I believe in your case you would need a modem and a cisco router (your current router provides the functionality of both). The router you would need must contain an NM that will terminate the connection to the modem. So the NM will need modem functionality. I'm sure they exist and I'm sure others have done it.
Can you tell me who your provider is? I think that's the best way to get started to find you a reasonably priced legacy router and modem that will work.
You may be able to ask your provider for a modem only solution to satisfy one piece of the puzzle. Below are the two options you might have. I beleive you'll be able to find the hardware to satisfy option #1 and less likely to find it for option #2.
This is a fiber to the home (FTTH) modem / router although I don't see the fiber termination point on the graphic in the link. I'm going out on a limb and guessing this is Google Fiber or equivalent.
You may need a modem before attaching your own router unless there's a network module available from Cisco that will allow you to terminate it in a cisco router. That module would need the modem capabilities.
GPON uplink
• 1490 nm wavelength downstream,
1310 nm wavelength upstream
• 2.488 Gb/s line rate downstream,
1.244 Gb/s line rate upstream
• GPON Encapsulation Method (GEM) mode
support for IP/Ethernet service traffic
support
• ITU-T G.984.3-compliant dynamic
bandwidth reporting
• ITU-T G.984.3-compliant Advanced
Encryption System (AES) in downstream
• ITU-T G.984.3-compliant FEC
• ITU-T G.988 Appendix 1 and Appendix 2
ONT Management Control Interface (OMCI)
• Remote software image download
• Small form factor (SFF) type laser, SC/APC
connector
