Armed with your sagittae, you can begin to grind your glass. If you did not get your surface curves generated, then you'll need to hog out the curves. If you did get them generated, then you should be able to start grinding with 40 micron aluminum oxide. Do that first and then dewedge as explained below. But before starting anything you should mark on the edges of your lenses with indelible marker the numbers 1, 2, 3, and 4 in several places, signifying which surface is which. This may save you from painful mistakes later.
For hoggers some words of advice: both crown and flint glasses grind easily and rather quickly. They are mechanically much weaker than Pyrex, so I would advise you not to use any abrasive courser than #120 Carbo. Even this is pretty course for the flint, so you might want to begin with #220 Carbo and see how your flint disk takes it. The appearance of the grind on the flint will always look more severe than that on the crown. It is more severe! Flint is soft and shatters easier than the crown does and a lot more than Pyrex does.
You should also be wary of applying too much pressure to your disks. Hogging out Pyrex for mirrors requires a lot of pressure on the center of your mirror disk. You may pile on 30 or 40 lbs. of weights to do the job and nothing bad happens. This would be a total disaster for crown and flint glasses. I have chipped the edges of disks through excessive force, and have seen a huge gouge in a commercial lens which probably got there through overly forceful grinding. So take your time and treat the glass gently as it will break a lot easier than Pyrex. Making a refractor lens by hand is a long-term project.
Keep a good bevel on the glass at all times, but otherwise grinding this glass requires much the same technique as the grinding out of a mirror. It is possible to hog out R2 on R3, but after that I recommend that you make 4 separate grinding and polishing tools. These tools are most conveniently made of dental plaster, and faced with pitch and rather soft ceramic tile 1" or 3/4" square. Hexagonal tiles will also work. For the method of constructing a plaster tool, see the article by Bertold Hamberger also on this website.
In the optics shop where I work, we frequently make grinding tools on a plaster, aluminum, or glass base, by covering them with a 1/4" layer of pitch and then imbedding grinding tiles into the pitch. The pitch we generally use is #64 Gugolz, which is fully adequate for all aspects of the present job. I recommend it. If you do use pitch for grinding, don't ever use it again for polishing work as it will have embedded grit in it!
This method of making grinding tools has several advantages over epoxying the tiles directly to their base:
You might worry that particles of course grit could get caught in the pitch and then come loose during fine grinding to scratch your optic. But if you scrub the lap carefully with a plastic brush under running water between changes of grit then this won't happen. I have ground many surfaces from #80 Carbo to 9 micron AlO using the same pitch and tile tool for the whole process and never scratched a surface due to contamination. It's a safe method.
You also don't need to bevel all the edges of your tiles to keep them from scratching your optics. The edges which border the outside of your lap should be bevelled so that you don't cut yourself on them when you lift the lap. But even unbevelled edges should never scratch your optic surface. The beveled edge on the glass is to keep the glass from cracking off a piece when a high force is placed on the edge.
As a general procedure for making your lens, I would advise you to hog out R3 on R2 first, and then cast two plaster laps for each surface, one to serve as a tool and the other as a support base when that side of your lens lies downward. It is convenient to set a support base down on your grinding stand, to cover it with a circle of rubberized shelf liner (the thick, stringy and beady, anti-skid kind sold in supermarkets and hardware stores nowadays), and to carefully tape the lens by its sides down to the base. Good elastic electrical tape works well here. 3M sells some excellent tape. Don't pull hard on the tape, because you could bend the lens, but by using moderate pull you'll help the lens stay in place and keep grit out from under it. The shelf liner has a lot of friction, and so will help to keep the lens from moving, but is quite springy which will fight any tendency of the lens to become astigmatic. You should, of course, exercise the usual cautions against astigmatism by turning the disk in its support bed often.
Once the laps are cast, let them dry for a week and then seal them with shellac or polyurethane if you are afraid of the laps changing from the water. In the mean time you can cast several more flat tools for the R1 and R4 surfaces. These surfaces are so weak that their tools and support bases can remain flat. If you have curves already generated then the tools will work just fine.
