-dFND53CKsNVo2LBmuLbtpWgj1bBtJK2f6LxOOdqphbzUQ6qOwlkSsZJdirSrvIWKGhkcXLyeEsk2ecH2U9_hjT1hGLQEYFQ4A6Q3Z1TUrywPnR96RyHTU58uYfU6OfW8WSEyCQ80KC9iKzNQVlFmRoataO7L8fC8V87KzqTChBeRiva8eiKWvOOWxU5hwsLjjPhE-nHrbU3I3vLyHI0QufZ0_wqRQrvjngTdghdCcFjoXLuEMz9Hf8RNAH5WCohlCUBB09bxMh6CnXGGw7NXOz5_LTiWUgvdmLV-pEnuEaqBhUw838av5C8JnqUJEyfORtkDnpsbHzTZAskwN_4Kzb7HS3bcdQeNO49wKwE_bF-0KefzG0x8ZoneA2JbsHCSGSHzMrykCpqcu2NFqmbdvh1-joWICS58klpC2QpsMDQpWhIcLM86b8c_q01tu=w1233-h996-no

Figure 2 is my guess. 2B follows rays from both edges of a slit object to their positions in the image plane, via a rather dumpy objective represented by its two equivalent planes. 2A notes that there is a path difference

**Pq**between the rays from each pair of higher order diffraction maxima, in the interference plane (aka objective rear focal plane) and the image plane. The bit of elementary trigonometry below shows that this path difference is equal to the path difference between rays from opposite edges of the slit at the angle**Theta**. The trig. looks pretty outrageous but angle**mnq**is <2 degrees with a x40 objective so the small angle approximation should hold well. Supposing the outer rays to be from 3rd order maxima the phase difference across the slit will be 3L/2 (L being wavelength) so the rays**mn**and**qn**uniting in the image plane also have a phase difference of 3L/2. They will thus interfere destructively giving a minimum of intensity at this point. Rays from the centre of the object slit will unit in the image on axis,**a**will be zero, there will be no phase difference between them, and they will interfere constructively giving a maximum. Between centre and edge the phase difference will change from 0 to 3L/2 ‘writing in’ 1 ½ periods of a sine wave. The same thing happens in the other half of the image so a total of 3 waves will be ‘written in’ as required by theory (for the 3rd order maximum).All of which is great, but they are the wrong way up. See fig. 1. As will be the 1st order rays, here in antiphase to the zero order. If they interfere constructively on axis then there would be a dark spot in the brightfield image where a bright spot should be. What please is wrong with my picture? (fig. 2) What have I ignored / misunderstood / got mixed up? Or, is fig. 2 right and there is something wrong with the account in Fig. 1? I don’t think there is: I am a long time microscope user and it seems to ‘add up’.

Any help

**very**much appreciated. I can’t stop trying to figure this out and get on with the rest of my life… Thanks for reading this far, John SW.
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