The Decaffeination Process
The method for decaffeination is rather complex. The simple way to look at it is a method to open up the green bean. Once the bean is open, extract the caffeine. Once the caffeine is extracted, dry the beans to contain the proper moisture content.
The decaffeination process involves the following common methods. At Decaffeinated in Canada, we will not carry the direct solvent method of decaffeination, as we only carry products from natural decaffeination processes.
Direct Solvent Method
This would be the first method for decaffeination completed in Germany. Commonly called the European Method. This method involves soaking green coffee beans in a solvent, usually methylene chloride (dichloromethane) or ethyl acetate. These solvents bond with the caffeine molecules, extracting them from the beans. After a period of soaking, the beans are dried, and residual solvents are removed in the drying.
This infographic is from the National Coffee Association showing the European Method.

https://www.ncausa.org/Decaffeinated-Coffee
Swiss Water Process
In this method, the green coffee beans are soaked in water to expand the pores. It is placed into a caffeine-lean green coffee extract (GCE). Since the other water-soluble parts of the green coffee bean are in the GCE, the caffeine is the only part removed. The resulting water is a caffeine-rich GCE, which goes through a filtration process to remove the caffeine return to a caffeine-lean GCE. This process is repeated until the green beans are 99.9% caffeine-free and sent to be dried.
This infographic is from the Swiss Water Process explaining their decaffeination process.

https://www.swisswater.com/pages/coffee-decaffeination-process
Carbon Dioxide (CO2) Method
This is also called supercritical CO2 or subcritical CO2. In this method, green coffee beans are steamed swelling the bean and opening the pores. The beans are then placed in contact with a CO2 mixture, in which the caffeine dissolves in the CO2 and other compounds stay with the bean. The CO2 mixture at the subcritical or supercritical state is highly selective at only extracting the caffeine. The CO2 mixture with the caffeine is sent to decrease the pressure where the CO2 goes back to a gas state and the caffeine is extracted. The CO2 gas is then compressed again to go into a liquid state ready to extract caffeine. Once the beans have removed the caffeine, they are dried to return to their natural state.
Below is an infographic from Belco on an article about the CR3 plant in Germany which does CO2 decaffeination.

Sugar Cane Process
This is a common process to see in decaffeinated coffee beans lately. The sugar cane process uses a natural source of ethyl acetate from organic sugar cane. The green beans are soaked in hot water to expand and open up the pores. This is when the sugar cane derived from ethyl acetate is soaked with the beans extracting the caffeine. When the process is done, steam is used to remove residual ethyl acetate from the bean. Then the beans go through a drying process to return the beans to their natural state.
Below is an infographic from Decafesol in Colombia showing the sugar cane decaffeination process.

https://www.descafecol.com/en/decaffeinated-products/