To make a pitch and tile tool, first take your finished, shellacked tool and use it as a model to form aluminum foil around its sides and bottom. You will pour hot pitch into this aluminum mold, so make sure that it can't leak. Pretend that you're going to make a large flat pitch "cookie." The tool's bottom serves as a model for the cookie mold of aluminum foil. Once the mold is ready, take it off the plaster model and lay it down on a level flat surface. Smooth out and wrinkles in the mold. Melt the #64 pitch and pour a 3/16" or so layer into the mold. Once the pitch has solidified and can be handled, rip the foil away and gently bend the cookie so that it will fit the curved top face of the plaster tool. Lay it on the tool, which in the meantime has been in the oven or on the hotplate and gotten warm to the touch but not hot.
The pitch will quickly adhere to the warm shellac on the tool and not come off. You now have a uniform layer of curved pitch without the mess and bother of other methods. Pitch laps can be made just as easily in the same way. For a pitch lap the next step would be to let the plaster cool, and then cut the channels under cold running water with a razor blade. They don't need to be cut with mathematical precision or be beautiful to work well. Just try to space them with relative uniformity and avoid placing one smack dab in the middle. Pitch cuts easily and without enormous mess under cold running water. But some mess is inevitable. Be careful not to let pitch flakes run down your drain or it may get clogged!
As for a tile tool, once the cookie has been fixed to the plaster but before the mass cools off completely, set your tile in place on top of the pitch. If the smell can be ventilated, take a propane torch and GENTLY heat the tile till the pitch wicks up their sides. Don't apply too much heat or the tiles will sink way down into the pitch and begin to slide off the tool! Now, place a sheet of paper or aluminum foil over the tool and invert the lens surface to be ground onto its grinding tool. Carefully add some weights to the back of the lens or put its support tool on top of it and then add the weights, and place the whole mass in the oven at low heat, or much better on a heat diffuser on top of a hot plate (wrap the mass in foil to concentrate the heat).
Watch carefully. After a while the tile will begin to sink into the softening pitch. When they sink half their height or more into the pitch shut off the heat and remove the mass from the oven or hotplate. Leave the lens on top for a while until the pitch begins to harden. Once the pitch is solid again (you can prod it from the side with your fingernail), you can remove the lens and run the pitch/tile tool under cold water. The lens should feel no more than moderately warm. If it has gotten hot, then you took too long. Immediately wrap it well in several layers of foil and towels and set in on the stove to cool off slowly. You might see some marks on the glass from the aluminum. These are not scratches in the glass. In fact, the glass has scratched the aluminum, since it's harder than that metal.
Examine your newly formed tool. The pitch may have filled up the channels between the tiles. That's a good thing, not a bad one. The pitch has now captured the tile well and it cannot come loose and break free. Under cold water score channels in the pitch to assist the grit in spreading over the glass surface during grinding. But don't try to cut the pitch away from the tile. Pitch grinds very well and won't stick to the glass, so long as you keep a moist slurry of grinding compound between them. The ground pitch will make the grinding mud turn darker than usual, but have no undesirable effects.
This method of making lapping tools avoids the struggle of some other methods and yields fine results. I use it constantly at work and never have real troubles with it. I have come to prefer it to all other methods that I've tried.
After making your tools and plaster bases and hogging out your curves. Incidentally, you should try to stop perhaps 100 or so microns of sagitta short of your final radii, because thickness matters with lenses and so you want to grind as little glass away as you possibly can, You can simultaneously smooth the hogged out glass and advance toward your final radius with the finer abrasives when it is time to switch to the next finer abrasive . Gradually work your way down to 40 micron aluminum oxide or #400 Carbo, smoothing the curves and coming closer to final radius. When you have done this, stop as it's time for a big departure from mirror making! It's now time to insure that you don't have any remaining wedge in the glass.
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